HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 22.2s

Alternatives: 15

Speedup: 1.8×

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: 98.6% 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: 98.8% accurate, 1.8× speedup

Math

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

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_i_m (* cosTheta_i_m (* sinTheta_i (/ sinTheta_O v))))
     (* cosTheta_O_m (/ 1.0 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_i_m - (cosTheta_i_m * (sinTheta_i * (sinTheta_O / v)))) * (cosTheta_O_m * (1.0f / 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_i_m - (costheta_i_m * (sintheta_i * (sintheta_o / v)))) * (costheta_o_m * (1.0e0 / 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(Float32(Float32(cosTheta_i_m - Float32(cosTheta_i_m * Float32(sinTheta_i * Float32(sinTheta_O / v)))) * Float32(cosTheta_O_m * Float32(Float32(1.0) / 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_i_m - (cosTheta_i_m * (sinTheta_i * (sinTheta_O / v)))) * (cosTheta_O_m * (single(1.0) / v))) / (sinh((single(1.0) / v)) / (single(0.5) / v))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

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

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

   8. associate-/l* N/A

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

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

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

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

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

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

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

   12. associate-/l* N/A

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

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

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

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

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

   15. *-lowering-*.f32 98.5%

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

5. Simplified 98.5%

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

   1. metadata-eval N/A

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

   2. div-inv N/A

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

   3. associate-/r/ N/A

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

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

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

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

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

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

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

   7. /-lowering-/.f32 98.7%

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

7. Applied egg-rr 98.7%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. *-commutative N/A

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

   5. div-inv N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. div-inv N/A

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

   10. *-commutative N/A

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

   11. clear-num N/A

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

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

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

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

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

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

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

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

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

9. Applied egg-rr 98.9%

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

Alternative 2: 98.7% accurate, 1.8× speedup

Math

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

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
   (*
    cosTheta_O_m
    (/
     (*
      (/ 0.5 v)
      (/ (- cosTheta_i_m (* cosTheta_i_m (* sinTheta_i (/ sinTheta_O v)))) 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 * (cosTheta_O_m * (((0.5f / v) * ((cosTheta_i_m - (cosTheta_i_m * (sinTheta_i * (sinTheta_O / v)))) / 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 * (costheta_o_m * (((0.5e0 / v) * ((costheta_i_m - (costheta_i_m * (sintheta_i * (sintheta_o / v)))) / 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(cosTheta_O_m * Float32(Float32(Float32(Float32(0.5) / v) * Float32(Float32(cosTheta_i_m - Float32(cosTheta_i_m * Float32(sinTheta_i * Float32(sinTheta_O / v)))) / 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 * (cosTheta_O_m * (((single(0.5) / v) * ((cosTheta_i_m - (cosTheta_i_m * (sinTheta_i * (sinTheta_O / v)))) / v)) / sinh((single(1.0) / v)))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

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

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

   8. associate-/l* N/A

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

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

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

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

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

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

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

   12. associate-/l* N/A

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

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

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

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

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

   15. *-lowering-*.f32 98.5%

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

5. Simplified 98.5%

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

   1. associate-/l* N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

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

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

   10. *-commutative N/A

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

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

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

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

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. associate-/r/ N/A

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

7. Applied egg-rr 98.9%

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

   1. associate-/r/ N/A

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

   2. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

   7. div-inv N/A

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

   8. *-commutative N/A

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

   9. clear-num N/A

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

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

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

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

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

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

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

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

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

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

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

9. Applied egg-rr 98.8%

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

10. Final simplification 98.8%

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

Alternative 3: 98.5% accurate, 1.9× speedup

Math

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

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_i_m (* (/ cosTheta_O_m v) (/ (/ 0.5 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 * (cosTheta_i_m * ((cosTheta_O_m / v) * ((0.5f / 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 * (costheta_i_m * ((costheta_o_m / v) * ((0.5e0 / 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(cosTheta_i_m * Float32(Float32(cosTheta_O_m / v) * Float32(Float32(Float32(0.5) / 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 * (cosTheta_i_m * ((cosTheta_O_m / v) * ((single(0.5) / v) / sinh((single(1.0) / v))))));
end

