HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 19.7s

Alternatives: 22

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\frac{1}{v} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (* (* (/ 1.0 v) cosTheta_O) cosTheta_i)
   (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (/ (sinh (/ 1.0 v)) (/ 0.5 v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((((1.0f / v) * cosTheta_O) * cosTheta_i) * expf(((sinTheta_O * sinTheta_i) / -v))) / (sinhf((1.0f / v)) / (0.5f / 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 = ((((1.0e0 / v) * costheta_o) * costheta_i) * exp(((sintheta_o * sintheta_i) / -v))) / (sinh((1.0e0 / v)) / (0.5e0 / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(Float32(Float32(1.0) / v) * cosTheta_O) * cosTheta_i) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(sinh(Float32(Float32(1.0) / v)) / Float32(Float32(0.5) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((((single(1.0) / v) * cosTheta_O) * cosTheta_i) * exp(((sinTheta_O * sinTheta_i) / -v))) / (sinh((single(1.0) / v)) / (single(0.5) / v));
end

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. clear-num N/A

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

   6. frac-2neg N/A

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

   7. associate-/r/ N/A

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

   8. metadata-eval N/A

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

   9. frac-2neg N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*r* N/A

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

   12. distribute-lft-neg-in N/A

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

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

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

   14. associate-*r* N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-neg.f32 99.1

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

6. Applied rewrites 99.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. div-inv N/A

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

   9. lift-/.f32 N/A

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

   10. neg-mul-1 N/A

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

   11. distribute-frac-neg N/A

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

   12. lower-neg.f32 N/A

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

   13. clear-num N/A

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

   14. div-inv N/A

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

   15. lift-/.f32 N/A

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

   16. remove-double-div N/A

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

   17. *-commutative N/A

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

   18. metadata-eval N/A

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

   19. div-inv N/A

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

   20. clear-num N/A

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

   21. lower-/.f32 99.1

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

8. Applied rewrites 99.1%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. neg-mul-1 N/A

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

   6. sinh-neg N/A

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

   7. lift-neg.f32 N/A

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

   8. frac-2neg N/A

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

   9. lower-/.f32 N/A

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

   10. lower-sinh.f32 N/A

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

   11. lower-/.f32 99.1

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

10. Applied rewrites 99.1%

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

11. Final simplification 99.1%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((((1.0f / v) * cosTheta_i) * cosTheta_O) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((2.0f * sinhf((1.0f / v))) * 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 = ((((1.0e0 / v) * costheta_i) * costheta_o) * exp(((sintheta_o * sintheta_i) / -v))) / ((2.0e0 * sinh((1.0e0 / v))) * v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.7

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 98.9

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

6. Applied rewrites 98.9%

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

7. Final simplification 98.9%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * (((-1.0f / v) / v) / (expf((-1.0f / v)) - expf((1.0f / 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 = (costheta_i * costheta_o) * ((((-1.0e0) / v) / v) / (exp(((-1.0e0) / v)) - exp((1.0e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lift-/.f32 N/A

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

   9. associate-/l/ N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/r/ N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f32 N/A

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. rec-exp N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower--.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-exp.f32 N/A

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

   14. lower-/.f32 98.7

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{cosTheta\_O}{v}}{v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_O / v) / v) * cosTheta_i) / (expf((1.0f / v)) - expf((-1.0f / 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 = (((costheta_o / v) / v) * costheta_i) / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 58.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 58.6

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

8. Applied rewrites 58.6%

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

9. Taylor expanded in sinTheta_i around 0

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

    1. times-frac N/A

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

    2. associate-*r/ N/A

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

    3. lower-/.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. unpow2 N/A

       \[\leadsto \frac{\frac{cosTheta\_O}{\color{blue}{v \cdot v}} \cdot cosTheta\_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

    6. associate-/r* N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{cosTheta\_O}{v}}{v}} \cdot cosTheta\_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

    7. lower-/.f32 N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{cosTheta\_O}{v}}{v}} \cdot cosTheta\_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

    8. lower-/.f32 N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta\_O}{v}}}{v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

    9. rec-exp N/A

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

    10. distribute-neg-frac N/A

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

    11. metadata-eval N/A

        \[\leadsto \frac{\frac{\frac{cosTheta\_O}{v}}{v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}} \]

    12. lower--.f32 N/A

        \[\leadsto \frac{\frac{\frac{cosTheta\_O}{v}}{v} \cdot cosTheta\_i}{\color{blue}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]

