HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.7%

Time: 46.2s

Alternatives: 24

Speedup: 1.9×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Alternative 1: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{\frac{cosTheta\_i\_m - \frac{cosTheta\_i\_m \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O\_m}}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (/
     (- cosTheta_i_m (/ (* cosTheta_i_m (* sinTheta_i sinTheta_O)) v))
     (/ (sinh (/ 1.0 v)) (/ 0.5 v)))
    (/ v cosTheta_O_m)))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (((cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)) / (sinhf((1.0f / v)) / (0.5f / v))) / (v / cosTheta_O_m)));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (((costheta_i_m - ((costheta_i_m * (sintheta_i * sintheta_o)) / v)) / (sinh((1.0e0 / v)) / (0.5e0 / v))) / (v / costheta_o_m)))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(cosTheta_i_m - Float32(Float32(cosTheta_i_m * Float32(sinTheta_i * sinTheta_O)) / v)) / Float32(sinh(Float32(Float32(1.0) / v)) / Float32(Float32(0.5) / v))) / Float32(v / cosTheta_O_m))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (((cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)) / (sinh((single(1.0) / v)) / (single(0.5) / v))) / (v / cosTheta_O_m)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) \cdot cosTheta\_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \cdot v} \]

   3. times-frac N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \color{blue}{\frac{cosTheta\_O}{v}} \]

   4. clear-num N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   5. un-div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right), \color{blue}{\left(\frac{v}{cosTheta\_O}\right)}\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}}} \]
10. Add Preprocessing

Alternative 2: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{\frac{\frac{0.5}{v} \cdot cosTheta\_O\_m}{\sinh \left(\frac{1}{v}\right)}}{\frac{v}{cosTheta\_i\_m - \frac{cosTheta\_i\_m \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (/ (* (/ 0.5 v) cosTheta_O_m) (sinh (/ 1.0 v)))
    (/ v (- cosTheta_i_m (/ (* cosTheta_i_m (* sinTheta_i sinTheta_O)) v)))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((((0.5f / v) * cosTheta_O_m) / sinhf((1.0f / v))) / (v / (cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((((0.5e0 / v) * costheta_o_m) / sinh((1.0e0 / v))) / (v / (costheta_i_m - ((costheta_i_m * (sintheta_i * sintheta_o)) / v)))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(Float32(0.5) / v) * cosTheta_O_m) / sinh(Float32(Float32(1.0) / v))) / Float32(v / Float32(cosTheta_i_m - Float32(Float32(cosTheta_i_m * Float32(sinTheta_i * sinTheta_O)) / v))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((((single(0.5) / v) * cosTheta_O_m) / sinh((single(1.0) / v))) / (v / (cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot v}} \]

   2. times-frac N/A

      \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \color{blue}{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{v}} \]

   3. clear-num N/A

      \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}} \]

   4. un-div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}}{\color{blue}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}} \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right), \color{blue}{\left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)}\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right), \left(\frac{\color{blue}{v}}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \frac{1}{\frac{1}{2}}\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   8. div-inv N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{v}{\frac{1}{2}}\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{1}{\frac{\frac{1}{2}}{v}}\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   10. div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{1}{2}}{v}}\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\sinh \left(\frac{1}{v}\right), \left(\frac{\frac{1}{2}}{v}\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   12. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\frac{\frac{1}{2}}{v}\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(\frac{\frac{1}{2}}{v}\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \color{blue}{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}\right)\right) \]

   16. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}\right)\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), \color{blue}{v}\right)\right)\right)\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}} \]
10. Step-by-step derivation

    1. associate-/r/ N/A

       \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{1}{2}}{v}\right), \mathsf{/.f32}\left(\color{blue}{v}, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right)\right)\right) \]

    2. associate-*l/ N/A

       \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}\right), \mathsf{/.f32}\left(\color{blue}{v}, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right)\right)\right) \]

    3. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \frac{\frac{1}{2}}{v}\right), \sinh \left(\frac{1}{v}\right)\right), \mathsf{/.f32}\left(\color{blue}{v}, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right)\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{\frac{1}{2}}{v}\right)\right), \sinh \left(\frac{1}{v}\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right)\right)\right) \]

    5. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \sinh \left(\frac{1}{v}\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right)\right)\right) \]

    6. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right)\right)\right) \]

    7. /-lowering-/.f32 98.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right)\right)\right) \]

11. Applied egg-rr 98.8%

    \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}} \]

12. Final simplification 98.8%

    \[\leadsto \frac{\frac{\frac{0.5}{v} \cdot cosTheta\_O}{\sinh \left(\frac{1}{v}\right)}}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}} \]
13. Add Preprocessing

Alternative 3: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{\frac{0.5}{v} \cdot \frac{cosTheta\_i\_m - cosTheta\_i\_m \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}{\frac{v}{cosTheta\_O\_m}}}{\sinh \left(\frac{1}{v}\right)}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (*
     (/ 0.5 v)
     (/
      (- cosTheta_i_m (* cosTheta_i_m (/ sinTheta_i (/ v sinTheta_O))))
      (/ v cosTheta_O_m)))
    (sinh (/ 1.0 v))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (((0.5f / v) * ((cosTheta_i_m - (cosTheta_i_m * (sinTheta_i / (v / sinTheta_O)))) / (v / cosTheta_O_m))) / sinhf((1.0f / v))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (((0.5e0 / v) * ((costheta_i_m - (costheta_i_m * (sintheta_i / (v / sintheta_o)))) / (v / costheta_o_m))) / sinh((1.0e0 / v))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(0.5) / v) * Float32(Float32(cosTheta_i_m - Float32(cosTheta_i_m * Float32(sinTheta_i / Float32(v / sinTheta_O)))) / Float32(v / cosTheta_O_m))) / sinh(Float32(Float32(1.0) / v)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (((single(0.5) / v) * ((cosTheta_i_m - (cosTheta_i_m * (sinTheta_i / (v / sinTheta_O)))) / (v / cosTheta_O_m))) / sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) \cdot cosTheta\_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \cdot v} \]

   3. times-frac N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \color{blue}{\frac{cosTheta\_O}{v}} \]

   4. clear-num N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   5. un-div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right), \color{blue}{\left(\frac{v}{cosTheta\_O}\right)}\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}}} \]
10. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{v}{cosTheta\_O}}{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{1}{2}}{v}}}}}} \]

    2. div-inv N/A

       \[\leadsto \frac{1}{\frac{\frac{v}{cosTheta\_O}}{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right) \cdot \color{blue}{\frac{1}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{1}{2}}{v}}}}}} \]

    3. clear-num N/A

       \[\leadsto \frac{1}{\frac{\frac{v}{cosTheta\_O}}{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right) \cdot \frac{\frac{\frac{1}{2}}{v}}{\color{blue}{\sinh \left(\frac{1}{v}\right)}}}} \]

    4. associate-/r* N/A

       \[\leadsto \frac{1}{\frac{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}{\color{blue}{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}}} \]

    5. clear-num N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}{\color{blue}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}} \]

    6. div-inv N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}} \]

    7. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{1}{\frac{v}{\color{blue}{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right)}}} \]

    8. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{1}{\frac{v}{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}} \]

    9. clear-num N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{v}} \]

11. Applied egg-rr 98.8%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{v} \cdot \frac{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}{\frac{v}{cosTheta\_O}}}{\sinh \left(\frac{1}{v}\right)}} \]
12. Add Preprocessing

Alternative 4: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(cosTheta\_O\_m \cdot \frac{cosTheta\_i\_m - cosTheta\_i\_m \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}{v \cdot \frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    cosTheta_O_m
    (/
     (- cosTheta_i_m (* cosTheta_i_m (/ sinTheta_i (/ v sinTheta_O))))
     (* v (/ (sinh (/ 1.0 v)) (/ 0.5 v))))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (cosTheta_O_m * ((cosTheta_i_m - (cosTheta_i_m * (sinTheta_i / (v / sinTheta_O)))) / (v * (sinhf((1.0f / v)) / (0.5f / v))))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (costheta_o_m * ((costheta_i_m - (costheta_i_m * (sintheta_i / (v / sintheta_o)))) / (v * (sinh((1.0e0 / v)) / (0.5e0 / v))))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(cosTheta_O_m * Float32(Float32(cosTheta_i_m - Float32(cosTheta_i_m * Float32(sinTheta_i / Float32(v / sinTheta_O)))) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) / Float32(Float32(0.5) / v)))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (cosTheta_O_m * ((cosTheta_i_m - (cosTheta_i_m * (sinTheta_i / (v / sinTheta_O)))) / (v * (sinh((single(1.0) / v)) / (single(0.5) / v))))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) \cdot cosTheta\_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \cdot v} \]

