HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.9%

Time: 22.2s

Alternatives: 19

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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   (* (/ 1.0 v) cosTheta_O_m)
   (*
    cosTheta_i
    (/
     (/ (/ 0.5 v) (sinh (/ 1.0 v)))
     (exp (/ (* sinTheta_i sinTheta_O) v)))))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (((1.0e0 / v) * costheta_o_m) * (costheta_i * (((0.5e0 / v) / sinh((1.0e0 / v))) / exp(((sintheta_i * sintheta_o) / v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

   1. div-inv N/A

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

   2. associate-/l* N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. associate-/l/ N/A

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

   10. associate-/r* N/A

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

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

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

6. Applied egg-rr 98.8%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

   4. /-lowering-/.f32 98.9%

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

8. Applied egg-rr 98.9%

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

Alternative 2: 98.7% accurate, 1.6× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (+
    (*
     cosTheta_O_m
     (- cosTheta_i (/ cosTheta_i (/ v (* sinTheta_i sinTheta_O)))))
    (/
     (*
      (*
       (* cosTheta_O_m (* sinTheta_i sinTheta_i))
       (* (* cosTheta_i (* sinTheta_O sinTheta_O)) -0.5))
      (/ -1.0 v))
     v))
   (* v (/ (sinh (/ 1.0 v)) (/ 0.5 v))))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * (costheta_i - (costheta_i / (v / (sintheta_i * sintheta_o))))) + ((((costheta_o_m * (sintheta_i * sintheta_i)) * ((costheta_i * (sintheta_o * sintheta_o)) * (-0.5e0))) * ((-1.0e0) / v)) / v)) / (v * (sinh((1.0e0 / v)) / (0.5e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

9. Applied egg-rr 98.9%

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

10. Final simplification 98.9%

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

Alternative 3: 98.7% accurate, 1.8× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   (*
    cosTheta_O_m
    (- cosTheta_i (/ cosTheta_i (/ (/ v sinTheta_O) sinTheta_i))))
   (/ (/ 1.0 v) (/ (sinh (/ 1.0 v)) (/ 0.5 v))))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * (costheta_i - (costheta_i / ((v / sintheta_o) / sintheta_i)))) * ((1.0e0 / v) / (sinh((1.0e0 / v)) / (0.5e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.6%

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

7. Simplified 98.6%

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

   1. div-inv N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

   6. *-commutative N/A

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

   7. associate-*l/ N/A

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

   8. associate-/r/ N/A

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

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

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

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

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

9. Applied egg-rr 98.8%

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

Alternative 4: 98.8% accurate, 1.8× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   (*
    cosTheta_O_m
    (- cosTheta_i (/ cosTheta_i (/ (/ v sinTheta_O) sinTheta_i))))
   (* (/ 1.0 v) (/ (/ 0.5 v) (sinh (/ 1.0 v)))))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * (costheta_i - (costheta_i / ((v / sintheta_o) / sintheta_i)))) * ((1.0e0 / v) * ((0.5e0 / v) / sinh((1.0e0 / v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.6%

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

7. Simplified 98.6%

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

   1. div-inv N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. associate-*l/ N/A

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

   9. associate-/r/ N/A

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

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

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

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

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

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

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

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

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

9. Applied egg-rr 98.8%

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

Alternative 5: 98.7% accurate, 1.8× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   (/
    (- cosTheta_i (/ cosTheta_i (/ (/ v sinTheta_O) sinTheta_i)))
    (/ (sinh (/ 1.0 v)) (/ 0.5 v)))
   (/ cosTheta_O_m v))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 - (costheta_i / ((v / sintheta_o) / sintheta_i))) / (sinh((1.0e0 / v)) / (0.5e0 / v))) * (costheta_o_m / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.6%

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

7. Simplified 98.6%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

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

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

9. Applied egg-rr 98.7%

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

Alternative 6: 98.7% accurate, 1.8× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_O_m
   (*
    (/ (/ 0.5 v) (sinh (/ 1.0 v)))
    (/ (- cosTheta_i (/ cosTheta_i (/ (/ v sinTheta_O) sinTheta_i))) v)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * (((0.5e0 / v) / sinh((1.0e0 / v))) * ((costheta_i - (costheta_i / ((v / sintheta_o) / sintheta_i))) / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.6%