Derivation

1. Initial program 98.5%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. exp-neg N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

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

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

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

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in v around inf

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

   1. /-lowering-/.f32 98.7%

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

7. Simplified 98.7%

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

Alternative 4: 98.5% accurate, 1.9× speedup

Math

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

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 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 / 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 / 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 / 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 / v) / (sinh((single(1.0) / v)) / (single(0.5) / v)))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

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

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

   8. associate-/l* N/A

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

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

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

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

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

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

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

   12. associate-/l* N/A

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

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

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

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

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

   15. *-lowering-*.f32 98.5%

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

5. Simplified 98.5%

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

   1. associate-/l* N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

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

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

   10. *-commutative N/A

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

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

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

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

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. associate-/r/ N/A

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

7. Applied egg-rr 98.9%

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

8. Taylor expanded in v around inf

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

   1. /-lowering-/.f32 98.8%

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

10. Simplified 98.8%

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

Alternative 5: 76.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(cosTheta\_O\_m \cdot \frac{cosTheta\_i\_m - cosTheta\_i\_m \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}{v \cdot \left(\frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} + \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)\right)}\right)\right) \end{array} \]

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
    (/
     (- cosTheta_i_m (* cosTheta_i_m (* sinTheta_i (/ sinTheta_O v))))
     (*
      v
      (+
       (/
        (+ 0.016666666666666666 (/ 0.0003968253968253968 (* v v)))
        (* v (* v (* v v))))
       (+ 2.0 (/ 0.3333333333333333 (* 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 - (cosTheta_i_m * (sinTheta_i * (sinTheta_O / v)))) / (v * (((0.016666666666666666f + (0.0003968253968253968f / (v * v))) / (v * (v * (v * v)))) + (2.0f + (0.3333333333333333f / (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 - (costheta_i_m * (sintheta_i * (sintheta_o / v)))) / (v * (((0.016666666666666666e0 + (0.0003968253968253968e0 / (v * v))) / (v * (v * (v * v)))) + (2.0e0 + (0.3333333333333333e0 / (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(cosTheta_O_m * Float32(Float32(cosTheta_i_m - Float32(cosTheta_i_m * Float32(sinTheta_i * Float32(sinTheta_O / v)))) / Float32(v * Float32(Float32(Float32(Float32(0.016666666666666666) + Float32(Float32(0.0003968253968253968) / Float32(v * v))) / Float32(v * Float32(v * Float32(v * v)))) + Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / 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 - (cosTheta_i_m * (sinTheta_i * (sinTheta_O / v)))) / (v * (((single(0.016666666666666666) + (single(0.0003968253968253968) / (v * v))) / (v * (v * (v * v)))) + (single(2.0) + (single(0.3333333333333333) / (v * v))))))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

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

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

   8. associate-/l* N/A

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

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

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

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

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

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

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

   12. associate-/l* N/A

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

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

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

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

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

   15. *-lowering-*.f32 98.5%

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

5. Simplified 98.5%

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

   1. metadata-eval N/A

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

   2. div-inv N/A

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

   3. associate-/r/ N/A

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

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

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

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

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

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

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

   7. /-lowering-/.f32 98.7%

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

7. Applied egg-rr 98.7%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. *-commutative N/A

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

   4. div-inv N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

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

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

9. Applied egg-rr 98.9%

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

10. Taylor expanded in v around -inf

    \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(sinTheta\_O, v\right)\right)\right)\right), \color{blue}{\left(-1 \cdot \left(v \cdot \left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right), cosTheta\_O\right) \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(sinTheta\_O, v\right)\right)\right)\right), \left(\mathsf{neg}\left(v \cdot \left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right), cosTheta\_O\right) \]

    2. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(sinTheta\_O, v\right)\right)\right)\right), \left(v \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right), cosTheta\_O\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(sinTheta\_O, v\right)\right)\right)\right), \mathsf{*.f32}\left(v, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right), cosTheta\_O\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(sinTheta\_O, v\right)\right)\right)\right), \mathsf{*.f32}\left(v, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right)\right)\right), cosTheta\_O\right) \]

    5. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(sinTheta\_O, v\right)\right)\right)\right), \mathsf{*.f32}\left(v, \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right)\right) + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right)\right)\right), cosTheta\_O\right) \]