   3. times-frac N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \color{blue}{\frac{cosTheta\_O}{v}} \]

   4. clear-num N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   5. un-div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right), \color{blue}{\left(\frac{v}{cosTheta\_O}\right)}\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}}} \]
10. Step-by-step derivation

    1. associate-/r/ N/A

       \[\leadsto \frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{1}{2}}{v}}}}{v} \cdot \color{blue}{cosTheta\_O} \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{1}{2}}{v}}}}{v}\right), \color{blue}{cosTheta\_O}\right) \]

11. Applied egg-rr 98.8%

    \[\leadsto \color{blue}{\frac{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}{v \cdot \frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}} \cdot cosTheta\_O} \]

12. Final simplification 98.8%

    \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}{v \cdot \frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}} \]
13. Add Preprocessing

Alternative 5: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\frac{cosTheta\_O\_m}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{0.5}{v} \cdot \frac{cosTheta\_i\_m - \frac{cosTheta\_i\_m \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v}\right)\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/ cosTheta_O_m (sinh (/ 1.0 v)))
    (*
     (/ 0.5 v)
     (/
      (- cosTheta_i_m (/ (* cosTheta_i_m (* sinTheta_i sinTheta_O)) v))
      v))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m / sinhf((1.0f / v))) * ((0.5f / v) * ((cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)) / v))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((costheta_o_m / sinh((1.0e0 / v))) * ((0.5e0 / v) * ((costheta_i_m - ((costheta_i_m * (sintheta_i * sintheta_o)) / v)) / v))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_O_m / sinh(Float32(Float32(1.0) / v))) * Float32(Float32(Float32(0.5) / v) * Float32(Float32(cosTheta_i_m - Float32(Float32(cosTheta_i_m * Float32(sinTheta_i * sinTheta_O)) / v)) / v)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m / sinh((single(1.0) / v))) * ((single(0.5) / v) * ((cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)) / v))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot v}} \]

   2. times-frac N/A

      \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \color{blue}{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{v}} \]

   3. clear-num N/A

      \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}} \]

   4. un-div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}}{\color{blue}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}} \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right), \color{blue}{\left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)}\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right), \left(\frac{\color{blue}{v}}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \frac{1}{\frac{1}{2}}\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   8. div-inv N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{v}{\frac{1}{2}}\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{1}{\frac{\frac{1}{2}}{v}}\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   10. div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{1}{2}}{v}}\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\sinh \left(\frac{1}{v}\right), \left(\frac{\frac{1}{2}}{v}\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   12. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\frac{\frac{1}{2}}{v}\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(\frac{\frac{1}{2}}{v}\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \color{blue}{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}\right)\right) \]

   16. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}\right)\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), \color{blue}{v}\right)\right)\right)\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}} \]
10. Step-by-step derivation

    1. div-inv N/A

       \[\leadsto \frac{cosTheta\_O}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{1}{2}}{v}}} \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}} \]

    2. associate-/r/ N/A

       \[\leadsto \left(\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{1}{2}}{v}\right) \cdot \frac{\color{blue}{1}}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}} \]

    3. clear-num N/A

       \[\leadsto \left(\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{1}{2}}{v}\right) \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\color{blue}{v}} \]

    4. associate-*l* N/A

       \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\left(\frac{\frac{1}{2}}{v} \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v}\right)} \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)}\right), \color{blue}{\left(\frac{\frac{1}{2}}{v} \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v}\right)}\right) \]

    6. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \sinh \left(\frac{1}{v}\right)\right), \left(\color{blue}{\frac{\frac{1}{2}}{v}} \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v}\right)\right) \]

    7. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right)\right), \left(\frac{\frac{1}{2}}{\color{blue}{v}} \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v}\right)\right) \]

    8. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \left(\frac{\frac{1}{2}}{v} \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v}\right)\right) \]

    9. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{*.f32}\left(\left(\frac{\frac{1}{2}}{v}\right), \color{blue}{\left(\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v}\right)}\right)\right) \]

    10. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\frac{\color{blue}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}{v}\right)\right)\right) \]

    11. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right), \color{blue}{v}\right)\right)\right) \]

11. Applied egg-rr 98.8%

    \[\leadsto \color{blue}{\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{0.5}{v} \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{v}\right)} \]

12. Final simplification 98.8%

    \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{0.5}{v} \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v}\right) \]
13. Add Preprocessing

Alternative 6: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\left(cosTheta\_O\_m \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \cdot \frac{cosTheta\_i\_m - cosTheta\_i\_m \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}{v}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (* cosTheta_O_m (/ (/ 0.5 v) (sinh (/ 1.0 v))))
    (/ (- cosTheta_i_m (* cosTheta_i_m (/ sinTheta_i (/ v sinTheta_O)))) v)))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * ((0.5f / v) / sinhf((1.0f / v)))) * ((cosTheta_i_m - (cosTheta_i_m * (sinTheta_i / (v / sinTheta_O)))) / v)));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((costheta_o_m * ((0.5e0 / v) / sinh((1.0e0 / v)))) * ((costheta_i_m - (costheta_i_m * (sintheta_i / (v / sintheta_o)))) / v)))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_O_m * Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v)))) * Float32(Float32(cosTheta_i_m - Float32(cosTheta_i_m * Float32(sinTheta_i / Float32(v / sinTheta_O)))) / v))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * ((single(0.5) / v) / sinh((single(1.0) / v)))) * ((cosTheta_i_m - (cosTheta_i_m * (sinTheta_i / (v / sinTheta_O)))) / v)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) \cdot cosTheta\_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \cdot v} \]

   3. times-frac N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \color{blue}{\frac{cosTheta\_O}{v}} \]

   4. clear-num N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   5. un-div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right), \color{blue}{\left(\frac{v}{cosTheta\_O}\right)}\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}}} \]
10. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{v}{cosTheta\_O}}{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{1}{2}}{v}}}}}} \]

    2. div-inv N/A

       \[\leadsto \frac{1}{\frac{\frac{v}{cosTheta\_O}}{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right) \cdot \color{blue}{\frac{1}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{1}{2}}{v}}}}}} \]

    3. clear-num N/A

       \[\leadsto \frac{1}{\frac{\frac{v}{cosTheta\_O}}{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right) \cdot \frac{\frac{\frac{1}{2}}{v}}{\color{blue}{\sinh \left(\frac{1}{v}\right)}}}} \]

    4. associate-/r* N/A

       \[\leadsto \frac{1}{\frac{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}{\color{blue}{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}}} \]

    5. clear-num N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}{\color{blue}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}} \]

    6. div-inv N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}} \]

    7. clear-num N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

    8. associate-/r/ N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v} \cdot \color{blue}{cosTheta\_O}\right) \]

    9. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{v} \cdot cosTheta\_O\right) \]

    10. *-commutative N/A

        \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta\_O \cdot \color{blue}{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{v}}\right) \]

11. Applied egg-rr 98.8%

    \[\leadsto \color{blue}{\left(\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}{v}} \]

12. Final simplification 98.8%

    \[\leadsto \left(cosTheta\_O \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \cdot \frac{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}{v} \]
13. Add Preprocessing