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

7. Simplified 98.6%

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

   1. div-inv N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

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

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

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

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

9. Applied egg-rr 98.7%

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

10. Final simplification 98.7%

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

Alternative 7: 98.5% accurate, 1.9× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/ (* cosTheta_O_m cosTheta_i) (* v (/ (sinh (/ 1.0 v)) (/ 0.5 v))))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * costheta_i) / (v * (sinh((1.0e0 / v)) / (0.5e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

9. Applied egg-rr 98.9%

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

10. Taylor expanded in v around inf

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

    1. mul-1-neg N/A

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

    2. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\left(0 - cosTheta\_O \cdot cosTheta\_i\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)}, \mathsf{neg.f32}\left(v\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)}, \mathsf{neg.f32}\left(v\right)\right)\right) \]

    4. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}\right), \mathsf{neg.f32}\left(v\right)\right)\right) \]

12. Simplified 98.6%

    \[\leadsto \frac{\color{blue}{0 - cosTheta\_O \cdot cosTheta\_i}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}} \cdot \left(-v\right)} \]

13. Final simplification 98.6%

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

Alternative 8: 98.3% accurate, 1.9× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/ (/ (* cosTheta_O_m cosTheta_i) v) (* (sinh (/ 1.0 v)) (* v 2.0)))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (((cosTheta_O_m * cosTheta_i) / v) / (sinhf((1.0f / v)) * (v * 2.0f)));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * costheta_i) / v) / (sinh((1.0e0 / v)) * (v * 2.0e0)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(Float32(cosTheta_O_m * cosTheta_i) / v) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(2.0)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (((cosTheta_O_m * cosTheta_i) / v) / (sinh((single(1.0) / v)) * (v * single(2.0))));
end

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

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

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

   2. *-lowering-*.f32 98.4%

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

7. Simplified 98.4%

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

Alternative 9: 74.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \frac{1}{\frac{\frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v} \cdot \left(v \cdot \frac{v}{0.5}\right)}{cosTheta\_O\_m \cdot \left(cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right) + \frac{\left(\left(cosTheta\_O\_m \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right) \cdot \left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot -0.5\right)\right) \cdot \frac{-1}{v}}{v}}} \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   1.0
   (/
    (*
     (/
      (+
       (/
        (+ 0.008333333333333333 (/ 0.0001984126984126984 (* v v)))
        (* (* v v) (* v v)))
       (+ 1.0 (/ 0.16666666666666666 (* v v))))
      v)
     (* v (/ v 0.5)))
    (+
     (*
      cosTheta_O_m
      (- cosTheta_i (/ cosTheta_i (/ v (* sinTheta_i sinTheta_O)))))
     (/
      (*
       (*
        (* cosTheta_O_m (* sinTheta_i sinTheta_i))
        (* (* cosTheta_i (* sinTheta_O sinTheta_O)) -0.5))
       (/ -1.0 v))
      v))))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (1.0f / ((((((0.008333333333333333f + (0.0001984126984126984f / (v * v))) / ((v * v) * (v * v))) + (1.0f + (0.16666666666666666f / (v * v)))) / v) * (v * (v / 0.5f))) / ((cosTheta_O_m * (cosTheta_i - (cosTheta_i / (v / (sinTheta_i * sinTheta_O))))) + ((((cosTheta_O_m * (sinTheta_i * sinTheta_i)) * ((cosTheta_i * (sinTheta_O * sinTheta_O)) * -0.5f)) * (-1.0f / v)) / v))));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (1.0e0 / ((((((0.008333333333333333e0 + (0.0001984126984126984e0 / (v * v))) / ((v * v) * (v * v))) + (1.0e0 + (0.16666666666666666e0 / (v * v)))) / v) * (v * (v / 0.5e0))) / ((costheta_o_m * (costheta_i - (costheta_i / (v / (sintheta_i * sintheta_o))))) + ((((costheta_o_m * (sintheta_i * sintheta_i)) * ((costheta_i * (sintheta_o * sintheta_o)) * (-0.5e0))) * ((-1.0e0) / v)) / v))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.008333333333333333) + Float32(Float32(0.0001984126984126984) / Float32(v * v))) / Float32(Float32(v * v) * Float32(v * v))) + Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) / Float32(v * v)))) / v) * Float32(v * Float32(v / Float32(0.5)))) / Float32(Float32(cosTheta_O_m * Float32(cosTheta_i - Float32(cosTheta_i / Float32(v / Float32(sinTheta_i * sinTheta_O))))) + Float32(Float32(Float32(Float32(cosTheta_O_m * Float32(sinTheta_i * sinTheta_i)) * Float32(Float32(cosTheta_i * Float32(sinTheta_O * sinTheta_O)) * Float32(-0.5))) * Float32(Float32(-1.0) / v)) / v)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (single(1.0) / ((((((single(0.008333333333333333) + (single(0.0001984126984126984) / (v * v))) / ((v * v) * (v * v))) + (single(1.0) + (single(0.16666666666666666) / (v * v)))) / v) * (v * (v / single(0.5)))) / ((cosTheta_O_m * (cosTheta_i - (cosTheta_i / (v / (sinTheta_i * sinTheta_O))))) + ((((cosTheta_O_m * (sinTheta_i * sinTheta_i)) * ((cosTheta_i * (sinTheta_O * sinTheta_O)) * single(-0.5))) * (single(-1.0) / v)) / v))));
end