    6. distribute-neg-out N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(sinTheta\_O, v\right)\right)\right)\right), \mathsf{*.f32}\left(v, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right)\right)\right), cosTheta\_O\right) \]

    7. remove-double-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(sinTheta\_O, v\right)\right)\right)\right), \mathsf{*.f32}\left(v, \left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right), cosTheta\_O\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(sinTheta\_O, v\right)\right)\right)\right), \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right), \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right), cosTheta\_O\right) \]

12. Simplified 77.8%

    \[\leadsto \frac{cosTheta\_i - cosTheta\_i \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}{\color{blue}{v \cdot \left(\frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{\left(v \cdot \left(v \cdot v\right)\right) \cdot v} + \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)\right)}} \cdot cosTheta\_O \]

13. Final simplification 77.8%

    \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i - cosTheta\_i \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}{v \cdot \left(\frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} + \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)\right)} \]
14. Add Preprocessing

Alternative 6: 70.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{\frac{cosTheta\_O\_m}{\frac{\frac{-2}{v}}{\frac{cosTheta\_i\_m - \frac{cosTheta\_i\_m}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}{v}} \cdot \left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right)}}{v}\right) \end{array} \]

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
     (*
      (/
       (/ -2.0 v)
       (/ (- cosTheta_i_m (/ cosTheta_i_m (/ v (* sinTheta_i sinTheta_O)))) v))
      (-
       -1.0
       (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* 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_O_m / (((-2.0f / v) / ((cosTheta_i_m - (cosTheta_i_m / (v / (sinTheta_i * sinTheta_O)))) / v)) * (-1.0f - ((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (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_o_m / ((((-2.0e0) / v) / ((costheta_i_m - (costheta_i_m / (v / (sintheta_i * sintheta_o)))) / v)) * ((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (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(cosTheta_O_m / Float32(Float32(Float32(Float32(-2.0) / v) / Float32(Float32(cosTheta_i_m - Float32(cosTheta_i_m / Float32(v / Float32(sinTheta_i * sinTheta_O)))) / v)) * Float32(Float32(-1.0) - Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * 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 / (((single(-2.0) / v) / ((cosTheta_i_m - (cosTheta_i_m / (v / (sinTheta_i * sinTheta_O)))) / v)) * (single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v))))) / v));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

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

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

   8. associate-/l* N/A

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

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

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

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

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

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

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

   12. associate-/l* N/A

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

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

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

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

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

   15. *-lowering-*.f32 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

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

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

8. Simplified 71.4%

   \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{v}}{\left(\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot 2\right) \cdot v} \]

10. Applied egg-rr 71.4%

    \[\leadsto \color{blue}{\frac{cosTheta\_O}{v} \cdot \frac{\frac{cosTheta\_i - cosTheta\_i \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}{v}}{\left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right) \cdot \frac{-2}{v}}} \]
11. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \frac{cosTheta\_O \cdot \frac{\frac{cosTheta\_i - cosTheta\_i \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}{v}}{\left(-1 - \frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{v \cdot v}\right) \cdot \frac{-2}{v}}}{\color{blue}{v}} \]

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

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

12. Applied egg-rr 71.4%

    \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{\left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right) \cdot \frac{\frac{-2}{v}}{\frac{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}{v}}}}{v}} \]

13. Final simplification 71.4%

    \[\leadsto \frac{\frac{cosTheta\_O}{\frac{\frac{-2}{v}}{\frac{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}{v}} \cdot \left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right)}}{v} \]
14. Add Preprocessing

Alternative 7: 70.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{\frac{cosTheta\_i\_m \cdot cosTheta\_O\_m}{v}}{\frac{0.016666666666666666 + v \cdot \left(v \cdot \left(0.3333333333333333 + \left(v \cdot v\right) \cdot 2\right)\right)}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)}}\right) \end{array} \]