    13. lower-exp.f32 N/A

        \[\leadsto \frac{\frac{\frac{cosTheta\_O}{v}}{v} \cdot cosTheta\_i}{\color{blue}{e^{\frac{1}{v}}} - e^{\frac{-1}{v}}} \]

    14. lower-/.f32 N/A

        \[\leadsto \frac{\frac{\frac{cosTheta\_O}{v}}{v} \cdot cosTheta\_i}{e^{\color{blue}{\frac{1}{v}}} - e^{\frac{-1}{v}}} \]

    15. lower-exp.f32 N/A

        \[\leadsto \frac{\frac{\frac{cosTheta\_O}{v}}{v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - \color{blue}{e^{\frac{-1}{v}}}} \]

    16. lower-/.f32 98.6

        \[\leadsto \frac{\frac{\frac{cosTheta\_O}{v}}{v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}} \]

11. Applied rewrites 98.6%

    \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_O}{v}}{v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]
12. Add Preprocessing

Alternative 5: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) / (v * v)) / (expf((1.0f / v)) - expf((-1.0f / 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 = ((costheta_i * costheta_o) / (v * v)) / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + 2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + 2} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + 2} \]

   6. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + 2} \]

   7. lower-*.f32 64.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + 2} \]

7. Applied rewrites 64.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{0.3333333333333333}{v \cdot v} + 2}} \]

8. Taylor expanded in v around inf

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

10. Applied rewrites 52.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, 0.16666666666666666\right)}{v \cdot v}, -0.5, cosTheta\_O \cdot \mathsf{fma}\left(cosTheta\_i, 0.5, \left(cosTheta\_i \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) \cdot -0.5\right)\right)}{v}} \]

11. Taylor expanded in sinTheta_i around 0

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

    1. associate-*r/ N/A

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

    2. associate-/r* N/A

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

    3. lower-/.f32 N/A

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

    4. lower-/.f32 N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. lower-*.f32 N/A

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

    8. mul-1-neg N/A

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

    9. lower-neg.f32 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f32 N/A

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

    12. rec-exp N/A

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

    13. distribute-neg-frac N/A

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

    14. metadata-eval N/A

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

    15. lower--.f32 N/A

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

    16. lower-exp.f32 N/A

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

    17. lower-/.f32 N/A

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

    18. lower-exp.f32 N/A

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

    19. lower-/.f32 98.5

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

13. Applied rewrites 98.5%

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

14. Final simplification 98.5%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]
15. Add Preprocessing

Alternative 6: 71.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{\frac{-1 - \frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v}}{v}}{v} \cdot 2}{\frac{-1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (* (/ cosTheta_O v) cosTheta_i) (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (/
   (*
    (/
     (-
      -1.0
      (/ (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) v) v))
     v)
    2.0)
   (/ -1.0 v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_O / v) * cosTheta_i) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((((-1.0f - (((0.16666666666666666f + (0.008333333333333333f / (v * v))) / v) / v)) / v) * 2.0f) / (-1.0f / 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 = (((costheta_o / v) * costheta_i) * exp(((sintheta_o * sintheta_i) / -v))) / (((((-1.0e0) - (((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / v) / v)) / v) * 2.0e0) / ((-1.0e0) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_O / v) * cosTheta_i) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(Float32(Float32(-1.0) - Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / v) / v)) / v) * Float32(2.0)) / Float32(Float32(-1.0) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_O / v) * cosTheta_i) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((((single(-1.0) - (((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / v) / v)) / v) * single(2.0)) / (single(-1.0) / v));
end

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 98.9

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

9. Applied rewrites 71.6%

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

10. Final simplification 71.6%

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

Alternative 7: 71.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(\frac{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v}}{v} - -1}{v} \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (* (* cosTheta_i cosTheta_O) (/ 1.0 v))
   (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (*
   (*
    (/
     (-
      (/ (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) v) v)
      -1.0)
     v)
    2.0)
   v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) * (1.0f / v)) * expf(((sinTheta_O * sinTheta_i) / -v))) / (((((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / v) / v) - -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 = (((costheta_i * costheta_o) * (1.0e0 / v)) * exp(((sintheta_o * sintheta_i) / -v))) / (((((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / v) / v) - (-1.0e0)) / v) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(1.0) / v)) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / v) / v) - Float32(-1.0)) / v) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) * (single(1.0) / v)) * exp(((sinTheta_O * sinTheta_i) / -v))) / (((((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / v) / v) - single(-1.0)) / v) * single(2.0)) * v);
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.7