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

9. Applied egg-rr 95.0%

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

10. Taylor expanded in v around -inf

    \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)}, \mathsf{*.f32}\left(\mathsf{/.f32}\left(v, \frac{1}{2}\right), v\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \frac{-1}{2}\right)\right)\right), v\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(v, \frac{1}{2}\right), v\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\color{blue}{cosTheta\_O}, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \frac{-1}{2}\right)\right)\right), v\right)\right)\right)\right) \]

    2. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{\mathsf{neg}\left(v\right)}\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(v, \frac{1}{2}\right), v\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\color{blue}{cosTheta\_O}, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \frac{-1}{2}\right)\right)\right), v\right)\right)\right)\right) \]

    3. mul-1-neg N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{-1 \cdot v}\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(v, \frac{1}{2}\right), v\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \frac{-1}{2}\right)\right)\right), v\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right), \left(-1 \cdot v\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(v, \frac{1}{2}\right), v\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\color{blue}{cosTheta\_O}, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \frac{-1}{2}\right)\right)\right), v\right)\right)\right)\right) \]

12. Simplified 76.9%

    \[\leadsto \frac{1}{\frac{\color{blue}{\frac{-\left(\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)\right)}{-v}} \cdot \left(\frac{v}{0.5} \cdot v\right)}{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right) + \frac{\frac{-1}{v} \cdot \left(\left(cosTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right) \cdot \left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot -0.5\right)\right)}{v}}} \]

13. Final simplification 76.9%

    \[\leadsto \frac{1}{\frac{\frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v} \cdot \left(v \cdot \frac{v}{0.5}\right)}{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right) + \frac{\left(\left(cosTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right) \cdot \left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot -0.5\right)\right) \cdot \frac{-1}{v}}{v}}} \]
14. Add Preprocessing