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_i_m cosTheta_O_m) v)
    (/
     (+
      0.016666666666666666
      (* v (* v (+ 0.3333333333333333 (* (* v v) 2.0)))))
     (* v (* 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_O_m) / v) / ((0.016666666666666666f + (v * (v * (0.3333333333333333f + ((v * v) * 2.0f))))) / (v * (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_o_m) / v) / ((0.016666666666666666e0 + (v * (v * (0.3333333333333333e0 + ((v * v) * 2.0e0))))) / (v * (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 * cosTheta_O_m) / v) / Float32(Float32(Float32(0.016666666666666666) + Float32(v * Float32(v * Float32(Float32(0.3333333333333333) + Float32(Float32(v * v) * Float32(2.0)))))) / Float32(v * Float32(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_O_m) / v) / ((single(0.016666666666666666) + (v * (v * (single(0.3333333333333333) + ((v * v) * single(2.0)))))) / (v * (v * (v * v))))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

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

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

   8. associate-/l* N/A

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

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

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

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

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

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

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

   12. associate-/l* N/A

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

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

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

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

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

   15. *-lowering-*.f32 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

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

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

8. Simplified 71.4%

   \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{v}}{\left(\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot 2\right) \cdot v} \]

9. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \color{blue}{\left(\frac{\frac{1}{60} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)}{{v}^{4}}\right)}\right) \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\left(\frac{1}{60} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right), \color{blue}{\left({v}^{4}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \left({v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)\right), \left({\color{blue}{v}}^{4}\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right) \cdot {v}^{2}\right)\right), \left({v}^{4}\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right) \cdot \left(v \cdot v\right)\right)\right), \left({v}^{4}\right)\right)\right) \]

    5. associate-*r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \left(\left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right) \cdot v\right) \cdot v\right)\right), \left({v}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right) \cdot v\right), v\right)\right), \left({v}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right), v\right), v\right)\right), \left({v}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(2 \cdot {v}^{2}\right)\right), v\right), v\right)\right), \left({v}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \left({v}^{2}\right)\right)\right), v\right), v\right)\right), \left({v}^{4}\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \left(v \cdot v\right)\right)\right), v\right), v\right)\right), \left({v}^{4}\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left({v}^{4}\right)\right)\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left({v}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

    13. pow-sqr N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left({v}^{2} \cdot \color{blue}{{v}^{2}}\right)\right)\right) \]

    14. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left(\left(v \cdot v\right) \cdot {\color{blue}{v}}^{2}\right)\right)\right) \]

    15. associate-*l* N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left(v \cdot \color{blue}{\left(v \cdot {v}^{2}\right)}\right)\right)\right) \]

    16. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left(v \cdot \left(v \cdot \left(v \cdot \color{blue}{v}\right)\right)\right)\right)\right) \]

    17. cube-mult N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left(v \cdot {v}^{\color{blue}{3}}\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{\left({v}^{3}\right)}\right)\right)\right) \]

    19. cube-mult N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \left(v \cdot \color{blue}{\left(v \cdot v\right)}\right)\right)\right)\right) \]

    20. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \left(v \cdot {v}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \color{blue}{\left({v}^{2}\right)}\right)\right)\right)\right) \]

    22. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \left(v \cdot \color{blue}{v}\right)\right)\right)\right)\right) \]

    23. *-lowering-*.f32 71.4%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right)\right)\right) \]

11. Simplified 71.4%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{v}}{\color{blue}{\frac{0.016666666666666666 + \left(\left(0.3333333333333333 + 2 \cdot \left(v \cdot v\right)\right) \cdot v\right) \cdot v}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)}}} \]

12. Taylor expanded in sinTheta_O around 0

    \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right) \]
13. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right)}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right) \]

    2. *-lowering-*.f32 71.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\color{blue}{\frac{1}{60}}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right) \]

14. Simplified 71.4%

    \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}}{\frac{0.016666666666666666 + \left(\left(0.3333333333333333 + 2 \cdot \left(v \cdot v\right)\right) \cdot v\right) \cdot v}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)}} \]

15. Final simplification 71.4%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.016666666666666666 + v \cdot \left(v \cdot \left(0.3333333333333333 + \left(v \cdot v\right) \cdot 2\right)\right)}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)}} \]
16. Add Preprocessing