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 71.5%

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

8. Final simplification 71.5%

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

Alternative 8: 71.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(\frac{\frac{\frac{\frac{0.008333333333333333}{v \cdot v} - -0.16666666666666666}{v}}{v} - -1}{v} \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ (* cosTheta_i cosTheta_O) v) (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (*
   (*
    (/
     (-
      (/ (/ (- (/ 0.008333333333333333 (* v v)) -0.16666666666666666) v) v)
      -1.0)
     v)
    2.0)
   v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) / v) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((((((((0.008333333333333333f / (v * v)) - -0.16666666666666666f) / v) / v) - -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 = (((costheta_i * costheta_o) / v) * exp(((sintheta_o * sintheta_i) / -v))) / ((((((((0.008333333333333333e0 / (v * v)) - (-0.16666666666666666e0)) / v) / v) - (-1.0e0)) / v) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.008333333333333333) / Float32(v * v)) - Float32(-0.16666666666666666)) / v) / v) - Float32(-1.0)) / v) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) / v) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((((((((single(0.008333333333333333) / (v * v)) - single(-0.16666666666666666)) / v) / v) - single(-1.0)) / v) * single(2.0)) * v);
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around -inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft1-in N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-rgt-identity N/A

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

   10. lower-/.f32 N/A

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

5. Applied rewrites 71.5%

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

6. Final simplification 71.5%

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

Alternative 9: 65.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(\frac{\frac{1}{\frac{v}{\frac{0.16666666666666666}{v}}} - -1}{v} \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ (* cosTheta_i cosTheta_O) v) (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (* (* (/ (- (/ 1.0 (/ v (/ 0.16666666666666666 v))) -1.0) v) 2.0) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) / v) * expf(((sinTheta_O * sinTheta_i) / -v))) / (((((1.0f / (v / (0.16666666666666666f / v))) - -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 = (((costheta_i * costheta_o) / v) * exp(((sintheta_o * sintheta_i) / -v))) / (((((1.0e0 / (v / (0.16666666666666666e0 / v))) - (-1.0e0)) / v) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(Float32(Float32(Float32(1.0) / Float32(v / Float32(Float32(0.16666666666666666) / v))) - Float32(-1.0)) / v) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) / v) * exp(((sinTheta_O * sinTheta_i) / -v))) / (((((single(1.0) / (v / (single(0.16666666666666666) / v))) - single(-1.0)) / v) * single(2.0)) * v);
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 64.9

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

5. Applied rewrites 64.9%

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

7. Applied rewrites 64.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\frac{1}{\frac{v}{\frac{0.16666666666666666}{v}}} - -1}{v} \cdot 2\right) \cdot v} \]

8. Final simplification 64.9%

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

Alternative 10: 65.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{2 - \frac{-1}{v} \cdot \frac{0.3333333333333333}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ (* cosTheta_i cosTheta_O) v) (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (- 2.0 (* (/ -1.0 v) (/ 0.3333333333333333 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) / v) * expf(((sinTheta_O * sinTheta_i) / -v))) / (2.0f - ((-1.0f / v) * (0.3333333333333333f / 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 = (((costheta_i * costheta_o) / v) * exp(((sintheta_o * sintheta_i) / -v))) / (2.0e0 - (((-1.0e0) / v) * (0.3333333333333333e0 / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(2.0) - Float32(Float32(Float32(-1.0) / v) * Float32(Float32(0.3333333333333333) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) / v) * exp(((sinTheta_O * sinTheta_i) / -v))) / (single(2.0) - ((single(-1.0) / v) * (single(0.3333333333333333) / v)));
end

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + 2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + 2} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + 2} \]

   6. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + 2} \]

   7. lower-*.f32 64.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + 2} \]

7. Applied rewrites 64.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{0.3333333333333333}{v \cdot v} + 2}} \]
8. Step-by-step derivation

9. Applied rewrites 64.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{1}{v} \cdot \frac{0.3333333333333333}{v} + 2} \]

10. Final simplification 64.9%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{2 - \frac{-1}{v} \cdot \frac{0.3333333333333333}{v}} \]
11. Add Preprocessing

Alternative 11: 65.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{\frac{0.3333333333333333}{v}}{v} + 2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ (* cosTheta_i cosTheta_O) v) (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (+ (/ (/ 0.3333333333333333 v) v) 2.0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) / v) * expf(((sinTheta_O * sinTheta_i) / -v))) / (((0.3333333333333333f / v) / v) + 2.0f);
}