Alternative 10: 74.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \frac{1}{\frac{v \cdot \left(\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}\right)}{cosTheta\_O\_m \cdot \left(cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right) + \frac{\left(\left(cosTheta\_O\_m \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right) \cdot \left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot -0.5\right)\right) \cdot \frac{-1}{v}}{v}}} \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   1.0
   (/
    (*
     v
     (+
      (+ 2.0 (/ 0.3333333333333333 (* v v)))
      (/
       (+ 0.016666666666666666 (/ 0.0003968253968253968 (* v v)))
       (* (* v v) (* v v)))))
    (+
     (*
      cosTheta_O_m
      (- cosTheta_i (/ cosTheta_i (/ v (* sinTheta_i sinTheta_O)))))
     (/
      (*
       (*
        (* cosTheta_O_m (* sinTheta_i sinTheta_i))
        (* (* cosTheta_i (* sinTheta_O sinTheta_O)) -0.5))
       (/ -1.0 v))
      v))))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (1.0f / ((v * ((2.0f + (0.3333333333333333f / (v * v))) + ((0.016666666666666666f + (0.0003968253968253968f / (v * v))) / ((v * v) * (v * v))))) / ((cosTheta_O_m * (cosTheta_i - (cosTheta_i / (v / (sinTheta_i * sinTheta_O))))) + ((((cosTheta_O_m * (sinTheta_i * sinTheta_i)) * ((cosTheta_i * (sinTheta_O * sinTheta_O)) * -0.5f)) * (-1.0f / v)) / v))));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (1.0e0 / ((v * ((2.0e0 + (0.3333333333333333e0 / (v * v))) + ((0.016666666666666666e0 + (0.0003968253968253968e0 / (v * v))) / ((v * v) * (v * v))))) / ((costheta_o_m * (costheta_i - (costheta_i / (v / (sintheta_i * sintheta_o))))) + ((((costheta_o_m * (sintheta_i * sintheta_i)) * ((costheta_i * (sintheta_o * sintheta_o)) * (-0.5e0))) * ((-1.0e0) / v)) / v))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(1.0) / Float32(Float32(v * Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))) + Float32(Float32(Float32(0.016666666666666666) + Float32(Float32(0.0003968253968253968) / Float32(v * v))) / Float32(Float32(v * v) * Float32(v * v))))) / Float32(Float32(cosTheta_O_m * Float32(cosTheta_i - Float32(cosTheta_i / Float32(v / Float32(sinTheta_i * sinTheta_O))))) + Float32(Float32(Float32(Float32(cosTheta_O_m * Float32(sinTheta_i * sinTheta_i)) * Float32(Float32(cosTheta_i * Float32(sinTheta_O * sinTheta_O)) * Float32(-0.5))) * Float32(Float32(-1.0) / v)) / v)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (single(1.0) / ((v * ((single(2.0) + (single(0.3333333333333333) / (v * v))) + ((single(0.016666666666666666) + (single(0.0003968253968253968) / (v * v))) / ((v * v) * (v * v))))) / ((cosTheta_O_m * (cosTheta_i - (cosTheta_i / (v / (sinTheta_i * sinTheta_O))))) + ((((cosTheta_O_m * (sinTheta_i * sinTheta_i)) * ((cosTheta_i * (sinTheta_O * sinTheta_O)) * single(-0.5))) * (single(-1.0) / v)) / v))));
end

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

9. Applied egg-rr 95.0%

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

10. Taylor expanded in v around -inf

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

    1. mul-1-neg N/A

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

    2. *-commutative N/A

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

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

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

    4. mul-1-neg N/A

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

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

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

12. Simplified 76.9%

    \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{-\left(v \cdot v\right) \cdot \left(v \cdot v\right)} - \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)\right) \cdot \left(-v\right)}}{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right) + \frac{\frac{-1}{v} \cdot \left(\left(cosTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right) \cdot \left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot -0.5\right)\right)}{v}}} \]

13. Final simplification 76.9%

    \[\leadsto \frac{1}{\frac{v \cdot \left(\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}\right)}{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right) + \frac{\left(\left(cosTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right) \cdot \left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot -0.5\right)\right) \cdot \frac{-1}{v}}{v}}} \]
14. Add Preprocessing

Alternative 11: 70.2% accurate, 4.3× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   (/ (* v 0.5) v)
   (/
    (/
     (+
      (*
       cosTheta_O_m
       (- cosTheta_i (/ cosTheta_i (/ (/ v sinTheta_O) sinTheta_i))))
      (/
       (*
        (/ 0.5 v)
        (*
         (* sinTheta_i sinTheta_i)
         (* cosTheta_O_m (* sinTheta_O (* cosTheta_i sinTheta_O)))))
       v))
     v)
    (-
     (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
     -1.0)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (((v * 0.5e0) / v) * ((((costheta_o_m * (costheta_i - (costheta_i / ((v / sintheta_o) / sintheta_i)))) + (((0.5e0 / v) * ((sintheta_i * sintheta_i) * (costheta_o_m * (sintheta_o * (costheta_i * sintheta_o))))) / v)) / v) / (((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

10. Simplified 72.3%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}{\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot \left(v \cdot 2\right)} \]
11. Step-by-step derivation