Alternative 7: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{\frac{cosTheta\_i\_m}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O\_m}}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/ (/ cosTheta_i_m (/ (sinh (/ 1.0 v)) (/ 0.5 v))) (/ v cosTheta_O_m)))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_i_m / (sinhf((1.0f / v)) / (0.5f / v))) / (v / cosTheta_O_m)));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((costheta_i_m / (sinh((1.0e0 / v)) / (0.5e0 / v))) / (v / costheta_o_m)))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_i_m / Float32(sinh(Float32(Float32(1.0) / v)) / Float32(Float32(0.5) / v))) / Float32(v / cosTheta_O_m))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_i_m / (sinh((single(1.0) / v)) / (single(0.5) / v))) / (v / cosTheta_O_m)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) \cdot cosTheta\_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \cdot v} \]

   3. times-frac N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \color{blue}{\frac{cosTheta\_O}{v}} \]

   4. clear-num N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   5. un-div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right), \color{blue}{\left(\frac{v}{cosTheta\_O}\right)}\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}}} \]

10. Taylor expanded in sinTheta_i around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{cosTheta\_i}, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]
11. Step-by-step derivation

12. Simplified 98.7%

    \[\leadsto \frac{\frac{\color{blue}{cosTheta\_i}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}} \]
13. Add Preprocessing

Alternative 8: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\frac{0.5}{v} \cdot \frac{\frac{cosTheta\_i\_m \cdot cosTheta\_O\_m}{v}}{\sinh \left(\frac{1}{v}\right)}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (* (/ 0.5 v) (/ (/ (* cosTheta_i_m cosTheta_O_m) v) (sinh (/ 1.0 v)))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((0.5f / v) * (((cosTheta_i_m * cosTheta_O_m) / v) / sinhf((1.0f / v)))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((0.5e0 / v) * (((costheta_i_m * costheta_o_m) / v) / sinh((1.0e0 / v)))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(0.5) / v) * Float32(Float32(Float32(cosTheta_i_m * cosTheta_O_m) / v) / sinh(Float32(Float32(1.0) / v))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((single(0.5) / v) * (((cosTheta_i_m * cosTheta_O_m) / v) / sinh((single(1.0) / v)))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right)}}{\color{blue}{v \cdot 2}} \]

   2. div-inv N/A

      \[\leadsto \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{1}{v \cdot 2}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right)}\right), \color{blue}{\left(\frac{1}{v \cdot 2}\right)}\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \sinh \left(\frac{1}{v}\right)\right), \left(\frac{\color{blue}{1}}{v \cdot 2}\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \sinh \left(\frac{1}{v}\right)\right), \left(\frac{1}{v \cdot 2}\right)\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \sinh \left(\frac{1}{v}\right)\right), \left(\frac{1}{v \cdot 2}\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_O}{v}\right)\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \sinh \left(\frac{1}{v}\right)\right), \left(\frac{1}{v \cdot 2}\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, v\right)\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \sinh \left(\frac{1}{v}\right)\right), \left(\frac{1}{v \cdot 2}\right)\right) \]

   9. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, v\right)\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \sinh \left(\frac{1}{v}\right)\right), \left(\frac{1}{v \cdot 2}\right)\right) \]

   10. associate-/l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, v\right)\right), \mathsf{exp.f32}\left(\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right)\right), \sinh \left(\frac{1}{v}\right)\right), \left(\frac{1}{v \cdot 2}\right)\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, v\right)\right), \mathsf{exp.f32}\left(\left(sinTheta\_i \cdot \frac{1}{\frac{v}{sinTheta\_O}}\right)\right)\right), \sinh \left(\frac{1}{v}\right)\right), \left(\frac{1}{v \cdot 2}\right)\right) \]

   12. un-div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, v\right)\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}\right)\right)\right), \sinh \left(\frac{1}{v}\right)\right), \left(\frac{1}{v \cdot 2}\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, v\right)\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \left(\frac{v}{sinTheta\_O}\right)\right)\right)\right), \sinh \left(\frac{1}{v}\right)\right), \left(\frac{1}{v \cdot 2}\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, v\right)\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right)\right), \sinh \left(\frac{1}{v}\right)\right), \left(\frac{1}{v \cdot 2}\right)\right) \]

   15. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, v\right)\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right)\right), \mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right)\right), \left(\frac{1}{v \cdot 2}\right)\right) \]

   16. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, v\right)\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right)\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \left(\frac{1}{v \cdot 2}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, v\right)\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right)\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \left(\frac{1}{2 \cdot \color{blue}{v}}\right)\right) \]

6. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{0.5}{v}} \]

7. Taylor expanded in v around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right) \]

   2. *-lowering-*.f32 98.4%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right) \]

9. Simplified 98.4%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{0.5}{v} \]

10. Final simplification 98.4%

    \[\leadsto \frac{0.5}{v} \cdot \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]
11. Add Preprocessing

Alternative 9: 69.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{\frac{cosTheta\_i\_m - \frac{cosTheta\_i\_m \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\frac{1 + \frac{\frac{0.008333333333333333}{v \cdot v} - -0.16666666666666666}{v \cdot v}}{v}}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O\_m}}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (/
     (- cosTheta_i_m (/ (* cosTheta_i_m (* sinTheta_i sinTheta_O)) v))
     (/
      (/
       (+
        1.0
        (/ (- (/ 0.008333333333333333 (* v v)) -0.16666666666666666) (* v v)))
       v)
      (/ 0.5 v)))
    (/ v cosTheta_O_m)))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (((cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)) / (((1.0f + (((0.008333333333333333f / (v * v)) - -0.16666666666666666f) / (v * v))) / v) / (0.5f / v))) / (v / cosTheta_O_m)));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (((costheta_i_m - ((costheta_i_m * (sintheta_i * sintheta_o)) / v)) / (((1.0e0 + (((0.008333333333333333e0 / (v * v)) - (-0.16666666666666666e0)) / (v * v))) / v) / (0.5e0 / v))) / (v / costheta_o_m)))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(cosTheta_i_m - Float32(Float32(cosTheta_i_m * Float32(sinTheta_i * sinTheta_O)) / v)) / Float32(Float32(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(0.008333333333333333) / Float32(v * v)) - Float32(-0.16666666666666666)) / Float32(v * v))) / v) / Float32(Float32(0.5) / v))) / Float32(v / cosTheta_O_m))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (((cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)) / (((single(1.0) + (((single(0.008333333333333333) / (v * v)) - single(-0.16666666666666666)) / (v * v))) / v) / (single(0.5) / v))) / (v / cosTheta_O_m)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) \cdot cosTheta\_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \cdot v} \]

   3. times-frac N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \color{blue}{\frac{cosTheta\_O}{v}} \]

   4. clear-num N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   5. un-div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right), \color{blue}{\left(\frac{v}{cosTheta\_O}\right)}\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}}} \]

10. Taylor expanded in v around -inf

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right), \mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}, \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right), \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    2. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right), \mathsf{/.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    3. mul-1-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right), \mathsf{/.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{-1 \cdot v}\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right), \left(-1 \cdot v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

12. Simplified 72.2%

    \[\leadsto \frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\color{blue}{\frac{\frac{-0.16666666666666666 + -1 \cdot \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} + -1}{-v}}}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}} \]

13. Final simplification 72.2%

    \[\leadsto \frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\frac{1 + \frac{\frac{0.008333333333333333}{v \cdot v} - -0.16666666666666666}{v \cdot v}}{v}}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}} \]
14. Add Preprocessing

Alternative 10: 69.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{\frac{\frac{cosTheta\_i\_m \cdot \left(v - sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\frac{\frac{\frac{0.008333333333333333}{v \cdot v} + 0.16666666666666666}{v \cdot v} + 1}{v}}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O\_m}}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (/
     (/ (* cosTheta_i_m (- v (* sinTheta_i sinTheta_O))) v)
     (/
      (/
       (+
        (/ (+ (/ 0.008333333333333333 (* v v)) 0.16666666666666666) (* v v))
        1.0)
       v)
      (/ 0.5 v)))
    (/ v cosTheta_O_m)))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((((cosTheta_i_m * (v - (sinTheta_i * sinTheta_O))) / v) / ((((((0.008333333333333333f / (v * v)) + 0.16666666666666666f) / (v * v)) + 1.0f) / v) / (0.5f / v))) / (v / cosTheta_O_m)));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((((costheta_i_m * (v - (sintheta_i * sintheta_o))) / v) / ((((((0.008333333333333333e0 / (v * v)) + 0.16666666666666666e0) / (v * v)) + 1.0e0) / v) / (0.5e0 / v))) / (v / costheta_o_m)))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(cosTheta_i_m * Float32(v - Float32(sinTheta_i * sinTheta_O))) / v) / Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.008333333333333333) / Float32(v * v)) + Float32(0.16666666666666666)) / Float32(v * v)) + Float32(1.0)) / v) / Float32(Float32(0.5) / v))) / Float32(v / cosTheta_O_m))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((((cosTheta_i_m * (v - (sinTheta_i * sinTheta_O))) / v) / ((((((single(0.008333333333333333) / (v * v)) + single(0.16666666666666666)) / (v * v)) + single(1.0)) / v) / (single(0.5) / v))) / (v / cosTheta_O_m)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) \cdot cosTheta\_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \cdot v} \]