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 = (((costheta_i * costheta_o) / v) * exp(((sintheta_o * sintheta_i) / -v))) / (((0.3333333333333333e0 / v) / v) + 2.0e0)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(Float32(0.3333333333333333) / v) / v) + Float32(2.0)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) / v) * exp(((sinTheta_O * sinTheta_i) / -v))) / (((single(0.3333333333333333) / v) / v) + single(2.0));
end

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + 2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + 2} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + 2} \]

   6. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + 2} \]

   7. lower-*.f32 64.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + 2} \]

7. Applied rewrites 64.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{0.3333333333333333}{v \cdot v} + 2}} \]
8. Step-by-step derivation

9. Applied rewrites 64.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{0.3333333333333333}{v}}{v} + 2} \]

10. Final simplification 64.9%

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

Alternative 12: 65.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{0.3333333333333333}{v \cdot v} + 2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (* (* cosTheta_i cosTheta_O) (/ 1.0 v))
   (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (+ (/ 0.3333333333333333 (* v v)) 2.0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) * (1.0f / v)) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((0.3333333333333333f / (v * v)) + 2.0f);
}

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 = (((costheta_i * costheta_o) * (1.0e0 / v)) * exp(((sintheta_o * sintheta_i) / -v))) / ((0.3333333333333333e0 / (v * v)) + 2.0e0)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(1.0) / v)) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(2.0)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) * (single(1.0) / v)) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((single(0.3333333333333333) / (v * v)) + single(2.0));
end

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + 2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + 2} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + 2} \]

   6. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + 2} \]

   7. lower-*.f32 64.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + 2} \]

7. Applied rewrites 64.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{0.3333333333333333}{v \cdot v} + 2}} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   2. clear-num N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   3. associate-/r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{1}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   5. lower-*.f32 64.9

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

9. Applied rewrites 64.9%

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

10. Final simplification 64.9%

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

Alternative 13: 65.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{0.3333333333333333}{v \cdot v} + 2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ (* cosTheta_i cosTheta_O) v) (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (+ (/ 0.3333333333333333 (* v v)) 2.0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) / v) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((0.3333333333333333f / (v * v)) + 2.0f);
}

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 = (((costheta_i * costheta_o) / v) * exp(((sintheta_o * sintheta_i) / -v))) / ((0.3333333333333333e0 / (v * v)) + 2.0e0)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(2.0)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) / v) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((single(0.3333333333333333) / (v * v)) + single(2.0));
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + 2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + 2} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + 2} \]

   6. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + 2} \]

   7. lower-*.f32 64.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + 2} \]

5. Applied rewrites 64.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{0.3333333333333333}{v \cdot v} + 2}} \]

6. Final simplification 64.9%

   \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{0.3333333333333333}{v \cdot v} + 2} \]
7. Add Preprocessing

Alternative 14: 65.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{0.3333333333333333}{v \cdot v} + 2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (* (/ cosTheta_O v) cosTheta_i) (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (+ (/ 0.3333333333333333 (* v v)) 2.0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_O / v) * cosTheta_i) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((0.3333333333333333f / (v * v)) + 2.0f);
}

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 = (((costheta_o / v) * costheta_i) * exp(((sintheta_o * sintheta_i) / -v))) / ((0.3333333333333333e0 / (v * v)) + 2.0e0)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_O / v) * cosTheta_i) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(2.0)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_O / v) * cosTheta_i) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((single(0.3333333333333333) / (v * v)) + single(2.0));
end

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + 2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + 2} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + 2} \]

   6. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + 2} \]

   7. lower-*.f32 64.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + 2} \]

7. Applied rewrites 64.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{0.3333333333333333}{v \cdot v} + 2}} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v}}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   3. associate-/l* N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   4. div-inv N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{1}{v}\right)}\right)}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot \left(cosTheta\_O \cdot \color{blue}{\frac{1}{v}}\right)\right)}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   6. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot \color{blue}{\left(\frac{1}{v} \cdot cosTheta\_O\right)}\right)}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   7. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(\frac{1}{v} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right)}}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   8. lower-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(\frac{1}{v} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right)}}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   9. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\left(cosTheta\_O \cdot \frac{1}{v}\right)} \cdot cosTheta\_i\right)}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   10. lift-/.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(cosTheta\_O \cdot \color{blue}{\frac{1}{v}}\right) \cdot cosTheta\_i\right)}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   11. div-inv N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{cosTheta\_O}{v}} \cdot cosTheta\_i\right)}{\frac{\frac{1}{3}}{v \cdot v} + 2} \]