Alternative 8: 70.3% accurate, 8.8× speedup

Math

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

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 v)
    (/
     (* cosTheta_i_m 0.5)
     (+
      (+ 1.0 (/ 0.16666666666666666 (* v v)))
      (/ 0.008333333333333333 (* v (* 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_O_m / v) * ((cosTheta_i_m * 0.5f) / ((1.0f + (0.16666666666666666f / (v * v))) + (0.008333333333333333f / (v * (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_o_m / v) * ((costheta_i_m * 0.5e0) / ((1.0e0 + (0.16666666666666666e0 / (v * v))) + (0.008333333333333333e0 / (v * (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(cosTheta_O_m / v) * Float32(Float32(cosTheta_i_m * Float32(0.5)) / Float32(Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) / Float32(v * v))) + Float32(Float32(0.008333333333333333) / Float32(v * Float32(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_O_m / v) * ((cosTheta_i_m * single(0.5)) / ((single(1.0) + (single(0.16666666666666666) / (v * v))) + (single(0.008333333333333333) / (v * (v * (v * v))))))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

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

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

   8. associate-/l* N/A

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

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

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

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

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

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

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

   12. associate-/l* N/A

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

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

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

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

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

   15. *-lowering-*.f32 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

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

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

8. Simplified 71.4%

   \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{v}}{\left(\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot 2\right) \cdot v} \]

10. Applied egg-rr 71.4%

    \[\leadsto \color{blue}{\frac{cosTheta\_O}{v} \cdot \frac{\frac{cosTheta\_i - cosTheta\_i \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}{v}}{\left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right) \cdot \frac{-2}{v}}} \]

11. Taylor expanded in sinTheta_i around 0

    \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{cosTheta\_i}{1 + \left(\frac{1}{120} \cdot \frac{1}{{v}^{4}} + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}\right)}\right) \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \left(\frac{\frac{1}{2} \cdot cosTheta\_i}{\color{blue}{1 + \left(\frac{1}{120} \cdot \frac{1}{{v}^{4}} + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot cosTheta\_i\right), \color{blue}{\left(1 + \left(\frac{1}{120} \cdot \frac{1}{{v}^{4}} + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)}\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \frac{1}{2}\right), \left(\color{blue}{1} + \left(\frac{1}{120} \cdot \frac{1}{{v}^{4}} + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \left(\color{blue}{1} + \left(\frac{1}{120} \cdot \frac{1}{{v}^{4}} + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right) \]

    5. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \left(1 + \left(\frac{\frac{1}{120} \cdot 1}{{v}^{4}} + \color{blue}{\frac{1}{6}} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \left(1 + \left(\frac{\frac{1}{120}}{{v}^{4}} + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \left(1 + \left(\frac{1}{6} \cdot \frac{1}{{v}^{2}} + \color{blue}{\frac{\frac{1}{120}}{{v}^{4}}}\right)\right)\right)\right) \]

    8. associate-+r+ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right) + \color{blue}{\frac{\frac{1}{120}}{{v}^{4}}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \mathsf{+.f32}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right), \color{blue}{\left(\frac{\frac{1}{120}}{{v}^{4}}\right)}\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right), \left(\frac{\color{blue}{\frac{1}{120}}}{{v}^{4}}\right)\right)\right)\right) \]

    11. associate-*r/ N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6} \cdot 1}{{v}^{2}}\right)\right), \left(\frac{\frac{1}{120}}{{v}^{4}}\right)\right)\right)\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6}}{{v}^{2}}\right)\right), \left(\frac{\frac{1}{120}}{{v}^{4}}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left({v}^{2}\right)\right)\right), \left(\frac{\frac{1}{120}}{{v}^{4}}\right)\right)\right)\right) \]

    14. unpow2 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left(v \cdot v\right)\right)\right), \left(\frac{\frac{1}{120}}{{v}^{4}}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), \left(\frac{\frac{1}{120}}{{v}^{4}}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{120}, \color{blue}{\left({v}^{4}\right)}\right)\right)\right)\right) \]

    17. metadata-eval N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{120}, \left({v}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right)\right)\right) \]

    18. pow-plus N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{120}, \left({v}^{3} \cdot \color{blue}{v}\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{1}{2}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{120}, \mathsf{*.f32}\left(\left({v}^{3}\right), \color{blue}{v}\right)\right)\right)\right)\right) \]