   3. times-frac N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \color{blue}{\frac{cosTheta\_O}{v}} \]

   4. clear-num N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   5. un-div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right), \color{blue}{\left(\frac{v}{cosTheta\_O}\right)}\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}}} \]

10. Taylor expanded in v around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(\frac{cosTheta\_i \cdot v - cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot v - cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    2. distribute-lft-out-- N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(v - sinTheta\_O \cdot sinTheta\_i\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(v - sinTheta\_O \cdot sinTheta\_i\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    4. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(v, \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    5. *-lowering-*.f32 98.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

12. Simplified 98.8%

    \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta\_i \cdot \left(v - sinTheta\_O \cdot sinTheta\_i\right)}{v}}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}} \]

13. Taylor expanded in v around -inf

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right)\right), v\right), \mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}, \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]
14. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right)\right), v\right), \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    2. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right)\right), v\right), \mathsf{/.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    3. mul-1-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right)\right), v\right), \mathsf{/.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{-1 \cdot v}\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right), \left(-1 \cdot v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

15. Simplified 72.2%

    \[\leadsto \frac{\frac{\frac{cosTheta\_i \cdot \left(v - sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\frac{\color{blue}{\frac{-1 + \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v}}{-v}}}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}} \]

16. Final simplification 72.2%

    \[\leadsto \frac{\frac{\frac{cosTheta\_i \cdot \left(v - sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\frac{\frac{\frac{0.008333333333333333}{v \cdot v} + 0.16666666666666666}{v \cdot v} + 1}{v}}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}} \]
17. Add Preprocessing

Alternative 11: 69.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{\frac{cosTheta\_O\_m}{\frac{\frac{\frac{\frac{0.008333333333333333}{v \cdot v} + 0.16666666666666666}{v \cdot v} + 1}{v}}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_i\_m - \frac{cosTheta\_i\_m \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (/
     cosTheta_O_m
     (/
      (/
       (+
        (/ (+ (/ 0.008333333333333333 (* v v)) 0.16666666666666666) (* v v))
        1.0)
       v)
      (/ 0.5 v)))
    (/ v (- cosTheta_i_m (/ (* cosTheta_i_m (* sinTheta_i sinTheta_O)) v)))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m / ((((((0.008333333333333333f / (v * v)) + 0.16666666666666666f) / (v * v)) + 1.0f) / v) / (0.5f / v))) / (v / (cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((costheta_o_m / ((((((0.008333333333333333e0 / (v * v)) + 0.16666666666666666e0) / (v * v)) + 1.0e0) / v) / (0.5e0 / v))) / (v / (costheta_i_m - ((costheta_i_m * (sintheta_i * sintheta_o)) / v)))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_O_m / Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.008333333333333333) / Float32(v * v)) + Float32(0.16666666666666666)) / Float32(v * v)) + Float32(1.0)) / v) / Float32(Float32(0.5) / v))) / Float32(v / Float32(cosTheta_i_m - Float32(Float32(cosTheta_i_m * Float32(sinTheta_i * sinTheta_O)) / v))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m / ((((((single(0.008333333333333333) / (v * v)) + single(0.16666666666666666)) / (v * v)) + single(1.0)) / v) / (single(0.5) / v))) / (v / (cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot v}} \]

   2. times-frac N/A

      \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \color{blue}{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{v}} \]

   3. clear-num N/A

      \[\leadsto \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}} \]

   4. un-div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}}{\color{blue}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}} \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right), \color{blue}{\left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)}\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right), \left(\frac{\color{blue}{v}}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \frac{1}{\frac{1}{2}}\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   8. div-inv N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{v}{\frac{1}{2}}\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{1}{\frac{\frac{1}{2}}{v}}\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   10. div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{1}{2}}{v}}\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\sinh \left(\frac{1}{v}\right), \left(\frac{\frac{1}{2}}{v}\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   12. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\frac{\frac{1}{2}}{v}\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(\frac{\frac{1}{2}}{v}\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \color{blue}{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}\right)\right) \]

   16. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}\right)\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), \color{blue}{v}\right)\right)\right)\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}}} \]

10. Taylor expanded in v around -inf

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}, \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right)\right)\right) \]

    2. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right)\right)\right) \]

    3. mul-1-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{-1 \cdot v}\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right)\right)\right) \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right), \left(-1 \cdot v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right)\right)\right) \]

12. Simplified 72.2%

    \[\leadsto \frac{\frac{cosTheta\_O}{\frac{\color{blue}{\frac{-1 + \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v}}{-v}}}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}} \]

13. Final simplification 72.2%

    \[\leadsto \frac{\frac{cosTheta\_O}{\frac{\frac{\frac{\frac{0.008333333333333333}{v \cdot v} + 0.16666666666666666}{v \cdot v} + 1}{v}}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}} \]
14. Add Preprocessing

Alternative 12: 63.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\frac{cosTheta\_i\_m - \frac{cosTheta\_i\_m \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v} \cdot \frac{cosTheta\_O\_m}{\frac{0.3333333333333333 + \frac{v}{\frac{0.5}{v}}}{v \cdot v}}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/ (- cosTheta_i_m (/ (* cosTheta_i_m (* sinTheta_i sinTheta_O)) v)) v)
    (/ cosTheta_O_m (/ (+ 0.3333333333333333 (/ v (/ 0.5 v))) (* v v)))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (((cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)) / v) * (cosTheta_O_m / ((0.3333333333333333f + (v / (0.5f / v))) / (v * v)))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (((costheta_i_m - ((costheta_i_m * (sintheta_i * sintheta_o)) / v)) / v) * (costheta_o_m / ((0.3333333333333333e0 + (v / (0.5e0 / v))) / (v * v)))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(cosTheta_i_m - Float32(Float32(cosTheta_i_m * Float32(sinTheta_i * sinTheta_O)) / v)) / v) * Float32(cosTheta_O_m / Float32(Float32(Float32(0.3333333333333333) + Float32(v / Float32(Float32(0.5) / v))) / Float32(v * v))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (((cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)) / v) * (cosTheta_O_m / ((single(0.3333333333333333) + (v / (single(0.5) / v))) / (v * v)))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

8. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\color{blue}{\left(\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right), v\right), \mathsf{*.f32}\left(\color{blue}{v}, 2\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6} \cdot 1}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6}}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left({v}^{2}\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left(v \cdot v\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 66.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

10. Simplified 66.8%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\color{blue}{\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}} \cdot \left(v \cdot 2\right)} \]

11. Taylor expanded in v around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \color{blue}{\left(\frac{\frac{1}{3} + 2 \cdot {v}^{2}}{{v}^{2}}\right)}\right) \]
12. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right), \color{blue}{\left({v}^{2}\right)}\right)\right) \]

    2. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(2 \cdot {v}^{2}\right)\right), \left({\color{blue}{v}}^{2}\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left({v}^{2} \cdot 2\right)\right), \left({v}^{2}\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\left(v \cdot v\right) \cdot 2\right)\right), \left({v}^{2}\right)\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(v \cdot \left(v \cdot 2\right)\right)\right), \left({v}^{2}\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \left(v \cdot 2\right)\right)\right), \left({v}^{2}\right)\right)\right) \]

    7. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \left({v}^{2}\right)\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \left(v \cdot \color{blue}{v}\right)\right)\right) \]