    1. metadata-eval N/A

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

    2. div-inv N/A

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

    3. associate-*l/ N/A

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

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

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

12. Applied egg-rr 72.3%

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

14. Applied egg-rr 72.3%

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

15. Final simplification 72.3%

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

Alternative 12: 70.2% accurate, 4.3× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (+
    (*
     cosTheta_O_m
     (- cosTheta_i (/ cosTheta_i (/ (/ v sinTheta_O) sinTheta_i))))
    (/
     (*
      (/ 0.5 v)
      (*
       (* sinTheta_i sinTheta_i)
       (* cosTheta_O_m (* sinTheta_O (* cosTheta_i sinTheta_O)))))
     v))
   (*
    v
    (/
     (/
      (-
       -1.0
       (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v)))
      (/ -0.5 v))
     v)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * (costheta_i - (costheta_i / ((v / sintheta_o) / sintheta_i)))) + (((0.5e0 / v) * ((sintheta_i * sintheta_i) * (costheta_o_m * (sintheta_o * (costheta_i * sintheta_o))))) / v)) / (v * ((((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v))) / ((-0.5e0) / v)) / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

10. Simplified 72.3%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}{\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot \left(v \cdot 2\right)} \]
11. Step-by-step derivation

    1. metadata-eval N/A

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

    2. div-inv N/A

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

    3. associate-*l/ N/A

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

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

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

12. Applied egg-rr 72.3%

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

14. Applied egg-rr 72.3%

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

15. Final simplification 72.3%

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

Alternative 13: 70.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \frac{cosTheta\_O\_m \cdot \left(cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right) + \frac{\left(\left(cosTheta\_O\_m \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right) \cdot \left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot -0.5\right)\right) \cdot \frac{-1}{v}}{v}}{v \cdot \left(\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2\right)} \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (+
    (*
     cosTheta_O_m
     (- cosTheta_i (/ cosTheta_i (/ v (* sinTheta_i sinTheta_O)))))
    (/
     (*
      (*
       (* cosTheta_O_m (* sinTheta_i sinTheta_i))
       (* (* cosTheta_i (* sinTheta_O sinTheta_O)) -0.5))
      (/ -1.0 v))
     v))
   (*
    v
    (-
     (/ (+ 0.3333333333333333 (/ 0.016666666666666666 (* v v))) (* v v))
     -2.0)))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (((cosTheta_O_m * (cosTheta_i - (cosTheta_i / (v / (sinTheta_i * sinTheta_O))))) + ((((cosTheta_O_m * (sinTheta_i * sinTheta_i)) * ((cosTheta_i * (sinTheta_O * sinTheta_O)) * -0.5f)) * (-1.0f / v)) / v)) / (v * (((0.3333333333333333f + (0.016666666666666666f / (v * v))) / (v * v)) - -2.0f)));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * (costheta_i - (costheta_i / (v / (sintheta_i * sintheta_o))))) + ((((costheta_o_m * (sintheta_i * sintheta_i)) * ((costheta_i * (sintheta_o * sintheta_o)) * (-0.5e0))) * ((-1.0e0) / v)) / v)) / (v * (((0.3333333333333333e0 + (0.016666666666666666e0 / (v * v))) / (v * v)) - (-2.0e0))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(Float32(cosTheta_O_m * Float32(cosTheta_i - Float32(cosTheta_i / Float32(v / Float32(sinTheta_i * sinTheta_O))))) + Float32(Float32(Float32(Float32(cosTheta_O_m * Float32(sinTheta_i * sinTheta_i)) * Float32(Float32(cosTheta_i * Float32(sinTheta_O * sinTheta_O)) * Float32(-0.5))) * Float32(Float32(-1.0) / v)) / v)) / Float32(v * Float32(Float32(Float32(Float32(0.3333333333333333) + Float32(Float32(0.016666666666666666) / Float32(v * v))) / Float32(v * v)) - Float32(-2.0)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (((cosTheta_O_m * (cosTheta_i - (cosTheta_i / (v / (sinTheta_i * sinTheta_O))))) + ((((cosTheta_O_m * (sinTheta_i * sinTheta_i)) * ((cosTheta_i * (sinTheta_O * sinTheta_O)) * single(-0.5))) * (single(-1.0) / v)) / v)) / (v * (((single(0.3333333333333333) + (single(0.016666666666666666) / (v * v))) / (v * v)) - single(-2.0))));
end