13. Simplified 71.4%

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

14. Final simplification 71.4%

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

Alternative 9: 70.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{cosTheta\_O\_m \cdot \left(cosTheta\_i\_m \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}{0.016666666666666666 + v \cdot \left(v \cdot \left(0.3333333333333333 + v \cdot \left(v \cdot 2\right)\right)\right)}\right) \end{array} \]

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 (* cosTheta_i_m (* v (* v v))))
    (+
     0.016666666666666666
     (* v (* 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 * (v * v)))) / (0.016666666666666666f + (v * (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 * (v * v)))) / (0.016666666666666666e0 + (v * (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 * Float32(v * Float32(v * v)))) / Float32(Float32(0.016666666666666666) + Float32(v * Float32(v * Float32(Float32(0.3333333333333333) + Float32(v * Float32(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 * (v * v)))) / (single(0.016666666666666666) + (v * (v * (single(0.3333333333333333) + (v * (v * single(2.0)))))))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

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

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

   8. associate-/l* N/A

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

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

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

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

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

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

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

   12. associate-/l* N/A

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

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

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

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

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

   15. *-lowering-*.f32 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

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

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

8. Simplified 71.4%

   \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{v}}{\left(\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot 2\right) \cdot v} \]

9. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \color{blue}{\left(\frac{\frac{1}{60} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)}{{v}^{4}}\right)}\right) \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\left(\frac{1}{60} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right), \color{blue}{\left({v}^{4}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \left({v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)\right), \left({\color{blue}{v}}^{4}\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right) \cdot {v}^{2}\right)\right), \left({v}^{4}\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right) \cdot \left(v \cdot v\right)\right)\right), \left({v}^{4}\right)\right)\right) \]

    5. associate-*r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \left(\left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right) \cdot v\right) \cdot v\right)\right), \left({v}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right) \cdot v\right), v\right)\right), \left({v}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{3} + 2 \cdot {v}^{2}\right), v\right), v\right)\right), \left({v}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(2 \cdot {v}^{2}\right)\right), v\right), v\right)\right), \left({v}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \left({v}^{2}\right)\right)\right), v\right), v\right)\right), \left({v}^{4}\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \left(v \cdot v\right)\right)\right), v\right), v\right)\right), \left({v}^{4}\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left({v}^{4}\right)\right)\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left({v}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

    13. pow-sqr N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left({v}^{2} \cdot \color{blue}{{v}^{2}}\right)\right)\right) \]

    14. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left(\left(v \cdot v\right) \cdot {\color{blue}{v}}^{2}\right)\right)\right) \]

    15. associate-*l* N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left(v \cdot \color{blue}{\left(v \cdot {v}^{2}\right)}\right)\right)\right) \]

    16. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left(v \cdot \left(v \cdot \left(v \cdot \color{blue}{v}\right)\right)\right)\right)\right) \]

    17. cube-mult N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \left(v \cdot {v}^{\color{blue}{3}}\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{\left({v}^{3}\right)}\right)\right)\right) \]

    19. cube-mult N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \left(v \cdot \color{blue}{\left(v \cdot v\right)}\right)\right)\right)\right) \]

    20. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \left(v \cdot {v}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \color{blue}{\left({v}^{2}\right)}\right)\right)\right)\right) \]

    22. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \left(v \cdot \color{blue}{v}\right)\right)\right)\right)\right) \]

    23. *-lowering-*.f32 71.4%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right), v\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right)\right)\right) \]

11. Simplified 71.4%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{v}}{\color{blue}{\frac{0.016666666666666666 + \left(\left(0.3333333333333333 + 2 \cdot \left(v \cdot v\right)\right) \cdot v\right) \cdot v}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)}}} \]

12. Taylor expanded in sinTheta_O around 0

    \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot {v}^{3}\right)}{\frac{1}{60} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)}} \]
13. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot {v}^{3}\right)\right), \color{blue}{\left(\frac{1}{60} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)}\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i \cdot {v}^{3}\right)\right), \left(\color{blue}{\frac{1}{60}} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \left({v}^{3}\right)\right)\right), \left(\frac{1}{60} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)\right) \]