   12. lower-/.f32 64.9

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

9. Applied rewrites 64.9%

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

10. Final simplification 64.9%

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

Alternative 15: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(\frac{1}{cosTheta\_i \cdot cosTheta\_O}\right)}^{-1} \cdot \frac{1}{v}\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* (pow (/ 1.0 (* cosTheta_i cosTheta_O)) -1.0) (/ 1.0 v)) 0.5))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (powf((1.0f / (cosTheta_i * cosTheta_O)), -1.0f) * (1.0f / v)) * 0.5f;
}

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 = (((1.0e0 / (costheta_i * costheta_o)) ** (-1.0e0)) * (1.0e0 / v)) * 0.5e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32((Float32(Float32(1.0) / Float32(cosTheta_i * cosTheta_O)) ^ Float32(-1.0)) * Float32(Float32(1.0) / v)) * Float32(0.5))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((single(1.0) / (cosTheta_i * cosTheta_O)) ^ single(-1.0)) * (single(1.0) / v)) * single(0.5);
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 58.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 58.6

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

8. Applied rewrites 58.6%

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

10. Applied rewrites 59.4%

    \[\leadsto \left(\frac{1}{v} \cdot {\left(\frac{1}{cosTheta\_i \cdot cosTheta\_O}\right)}^{-1}\right) \cdot 0.5 \]

11. Final simplification 59.4%

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

Alternative 16: 60.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{2}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / (2.0f / ((cosTheta_i * cosTheta_O) / 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 = 1.0e0 / (2.0e0 / ((costheta_i * costheta_o) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 58.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 58.6

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

8. Applied rewrites 58.6%

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

10. Applied rewrites 59.2%

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

Alternative 17: 60.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{v}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot 0.5}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (/ v (* (* cosTheta_i cosTheta_O) 0.5))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / (v / ((cosTheta_i * cosTheta_O) * 0.5f));
}

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 = 1.0e0 / (v / ((costheta_i * costheta_o) * 0.5e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / Float32(v / Float32(Float32(cosTheta_i * cosTheta_O) * Float32(0.5))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / (v / ((cosTheta_i * cosTheta_O) * single(0.5)));
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 58.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 58.6

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

8. Applied rewrites 58.6%

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

10. Applied rewrites 59.2%

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

11. Final simplification 59.2%

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

Alternative 18: 60.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 0.5 (/ v (* cosTheta_i cosTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f / (v / (cosTheta_i * cosTheta_O));
}

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 = 0.5e0 / (v / (costheta_i * costheta_o))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) / Float32(v / Float32(cosTheta_i * cosTheta_O)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / (v / (cosTheta_i * cosTheta_O));
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 58.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 58.6

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

8. Applied rewrites 58.6%

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

10. Applied rewrites 59.1%

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

Alternative 19: 59.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (* cosTheta_i cosTheta_O)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * (cosTheta_i * cosTheta_O);
}

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 = (0.5e0 / v) * (costheta_i * costheta_o)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * Float32(cosTheta_i * cosTheta_O))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (cosTheta_i * cosTheta_O);
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lift-/.f32 N/A

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

   9. associate-/l/ N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/r/ N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f32 N/A

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in v around inf

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

   1. lower-/.f32 58.6

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

9. Applied rewrites 58.6%

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

10. Final simplification 58.6%

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

Alternative 20: 59.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ (* cosTheta_i cosTheta_O) v) 0.5))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) / v) * 0.5f;
}

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 = ((costheta_i * costheta_o) / v) * 0.5e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) * Float32(0.5))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / v) * single(0.5);
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 58.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 58.6

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

8. Applied rewrites 58.6%

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

Alternative 21: 59.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot cosTheta\_i\right) \cdot \frac{cosTheta\_O}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* 0.5 cosTheta_i) (/ cosTheta_O v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f * cosTheta_i) * (cosTheta_O / 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 = (0.5e0 * costheta_i) * (costheta_o / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) * cosTheta_i) * Float32(cosTheta_O / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) * cosTheta_i) * (cosTheta_O / v);
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 58.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 58.6

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

8. Applied rewrites 58.6%

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

10. Applied rewrites 58.6%

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

11. Final simplification 58.6%

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

Alternative 22: 59.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* (/ cosTheta_i v) cosTheta_O) 0.5))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i / v) * cosTheta_O) * 0.5f;
}

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 = ((costheta_i / v) * costheta_o) * 0.5e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i / v) * cosTheta_O) * Float32(0.5))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i / v) * cosTheta_O) * single(0.5);
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 58.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 58.6

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

8. Applied rewrites 58.6%

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

10. Applied rewrites 58.6%

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

Reproduce

herbie shell --seed 2024268 
(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)))