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

9. Applied egg-rr 98.9%

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

10. Taylor expanded in v around inf

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

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

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

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

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

    5. mul-1-neg N/A

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

    6. distribute-neg-frac2 N/A

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

    7. mul-1-neg N/A

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

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

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

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

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

    10. associate-*r/ N/A

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

    11. metadata-eval N/A

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

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

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

    13. unpow2 N/A

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

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

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

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

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

    16. unpow2 N/A

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

    17. *-lowering-*.f32 72.3%

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

12. Simplified 72.3%

    \[\leadsto \frac{-\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right) + \frac{\frac{-1}{v} \cdot \left(\left(cosTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right) \cdot \left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot -0.5\right)\right)}{v}\right)}{\color{blue}{v \cdot \left(\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{-1 \cdot \left(v \cdot v\right)} + -2\right)}} \]

13. Final simplification 72.3%

    \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right) + \frac{\left(\left(cosTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right) \cdot \left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot -0.5\right)\right) \cdot \frac{-1}{v}}{v}}{v \cdot \left(\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2\right)} \]
14. Add Preprocessing

Alternative 14: 70.2% accurate, 6.7× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (/
    (* (* cosTheta_O_m cosTheta_i) (- 1.0 (/ (* sinTheta_i sinTheta_O) v)))
    v)
   (/
    (/
     (-
      (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
      -1.0)
     (/ 0.5 v))
    v))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * costheta_i) * (1.0e0 - ((sintheta_i * sintheta_o) / v))) / v) / (((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0)) / (0.5e0 / v)) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

10. Simplified 72.3%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}{\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot \left(v \cdot 2\right)} \]
11. Step-by-step derivation

    1. metadata-eval N/A

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

    2. div-inv N/A

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

    3. associate-*l/ N/A

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

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

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

12. Applied egg-rr 72.3%

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

13. Taylor expanded in v around inf

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

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

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

15. Simplified 72.3%

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

16. Final simplification 72.3%

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

Alternative 15: 70.2% accurate, 8.8× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (/ (* cosTheta_O_m cosTheta_i) v)
   (/
    (/
     (-
      (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
      -1.0)
     (/ 0.5 v))
    v))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * costheta_i) / v) / (((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0)) / (0.5e0 / v)) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

10. Simplified 72.3%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}{\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot \left(v \cdot 2\right)} \]
11. Step-by-step derivation

    1. metadata-eval N/A

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

    2. div-inv N/A

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

    3. associate-*l/ N/A

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

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

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

12. Applied egg-rr 72.3%

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

13. Taylor expanded in sinTheta_O around 0

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

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

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

    2. *-lowering-*.f32 72.3%

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

15. Simplified 72.3%

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

16. Final simplification 72.3%

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

Alternative 16: 64.0% accurate, 10.5× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (/
    (*
     cosTheta_O_m
     (- cosTheta_i (/ (* cosTheta_i (* sinTheta_i sinTheta_O)) v)))
    v)
   (+ 2.0 (/ 0.3333333333333333 (* v v))))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (((cosTheta_O_m * (cosTheta_i - ((cosTheta_i * (sinTheta_i * sinTheta_O)) / v))) / v) / (2.0f + (0.3333333333333333f / (v * v))));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * (costheta_i - ((costheta_i * (sintheta_i * sintheta_o)) / v))) / v) / (2.0e0 + (0.3333333333333333e0 / (v * v))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(Float32(cosTheta_O_m * Float32(cosTheta_i - Float32(Float32(cosTheta_i * Float32(sinTheta_i * sinTheta_O)) / v))) / v) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (((cosTheta_O_m * (cosTheta_i - ((cosTheta_i * (sinTheta_i * sinTheta_O)) / v))) / v) / (single(2.0) + (single(0.3333333333333333) / (v * v))));
end

Derivation

1. Initial program 98.7%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in v around inf

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

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

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

   2. associate-*r/ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3} \cdot 1}{\color{blue}{{v}^{2}}}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3}}{{\color{blue}{v}}^{2}}\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \color{blue}{\left({v}^{2}\right)}\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \left(v \cdot \color{blue}{v}\right)\right)\right)\right) \]

   6. *-lowering-*.f32 66.2%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right)\right) \]

10. Simplified 66.2%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

11. Final simplification 66.2%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right)}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]
12. Add Preprocessing