    4. cube-mult N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \left(v \cdot \left(v \cdot v\right)\right)\right)\right), \left(\frac{1}{60} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \left(v \cdot {v}^{2}\right)\right)\right), \left(\frac{1}{60} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \left({v}^{2}\right)\right)\right)\right), \left(\frac{1}{60} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \left(v \cdot v\right)\right)\right)\right), \left(\frac{1}{60} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \left(\frac{1}{60} + {v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \color{blue}{\left({v}^{2} \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)}\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \left(\left(v \cdot v\right) \cdot \left(\color{blue}{\frac{1}{3}} + 2 \cdot {v}^{2}\right)\right)\right)\right) \]

    11. associate-*l* N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \left(v \cdot \color{blue}{\left(v \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)}\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(v, \color{blue}{\left(v \cdot \left(\frac{1}{3} + 2 \cdot {v}^{2}\right)\right)}\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{1}{3} + 2 \cdot {v}^{2}\right)}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\frac{1}{3}, \color{blue}{\left(2 \cdot {v}^{2}\right)}\right)\right)\right)\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\frac{1}{3}, \left({v}^{2} \cdot \color{blue}{2}\right)\right)\right)\right)\right)\right) \]

    16. unpow2 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\frac{1}{3}, \left(\left(v \cdot v\right) \cdot 2\right)\right)\right)\right)\right)\right) \]

    17. associate-*l* N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\frac{1}{3}, \left(v \cdot \color{blue}{\left(v \cdot 2\right)}\right)\right)\right)\right)\right)\right) \]

    18. *-commutative N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\frac{1}{3}, \left(v \cdot \left(2 \cdot \color{blue}{v}\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \color{blue}{\left(2 \cdot v\right)}\right)\right)\right)\right)\right)\right) \]

    20. *-commutative N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \left(v \cdot \color{blue}{2}\right)\right)\right)\right)\right)\right)\right) \]

    21. *-lowering-*.f32 71.3%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{+.f32}\left(\frac{1}{60}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right)\right)\right)\right)\right) \]

14. Simplified 71.3%

    \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}{0.016666666666666666 + v \cdot \left(v \cdot \left(0.3333333333333333 + v \cdot \left(v \cdot 2\right)\right)\right)}} \]
15. Add Preprocessing

Alternative 10: 70.2% accurate, 10.5× speedup

Math

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

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
   (/
    (* -0.5 (* cosTheta_i_m cosTheta_O_m))
    (*
     v
     (-
      -1.0
      (/
       (+ 0.16666666666666666 (/ 0.008333333333333333 (* v 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 * ((-0.5f * (cosTheta_i_m * cosTheta_O_m)) / (v * (-1.0f - ((0.16666666666666666f + (0.008333333333333333f / (v * 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 * (((-0.5e0) * (costheta_i_m * costheta_o_m)) / (v * ((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * 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(-0.5) * Float32(cosTheta_i_m * cosTheta_O_m)) / Float32(v * Float32(Float32(-1.0) - Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * 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 * ((single(-0.5) * (cosTheta_i_m * cosTheta_O_m)) / (v * (single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v))))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

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

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

   8. associate-/l* N/A

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

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

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

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

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

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

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

   12. associate-/l* N/A

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

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

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

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

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

   15. *-lowering-*.f32 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

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

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

8. Simplified 71.4%

   \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{v}}{\left(\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot 2\right) \cdot v} \]

9. Taylor expanded in sinTheta_O around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v \cdot \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)}} \]

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

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

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

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

    4. *-commutative N/A

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

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

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

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

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

    7. sub-neg N/A

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

    8. metadata-eval N/A

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

    9. +-commutative N/A

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

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

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

    11. mul-1-neg N/A

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

    12. distribute-neg-frac2 N/A

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

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

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

11. Simplified 71.3%

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

12. Final simplification 71.3%

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

Alternative 11: 64.1% accurate, 10.5× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(cosTheta\_O\_m \cdot \frac{\frac{cosTheta\_i\_m - \frac{cosTheta\_i\_m}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}}\right)\right) \end{array} \]