    9. *-lowering-*.f32 66.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right) \]

13. Simplified 66.8%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\color{blue}{\frac{0.3333333333333333 + v \cdot \left(v \cdot 2\right)}{v \cdot v}}} \]
14. Step-by-step derivation

    1. associate-/l/ N/A

       \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{\frac{\frac{1}{3} + v \cdot \left(v \cdot 2\right)}{v \cdot v} \cdot v}} \]

    2. associate-/l* N/A

       \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{\frac{\frac{1}{3} + v \cdot \left(v \cdot 2\right)}{v \cdot \color{blue}{v}} \cdot v} \]

    3. *-commutative N/A

       \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}{\frac{\frac{1}{3} + v \cdot \left(v \cdot 2\right)}{v \cdot v} \cdot v} \]

    4. associate-/l* N/A

       \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right)}{\frac{\frac{1}{3} + v \cdot \left(v \cdot 2\right)}{v \cdot \color{blue}{v}} \cdot v} \]

    5. times-frac N/A

       \[\leadsto \frac{cosTheta\_O}{\frac{\frac{1}{3} + v \cdot \left(v \cdot 2\right)}{v \cdot v}} \cdot \color{blue}{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v}} \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{cosTheta\_O}{\frac{\frac{1}{3} + v \cdot \left(v \cdot 2\right)}{v \cdot v}}\right), \color{blue}{\left(\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v}\right)}\right) \]

15. Applied egg-rr 66.8%

    \[\leadsto \color{blue}{\frac{cosTheta\_O}{\frac{0.3333333333333333 + \frac{v}{\frac{0.5}{v}}}{v \cdot v}} \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{v}} \]

16. Final simplification 66.8%

    \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{v} \cdot \frac{cosTheta\_O}{\frac{0.3333333333333333 + \frac{v}{\frac{0.5}{v}}}{v \cdot v}} \]
17. Add Preprocessing

Alternative 13: 63.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\frac{0.5}{v} \cdot \frac{cosTheta\_O\_m \cdot \left(cosTheta\_i\_m - cosTheta\_i\_m \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}\right)}{\frac{0.16666666666666666}{v \cdot v} + 1}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/ 0.5 v)
    (/
     (*
      cosTheta_O_m
      (- cosTheta_i_m (* cosTheta_i_m (/ sinTheta_i (/ v sinTheta_O)))))
     (+ (/ 0.16666666666666666 (* v v)) 1.0))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((0.5f / v) * ((cosTheta_O_m * (cosTheta_i_m - (cosTheta_i_m * (sinTheta_i / (v / sinTheta_O))))) / ((0.16666666666666666f / (v * v)) + 1.0f))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((0.5e0 / v) * ((costheta_o_m * (costheta_i_m - (costheta_i_m * (sintheta_i / (v / sintheta_o))))) / ((0.16666666666666666e0 / (v * v)) + 1.0e0))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(0.5) / v) * Float32(Float32(cosTheta_O_m * Float32(cosTheta_i_m - Float32(cosTheta_i_m * Float32(sinTheta_i / Float32(v / sinTheta_O))))) / Float32(Float32(Float32(0.16666666666666666) / Float32(v * v)) + Float32(1.0))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((single(0.5) / v) * ((cosTheta_O_m * (cosTheta_i_m - (cosTheta_i_m * (sinTheta_i / (v / sinTheta_O))))) / ((single(0.16666666666666666) / (v * v)) + single(1.0)))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

8. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\color{blue}{\left(\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right), v\right), \mathsf{*.f32}\left(\color{blue}{v}, 2\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6} \cdot 1}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6}}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left({v}^{2}\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left(v \cdot v\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 66.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

10. Simplified 66.8%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\color{blue}{\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}} \cdot \left(v \cdot 2\right)} \]
11. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\frac{1 + \frac{\frac{1}{6}}{v \cdot v}}{v}}}{\color{blue}{v \cdot 2}} \]

    2. div-inv N/A

       \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\frac{1 + \frac{\frac{1}{6}}{v \cdot v}}{v}} \cdot \color{blue}{\frac{1}{v \cdot 2}} \]

    3. metadata-eval N/A

       \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\frac{1 + \frac{\frac{1}{6}}{v \cdot v}}{v}} \cdot \frac{1}{v \cdot \frac{1}{\color{blue}{\frac{1}{2}}}} \]

    4. div-inv N/A

       \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\frac{1 + \frac{\frac{1}{6}}{v \cdot v}}{v}} \cdot \frac{1}{\frac{v}{\color{blue}{\frac{1}{2}}}} \]

    5. clear-num N/A

       \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\frac{1 + \frac{\frac{1}{6}}{v \cdot v}}{v}} \cdot \frac{\frac{1}{2}}{\color{blue}{v}} \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\frac{1 + \frac{\frac{1}{6}}{v \cdot v}}{v}}\right), \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right) \]

12. Applied egg-rr 66.8%

    \[\leadsto \color{blue}{\frac{\left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}\right) \cdot cosTheta\_O}{\left(1 + \frac{0.16666666666666666}{v \cdot v}\right) \cdot 1} \cdot \frac{0.5}{v}} \]

13. Final simplification 66.8%

    \[\leadsto \frac{0.5}{v} \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}\right)}{\frac{0.16666666666666666}{v \cdot v} + 1} \]
14. Add Preprocessing

Alternative 14: 63.5% accurate, 10.5× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{\frac{cosTheta\_i\_m - \frac{cosTheta\_i\_m \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}}}{\frac{v}{cosTheta\_O\_m}}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (/
     (- cosTheta_i_m (/ (* cosTheta_i_m (* sinTheta_i sinTheta_O)) v))
     (+ 2.0 (/ 0.3333333333333333 (* v v))))
    (/ v cosTheta_O_m)))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (((cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)) / (2.0f + (0.3333333333333333f / (v * v)))) / (v / cosTheta_O_m)));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (((costheta_i_m - ((costheta_i_m * (sintheta_i * sintheta_o)) / v)) / (2.0e0 + (0.3333333333333333e0 / (v * v)))) / (v / costheta_o_m)))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(cosTheta_i_m - Float32(Float32(cosTheta_i_m * Float32(sinTheta_i * sinTheta_O)) / v)) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))) / Float32(v / cosTheta_O_m))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (((cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)) / (single(2.0) + (single(0.3333333333333333) / (v * v)))) / (v / cosTheta_O_m)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) \cdot cosTheta\_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \cdot v} \]

   3. times-frac N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \color{blue}{\frac{cosTheta\_O}{v}} \]

   4. clear-num N/A

      \[\leadsto \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   5. un-div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right), \color{blue}{\left(\frac{v}{cosTheta\_O}\right)}\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}}{\frac{v}{cosTheta\_O}}} \]

10. Taylor expanded in v around inf

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right), \color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]
11. Step-by-step derivation

    1. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right), \mathsf{+.f32}\left(2, \left(\frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    2. associate-*r/ N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3} \cdot 1}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3}}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \left({v}^{2}\right)\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \left(v \cdot v\right)\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

    6. *-lowering-*.f32 66.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right), v\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(v, cosTheta\_O\right)\right) \]

12. Simplified 66.8%

    \[\leadsto \frac{\frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}}}{\frac{v}{cosTheta\_O}} \]
13. Add Preprocessing

Alternative 15: 63.5% accurate, 10.5× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\left(cosTheta\_i\_m - \frac{cosTheta\_i\_m \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right) \cdot \frac{cosTheta\_O\_m}{\frac{0.3333333333333333 + \frac{v}{\frac{0.5}{v}}}{v}}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (- cosTheta_i_m (/ (* cosTheta_i_m (* sinTheta_i sinTheta_O)) v))
    (/ cosTheta_O_m (/ (+ 0.3333333333333333 (/ v (/ 0.5 v))) v))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)) * (cosTheta_O_m / ((0.3333333333333333f + (v / (0.5f / v))) / v))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((costheta_i_m - ((costheta_i_m * (sintheta_i * sintheta_o)) / v)) * (costheta_o_m / ((0.3333333333333333e0 + (v / (0.5e0 / v))) / v))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_i_m - Float32(Float32(cosTheta_i_m * Float32(sinTheta_i * sinTheta_O)) / v)) * Float32(cosTheta_O_m / Float32(Float32(Float32(0.3333333333333333) + Float32(v / Float32(Float32(0.5) / v))) / v)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_i_m - ((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / v)) * (cosTheta_O_m / ((single(0.3333333333333333) + (v / (single(0.5) / v))) / v))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

8. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\color{blue}{\left(\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right), v\right), \mathsf{*.f32}\left(\color{blue}{v}, 2\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6} \cdot 1}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6}}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left({v}^{2}\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left(v \cdot v\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 66.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

10. Simplified 66.8%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\color{blue}{\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}} \cdot \left(v \cdot 2\right)} \]

11. Taylor expanded in v around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \color{blue}{\left(\frac{\frac{1}{3} + 2 \cdot {v}^{2}}{{v}^{2}}\right)}\right) \]
12. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right), \color{blue}{\left({v}^{2}\right)}\right)\right) \]

    2. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(2 \cdot {v}^{2}\right)\right), \left({\color{blue}{v}}^{2}\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left({v}^{2} \cdot 2\right)\right), \left({v}^{2}\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\left(v \cdot v\right) \cdot 2\right)\right), \left({v}^{2}\right)\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(v \cdot \left(v \cdot 2\right)\right)\right), \left({v}^{2}\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \left(v \cdot 2\right)\right)\right), \left({v}^{2}\right)\right)\right) \]

    7. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \left({v}^{2}\right)\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \left(v \cdot \color{blue}{v}\right)\right)\right) \]

    9. *-lowering-*.f32 66.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right) \]

13. Simplified 66.8%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\color{blue}{\frac{0.3333333333333333 + v \cdot \left(v \cdot 2\right)}{v \cdot v}}} \]

15. Applied egg-rr 66.8%

    \[\leadsto \color{blue}{\left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) \cdot \frac{cosTheta\_O}{\frac{0.3333333333333333 + \frac{v}{\frac{0.5}{v}}}{v}}} \]

16. Final simplification 66.8%

    \[\leadsto \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right) \cdot \frac{cosTheta\_O}{\frac{0.3333333333333333 + \frac{v}{\frac{0.5}{v}}}{v}} \]
17. Add Preprocessing

Alternative 16: 63.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{cosTheta\_O\_m \cdot \frac{cosTheta\_i\_m}{v}}{\frac{0.3333333333333333 + v \cdot \left(v \cdot 2\right)}{v \cdot v}}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (* cosTheta_O_m (/ cosTheta_i_m v))
    (/ (+ 0.3333333333333333 (* v (* v 2.0))) (* v v))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * (cosTheta_i_m / v)) / ((0.3333333333333333f + (v * (v * 2.0f))) / (v * v))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((costheta_o_m * (costheta_i_m / v)) / ((0.3333333333333333e0 + (v * (v * 2.0e0))) / (v * v))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_O_m * Float32(cosTheta_i_m / v)) / Float32(Float32(Float32(0.3333333333333333) + Float32(v * Float32(v * Float32(2.0)))) / Float32(v * v)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * (cosTheta_i_m / v)) / ((single(0.3333333333333333) + (v * (v * single(2.0)))) / (v * v))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

8. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\color{blue}{\left(\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right), v\right), \mathsf{*.f32}\left(\color{blue}{v}, 2\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6} \cdot 1}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6}}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left({v}^{2}\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left(v \cdot v\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 66.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

10. Simplified 66.8%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\color{blue}{\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}} \cdot \left(v \cdot 2\right)} \]

11. Taylor expanded in v around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \color{blue}{\left(\frac{\frac{1}{3} + 2 \cdot {v}^{2}}{{v}^{2}}\right)}\right) \]
12. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right), \color{blue}{\left({v}^{2}\right)}\right)\right) \]

    2. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(2 \cdot {v}^{2}\right)\right), \left({\color{blue}{v}}^{2}\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left({v}^{2} \cdot 2\right)\right), \left({v}^{2}\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\left(v \cdot v\right) \cdot 2\right)\right), \left({v}^{2}\right)\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(v \cdot \left(v \cdot 2\right)\right)\right), \left({v}^{2}\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \left(v \cdot 2\right)\right)\right), \left({v}^{2}\right)\right)\right) \]

    7. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \left({v}^{2}\right)\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \left(v \cdot \color{blue}{v}\right)\right)\right) \]

    9. *-lowering-*.f32 66.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right) \]

13. Simplified 66.8%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\color{blue}{\frac{0.3333333333333333 + v \cdot \left(v \cdot 2\right)}{v \cdot v}}} \]

14. Taylor expanded in sinTheta_O around 0

    \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right) \]
15. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right)}, \mathsf{*.f32}\left(v, v\right)\right)\right) \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right)}, \mathsf{*.f32}\left(v, v\right)\right)\right) \]

    3. /-lowering-/.f32 66.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \color{blue}{\mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)}\right), \mathsf{*.f32}\left(v, v\right)\right)\right) \]

16. Simplified 66.8%

    \[\leadsto \frac{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}}{\frac{0.3333333333333333 + v \cdot \left(v \cdot 2\right)}{v \cdot v}} \]
17. Add Preprocessing

Alternative 17: 63.5% accurate, 14.7× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{cosTheta\_i\_m \cdot \left(0.5 \cdot cosTheta\_O\_m\right)}{v \cdot \left(\frac{0.16666666666666666}{v \cdot v} + 1\right)}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (* cosTheta_i_m (* 0.5 cosTheta_O_m))
    (* v (+ (/ 0.16666666666666666 (* v v)) 1.0))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_i_m * (0.5f * cosTheta_O_m)) / (v * ((0.16666666666666666f / (v * v)) + 1.0f))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((costheta_i_m * (0.5e0 * costheta_o_m)) / (v * ((0.16666666666666666e0 / (v * v)) + 1.0e0))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_i_m * Float32(Float32(0.5) * cosTheta_O_m)) / Float32(v * Float32(Float32(Float32(0.16666666666666666) / Float32(v * v)) + Float32(1.0))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_i_m * (single(0.5) * cosTheta_O_m)) / (v * ((single(0.16666666666666666) / (v * v)) + single(1.0)))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

8. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\color{blue}{\left(\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right), v\right), \mathsf{*.f32}\left(\color{blue}{v}, 2\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6} \cdot 1}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6}}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left({v}^{2}\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left(v \cdot v\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 66.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

10. Simplified 66.8%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\color{blue}{\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}} \cdot \left(v \cdot 2\right)} \]

11. Taylor expanded in sinTheta_O around 0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v \cdot \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}} \]

    2. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{\left(v \cdot \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)}\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{1}{2}\right), \left(\color{blue}{v} \cdot \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{2}\right), \left(v \cdot \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), \left(\color{blue}{v} \cdot \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), \left(v \cdot \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right) \]

    7. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), \left(\color{blue}{v} \cdot \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), \left(v \cdot \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right) \]

    9. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), \left(v \cdot \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right) \]

    10. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), \mathsf{*.f32}\left(v, \color{blue}{\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}\right)\right) \]

    11. +-lowering-+.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}\right)\right)\right) \]

    12. associate-*r/ N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6} \cdot 1}{\color{blue}{{v}^{2}}}\right)\right)\right)\right) \]

    13. metadata-eval N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6}}{{\color{blue}{v}}^{2}}\right)\right)\right)\right) \]

    14. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \color{blue}{\left({v}^{2}\right)}\right)\right)\right)\right) \]

    15. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left(v \cdot \color{blue}{v}\right)\right)\right)\right)\right) \]

    16. *-lowering-*.f32 66.8%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right)\right)\right) \]

13. Simplified 66.8%

    \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v \cdot \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}} \]

14. Final simplification 66.8%

    \[\leadsto \frac{cosTheta\_i \cdot \left(0.5 \cdot cosTheta\_O\right)}{v \cdot \left(\frac{0.16666666666666666}{v \cdot v} + 1\right)} \]
15. Add Preprocessing