Alternative 17: 58.8% accurate, 24.4× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \frac{1}{\frac{\frac{v}{cosTheta\_O\_m \cdot cosTheta\_i}}{0.5}} \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (/ 1.0 (/ (/ v (* cosTheta_O_m cosTheta_i)) 0.5))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (1.0f / ((v / (cosTheta_O_m * cosTheta_i)) / 0.5f));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (1.0e0 / ((v / (costheta_o_m * costheta_i)) / 0.5e0))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(1.0) / Float32(Float32(v / Float32(cosTheta_O_m * cosTheta_i)) / Float32(0.5))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (single(1.0) / ((v / (cosTheta_O_m * cosTheta_i)) / single(0.5)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{v}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 60.6%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 60.6%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{v}{cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{1}{2}\right)}}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{v}{cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{1}{2}\right)}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{v}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{1}{2}}}\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}{\color{blue}{\frac{1}{2}}}\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{v}{cosTheta\_i \cdot cosTheta\_O}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, \left(cosTheta\_i \cdot cosTheta\_O\right)\right), \frac{1}{2}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \frac{1}{2}\right)\right) \]

   8. *-lowering-*.f32 61.3%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \frac{1}{2}\right)\right) \]

9. Applied egg-rr 61.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta\_O \cdot cosTheta\_i}}{0.5}}} \]
10. Add Preprocessing

Alternative 18: 58.2% accurate, 24.4× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(0.5 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O\_m \cdot cosTheta\_i\right)\right)\right) \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* 0.5 (* (/ 1.0 v) (* cosTheta_O_m cosTheta_i)))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (0.5f * ((1.0f / v) * (cosTheta_O_m * cosTheta_i)));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (0.5e0 * ((1.0e0 / v) * (costheta_o_m * costheta_i)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(0.5) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_O_m * cosTheta_i))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (single(0.5) * ((single(1.0) / v) * (cosTheta_O_m * cosTheta_i)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{v}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 60.6%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 60.6%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{2}}{v} \]

   2. associate-/l* N/A

      \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \left(\frac{\color{blue}{\frac{1}{2}}}{v}\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \left(\frac{\color{blue}{\frac{1}{2}}}{v}\right)\right) \]

   6. /-lowering-/.f32 60.6%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right)\right) \]

9. Applied egg-rr 60.6%

   \[\leadsto \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{0.5}{v}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)} \]

    2. div-inv N/A

       \[\leadsto \left(\frac{1}{2} \cdot \frac{1}{v}\right) \cdot \left(\color{blue}{cosTheta\_O} \cdot cosTheta\_i\right) \]

    3. associate-*l* N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)} \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(\frac{1}{v}\right), \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)\right) \]

    6. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, v\right), \left(\color{blue}{cosTheta\_O} \cdot cosTheta\_i\right)\right)\right) \]

    7. *-lowering-*.f32 60.6%

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{cosTheta\_i}\right)\right)\right) \]

11. Applied egg-rr 60.6%

    \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)} \]
12. Add Preprocessing

Alternative 19: 58.2% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(0.5 \cdot \frac{cosTheta\_O\_m \cdot cosTheta\_i}{v}\right) \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* 0.5 (/ (* cosTheta_O_m cosTheta_i) v))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (0.5f * ((cosTheta_O_m * cosTheta_i) / v));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (0.5e0 * ((costheta_o_m * costheta_i) / v))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(0.5) * Float32(Float32(cosTheta_O_m * cosTheta_i) / v)))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (single(0.5) * ((cosTheta_O_m * cosTheta_i) / v));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{\left(-1 \cdot \frac{-1 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right)\right)\right) + \frac{1}{2} \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right)\right)\right)}{{v}^{2}} + cosTheta\_O \cdot cosTheta\_i\right) - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) + \frac{-1}{v} \cdot \left(\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot cosTheta\_O\right)\right) \cdot \frac{-0.5}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

9. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\frac{-\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right) + \frac{\frac{-1}{v} \cdot \left(\left(cosTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right) \cdot \left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot -0.5\right)\right)}{v}\right)}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}} \cdot \left(-v\right)}} \]

10. Taylor expanded in v around inf

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
11. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]

    2. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right)\right) \]

    3. *-lowering-*.f32 60.6%

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right)\right) \]

12. Simplified 60.6%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(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)))