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_O_m
    (/
     (/ (- cosTheta_i_m (/ cosTheta_i_m (/ v (* sinTheta_i sinTheta_O)))) v)
     (+ 2.0 (/ 0.3333333333333333 (* 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 - (cosTheta_i_m / (v / (sinTheta_i * sinTheta_O)))) / v) / (2.0f + (0.3333333333333333f / (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 - (costheta_i_m / (v / (sintheta_i * sintheta_o)))) / v) / (2.0e0 + (0.3333333333333333e0 / (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(cosTheta_O_m * Float32(Float32(Float32(cosTheta_i_m - Float32(cosTheta_i_m / Float32(v / Float32(sinTheta_i * sinTheta_O)))) / v) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / 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 - (cosTheta_i_m / (v / (sinTheta_i * sinTheta_O)))) / v) / (single(2.0) + (single(0.3333333333333333) / (v * v))))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

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

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

   8. associate-/l* N/A

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

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

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

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

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

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

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

   12. associate-/l* N/A

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

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

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

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

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

   15. *-lowering-*.f32 98.5%

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

5. Simplified 98.5%

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

   1. associate-/l* N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

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

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

   10. *-commutative N/A

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

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

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

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

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. associate-/r/ N/A

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

7. Applied egg-rr 98.9%

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

8. Taylor expanded in v around inf

   \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right), v\right), \color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right), v\right), \mathsf{+.f32}\left(2, \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right), v\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3} \cdot 1}{\color{blue}{{v}^{2}}}\right)\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right), v\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3}}{{\color{blue}{v}}^{2}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right), v\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \color{blue}{\left({v}^{2}\right)}\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right), v\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \left(v \cdot \color{blue}{v}\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 65.2%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right), v\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right)\right)\right) \]

10. Simplified 65.2%

    \[\leadsto cosTheta\_O \cdot \frac{\frac{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]
11. Add Preprocessing

Alternative 12: 58.9% 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 98.5%

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

3. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \color{blue}{\frac{1}{2}} \]

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

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

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

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

   4. *-lowering-*.f32 59.3%

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

5. Simplified 59.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot 0.5} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v} \]

   3. clear-num N/A

      \[\leadsto \frac{1}{2} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]

   4. un-div-inv N/A

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]

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

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

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

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

   7. *-lowering-*.f32 60.0%

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

7. Applied egg-rr 60.0%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]
8. Add Preprocessing

Alternative 13: 58.4% 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 98.5%

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

3. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \color{blue}{\frac{1}{2}} \]

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

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

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

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

   4. *-lowering-*.f32 59.3%

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

5. Simplified 59.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot 0.5} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot \frac{1}{2} \]

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

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

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

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

   7. /-lowering-/.f32 59.3%

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

7. Applied egg-rr 59.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}} \cdot 0.5} \]

8. Final simplification 59.3%

   \[\leadsto 0.5 \cdot \frac{cosTheta\_i}{\frac{v}{cosTheta\_O}} \]
9. Add Preprocessing

Alternative 14: 58.4% 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 98.5%

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

3. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \color{blue}{\frac{1}{2}} \]

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

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

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

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

   4. *-lowering-*.f32 59.3%

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

5. Simplified 59.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot 0.5} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

   5. /-lowering-/.f32 59.3%

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

7. Applied egg-rr 59.3%

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

8. Final simplification 59.3%

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

Alternative 15: 58.4% accurate, 31.4× speedup

Math

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

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_i_m (* cosTheta_O_m (/ 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_i_m * (cosTheta_O_m * (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_i_m * (costheta_o_m * (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_i_m * Float32(cosTheta_O_m * 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_i_m * (cosTheta_O_m * (single(0.5) / v))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \color{blue}{\frac{1}{2}} \]

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

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

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

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

   4. *-lowering-*.f32 59.3%

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

5. Simplified 59.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot 0.5} \]
6. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

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

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

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

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

   6. /-lowering-/.f32 59.3%

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

7. Applied egg-rr 59.3%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. /-lowering-/.f32 59.3%

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

9. Applied egg-rr 59.3%

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

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

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

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

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

    5. /-lowering-/.f32 59.3%

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

11. Applied egg-rr 59.3%

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

12. Final simplification 59.3%

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

Reproduce

herbie shell --seed 2024191 
(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)))