Alternative 18: 63.5% accurate, 16.9× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{cosTheta\_O\_m \cdot \left(cosTheta\_i\_m \cdot v\right)}{0.3333333333333333 + \left(v \cdot v\right) \cdot 2}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (* cosTheta_O_m (* cosTheta_i_m v))
    (+ 0.3333333333333333 (* (* v v) 2.0))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * (cosTheta_i_m * v)) / (0.3333333333333333f + ((v * v) * 2.0f))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((costheta_o_m * (costheta_i_m * v)) / (0.3333333333333333e0 + ((v * v) * 2.0e0))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_O_m * Float32(cosTheta_i_m * v)) / Float32(Float32(0.3333333333333333) + Float32(Float32(v * v) * Float32(2.0))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * (cosTheta_i_m * v)) / (single(0.3333333333333333) + ((v * v) * single(2.0)))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

8. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\color{blue}{\left(\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right), v\right), \mathsf{*.f32}\left(\color{blue}{v}, 2\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6} \cdot 1}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6}}{{v}^{2}}\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left({v}^{2}\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left(v \cdot v\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 66.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

10. Simplified 66.8%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\color{blue}{\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}} \cdot \left(v \cdot 2\right)} \]

11. Taylor expanded in v around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \color{blue}{\left(\frac{\frac{1}{3} + 2 \cdot {v}^{2}}{{v}^{2}}\right)}\right) \]
12. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right), \color{blue}{\left({v}^{2}\right)}\right)\right) \]

    2. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(2 \cdot {v}^{2}\right)\right), \left({\color{blue}{v}}^{2}\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left({v}^{2} \cdot 2\right)\right), \left({v}^{2}\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\left(v \cdot v\right) \cdot 2\right)\right), \left({v}^{2}\right)\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(v \cdot \left(v \cdot 2\right)\right)\right), \left({v}^{2}\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \left(v \cdot 2\right)\right)\right), \left({v}^{2}\right)\right)\right) \]

    7. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \left({v}^{2}\right)\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \left(v \cdot \color{blue}{v}\right)\right)\right) \]

    9. *-lowering-*.f32 66.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, 2\right)\right)\right), \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right) \]

13. Simplified 66.8%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\color{blue}{\frac{0.3333333333333333 + v \cdot \left(v \cdot 2\right)}{v \cdot v}}} \]

14. Taylor expanded in sinTheta_O around 0

    \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)}{\frac{1}{3} + 2 \cdot {v}^{2}}} \]
15. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)\right), \color{blue}{\left(\frac{1}{3} + 2 \cdot {v}^{2}\right)}\right) \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i \cdot v\right)\right), \left(\color{blue}{\frac{1}{3}} + 2 \cdot {v}^{2}\right)\right) \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, v\right)\right), \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right) \]

    4. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{1}{3}, \color{blue}{\left(2 \cdot {v}^{2}\right)}\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \color{blue}{\left({v}^{2}\right)}\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \left(v \cdot \color{blue}{v}\right)\right)\right)\right) \]

    7. *-lowering-*.f32 66.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right)\right) \]

16. Simplified 66.8%

    \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)}{0.3333333333333333 + 2 \cdot \left(v \cdot v\right)}} \]

17. Final simplification 66.8%

    \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)}{0.3333333333333333 + \left(v \cdot v\right) \cdot 2} \]
18. Add Preprocessing

Alternative 19: 58.2% accurate, 24.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{1}{\frac{\frac{v}{cosTheta\_O\_m}}{cosTheta\_i\_m \cdot 0.5}}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (* cosTheta_i_s (/ 1.0 (/ (/ v cosTheta_O_m) (* cosTheta_i_m 0.5))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (1.0f / ((v / cosTheta_O_m) / (cosTheta_i_m * 0.5f))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (1.0e0 / ((v / costheta_o_m) / (costheta_i_m * 0.5e0))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(1.0) / Float32(Float32(v / cosTheta_O_m) / Float32(cosTheta_i_m * Float32(0.5))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (single(1.0) / ((v / cosTheta_O_m) / (cosTheta_i_m * single(0.5)))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{v}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 61.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 61.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_O \cdot \frac{1}{2}\right) \cdot cosTheta\_i}{v} \]

   2. associate-/l* N/A

      \[\leadsto \left(cosTheta\_O \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{cosTheta\_i}{v}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_O \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{cosTheta\_i}{v}\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right), \left(\frac{\color{blue}{cosTheta\_i}}{v}\right)\right) \]

   5. /-lowering-/.f32 61.8%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right), \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right)\right) \]

9. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{\left(cosTheta\_O \cdot 0.5\right) \cdot \frac{cosTheta\_i}{v}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{cosTheta\_i}}{v} \]

    2. associate-*l* N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

    3. clear-num N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_O \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i}}}\right) \]

    4. associate-/r/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_O \cdot \left(\frac{1}{v} \cdot \color{blue}{cosTheta\_i}\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(cosTheta\_O \cdot \frac{1}{v}\right) \cdot \color{blue}{cosTheta\_i}\right) \]

    6. div-inv N/A

       \[\leadsto \frac{1}{2} \cdot \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \]

    7. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{v}}\right) \]

    8. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}\right) \]

    9. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_i \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_O}}}\right)\right) \]

    10. un-div-inv N/A

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{cosTheta\_i}{\color{blue}{\frac{v}{cosTheta\_O}}}\right)\right) \]

    11. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{v}{cosTheta\_O}\right)}\right)\right) \]

    12. /-lowering-/.f32 61.8%

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \color{blue}{cosTheta\_O}\right)\right)\right) \]

11. Applied egg-rr 61.8%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{2} \cdot cosTheta\_i}{\color{blue}{\frac{v}{cosTheta\_O}}} \]

    2. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{v}{cosTheta\_O}}{\frac{1}{2} \cdot cosTheta\_i}}} \]

    3. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{\frac{v}{cosTheta\_O}}{\frac{1}{2} \cdot cosTheta\_i}\right)}\right) \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{v}{cosTheta\_O}\right), \color{blue}{\left(\frac{1}{2} \cdot cosTheta\_i\right)}\right)\right) \]

    5. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, cosTheta\_O\right), \left(\color{blue}{\frac{1}{2}} \cdot cosTheta\_i\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, cosTheta\_O\right), \left(cosTheta\_i \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

    7. *-lowering-*.f32 62.3%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, cosTheta\_O\right), \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

13. Applied egg-rr 62.3%

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i \cdot 0.5}}} \]
14. Add Preprocessing

Alternative 20: 58.2% accurate, 24.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{1}{\frac{v}{cosTheta\_i\_m \cdot \left(0.5 \cdot cosTheta\_O\_m\right)}}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (* cosTheta_i_s (/ 1.0 (/ v (* cosTheta_i_m (* 0.5 cosTheta_O_m)))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (1.0f / (v / (cosTheta_i_m * (0.5f * cosTheta_O_m)))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (1.0e0 / (v / (costheta_i_m * (0.5e0 * costheta_o_m)))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(1.0) / Float32(v / Float32(cosTheta_i_m * Float32(Float32(0.5) * cosTheta_O_m))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (single(1.0) / (v / (cosTheta_i_m * (single(0.5) * cosTheta_O_m)))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{v}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 61.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 61.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{v}{cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{1}{2}\right)}}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{v}{cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{1}{2}\right)}\right)}\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \color{blue}{\left(cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{1}{2}\right)\right)}\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(cosTheta\_O \cdot \frac{1}{2}\right)}\right)\right)\right) \]

   5. *-lowering-*.f32 62.3%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

9. Applied egg-rr 62.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}}} \]

10. Final simplification 62.3%

    \[\leadsto \frac{1}{\frac{v}{cosTheta\_i \cdot \left(0.5 \cdot cosTheta\_O\right)}} \]
11. Add Preprocessing

Alternative 21: 58.2% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{0.5}{\frac{v}{cosTheta\_i\_m \cdot cosTheta\_O\_m}}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (/ 0.5 (/ v (* cosTheta_i_m cosTheta_O_m))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (0.5f / (v / (cosTheta_i_m * cosTheta_O_m))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (0.5e0 / (v / (costheta_i_m * costheta_o_m))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(0.5) / Float32(v / Float32(cosTheta_i_m * cosTheta_O_m)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (single(0.5) / (v / (cosTheta_i_m * cosTheta_O_m))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{v}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 61.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 61.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_O \cdot \frac{1}{2}\right) \cdot cosTheta\_i}{v} \]

   2. associate-/l* N/A

      \[\leadsto \left(cosTheta\_O \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{cosTheta\_i}{v}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_O \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{cosTheta\_i}{v}\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right), \left(\frac{\color{blue}{cosTheta\_i}}{v}\right)\right) \]

   5. /-lowering-/.f32 61.8%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right), \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right)\right) \]

9. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{\left(cosTheta\_O \cdot 0.5\right) \cdot \frac{cosTheta\_i}{v}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{cosTheta\_i}}{v} \]

    2. associate-*l* N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

    3. clear-num N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_O \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i}}}\right) \]

    4. associate-/r/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_O \cdot \left(\frac{1}{v} \cdot \color{blue}{cosTheta\_i}\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(cosTheta\_O \cdot \frac{1}{v}\right) \cdot \color{blue}{cosTheta\_i}\right) \]

    6. div-inv N/A

       \[\leadsto \frac{1}{2} \cdot \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \]

    7. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{v}}\right) \]

    8. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}\right) \]

    9. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_i \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_O}}}\right)\right) \]

    10. un-div-inv N/A

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{cosTheta\_i}{\color{blue}{\frac{v}{cosTheta\_O}}}\right)\right) \]

    11. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{v}{cosTheta\_O}\right)}\right)\right) \]

    12. /-lowering-/.f32 61.8%

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \color{blue}{cosTheta\_O}\right)\right)\right) \]

11. Applied egg-rr 61.8%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}} \]
12. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{1}{2} \cdot \frac{1}{\color{blue}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i}}} \]

    2. un-div-inv N/A

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i}}} \]

    3. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i}\right)}\right) \]

    4. associate-/l/ N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \left(\frac{v}{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}\right)\right) \]

    5. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)\right) \]

    6. *-lowering-*.f32 62.3%

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{cosTheta\_O}\right)\right)\right) \]

13. Applied egg-rr 62.3%

    \[\leadsto \color{blue}{\frac{0.5}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]
14. Add Preprocessing

Alternative 22: 57.7% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\frac{0.5}{v} \cdot \left(cosTheta\_i\_m \cdot cosTheta\_O\_m\right)\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (* (/ 0.5 v) (* cosTheta_i_m cosTheta_O_m)))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((0.5f / v) * (cosTheta_i_m * cosTheta_O_m)));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((0.5e0 / v) * (costheta_i_m * costheta_o_m)))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(0.5) / v) * Float32(cosTheta_i_m * cosTheta_O_m))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((single(0.5) / v) * (cosTheta_i_m * cosTheta_O_m)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{v}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 61.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 61.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{2}}{v} \]

   2. associate-/l* N/A

      \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(\frac{\color{blue}{\frac{1}{2}}}{v}\right)\right) \]

   5. /-lowering-/.f32 61.8%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right)\right) \]

9. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{0.5}{v}} \]

10. Final simplification 61.8%

    \[\leadsto \frac{0.5}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]
11. Add Preprocessing

Alternative 23: 57.7% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(0.5 \cdot \frac{cosTheta\_i\_m}{\frac{v}{cosTheta\_O\_m}}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (* 0.5 (/ cosTheta_i_m (/ v cosTheta_O_m))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (0.5f * (cosTheta_i_m / (v / cosTheta_O_m))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (0.5e0 * (costheta_i_m / (v / costheta_o_m))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(0.5) * Float32(cosTheta_i_m / Float32(v / cosTheta_O_m)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (single(0.5) * (cosTheta_i_m / (v / cosTheta_O_m))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{v}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 61.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 61.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_O \cdot \frac{1}{2}\right) \cdot cosTheta\_i}{v} \]

   2. associate-/l* N/A

      \[\leadsto \left(cosTheta\_O \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{cosTheta\_i}{v}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_O \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{cosTheta\_i}{v}\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right), \left(\frac{\color{blue}{cosTheta\_i}}{v}\right)\right) \]

   5. /-lowering-/.f32 61.8%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right), \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right)\right) \]

9. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{\left(cosTheta\_O \cdot 0.5\right) \cdot \frac{cosTheta\_i}{v}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{cosTheta\_i}}{v} \]

    2. associate-*l* N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

    3. clear-num N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_O \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i}}}\right) \]

    4. associate-/r/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_O \cdot \left(\frac{1}{v} \cdot \color{blue}{cosTheta\_i}\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(cosTheta\_O \cdot \frac{1}{v}\right) \cdot \color{blue}{cosTheta\_i}\right) \]

    6. div-inv N/A

       \[\leadsto \frac{1}{2} \cdot \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \]

    7. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{v}}\right) \]

    8. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}\right) \]

    9. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_i \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_O}}}\right)\right) \]

    10. un-div-inv N/A

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{cosTheta\_i}{\color{blue}{\frac{v}{cosTheta\_O}}}\right)\right) \]

    11. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{v}{cosTheta\_O}\right)}\right)\right) \]

    12. /-lowering-/.f32 61.8%

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \color{blue}{cosTheta\_O}\right)\right)\right) \]

11. Applied egg-rr 61.8%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}} \]
12. Add Preprocessing

Alternative 24: 57.7% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(0.5 \cdot \left(cosTheta\_i\_m \cdot \frac{cosTheta\_O\_m}{v}\right)\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (* 0.5 (* cosTheta_i_m (/ cosTheta_O_m v))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (0.5f * (cosTheta_i_m * (cosTheta_O_m / v))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (0.5e0 * (costheta_i_m * (costheta_o_m / v))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(0.5) * Float32(cosTheta_i_m * Float32(cosTheta_O_m / v)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (single(0.5) * (cosTheta_i_m * (cosTheta_O_m / v))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{v}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 61.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 61.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_O \cdot \frac{1}{2}\right) \cdot cosTheta\_i}{v} \]

   2. associate-/l* N/A

      \[\leadsto \left(cosTheta\_O \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{cosTheta\_i}{v}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_O \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{cosTheta\_i}{v}\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right), \left(\frac{\color{blue}{cosTheta\_i}}{v}\right)\right) \]

   5. /-lowering-/.f32 61.8%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right), \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right)\right) \]

9. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{\left(cosTheta\_O \cdot 0.5\right) \cdot \frac{cosTheta\_i}{v}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{cosTheta\_i}}{v} \]

    2. associate-*l* N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

    3. clear-num N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_O \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i}}}\right) \]

    4. associate-/r/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_O \cdot \left(\frac{1}{v} \cdot \color{blue}{cosTheta\_i}\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \frac{1}{2} \cdot \left(\left(cosTheta\_O \cdot \frac{1}{v}\right) \cdot \color{blue}{cosTheta\_i}\right) \]

    6. div-inv N/A

       \[\leadsto \frac{1}{2} \cdot \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \]

    7. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{v}}\right) \]

    8. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}\right) \]

    9. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_i \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_O}}}\right)\right) \]

    10. un-div-inv N/A

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{cosTheta\_i}{\color{blue}{\frac{v}{cosTheta\_O}}}\right)\right) \]

    11. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{v}{cosTheta\_O}\right)}\right)\right) \]

    12. /-lowering-/.f32 61.8%

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \color{blue}{cosTheta\_O}\right)\right)\right) \]

11. Applied egg-rr 61.8%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}} \]
12. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{1}{\color{blue}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i}}}\right)\right) \]

    2. associate-/r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{1}{\frac{v}{cosTheta\_O}} \cdot \color{blue}{cosTheta\_i}\right)\right) \]

    3. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(\frac{cosTheta\_O}{v}\right), \color{blue}{cosTheta\_i}\right)\right) \]

    5. /-lowering-/.f32 61.8%

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), cosTheta\_i\right)\right) \]

13. Applied egg-rr 61.8%

    \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)} \]

14. Final simplification 61.8%

    \[\leadsto 0.5 \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))