HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 15.8s

Alternatives: 17

Speedup: 1.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i (- sinTheta_O)) v))
   (* (/ 1.0 v) (* cosTheta_i cosTheta_O)))
  (/ (* (sinh (/ 1.0 v)) 4.0) (/ 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)) * ((1.0f / v) * (cosTheta_i * cosTheta_O))) / ((sinhf((1.0f / v)) * 4.0f) / (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)) * ((1.0e0 / v) * (costheta_i * costheta_o))) / ((sinh((1.0e0 / v)) * 4.0e0) / (2.0e0 / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(sinTheta_i * Float32(-sinTheta_O)) / v)) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O))) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(4.0)) / Float32(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)) * ((single(1.0) / v) * (cosTheta_i * cosTheta_O))) / ((sinh((single(1.0) / v)) * single(4.0)) / (single(2.0) / v));
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. un-div-inv N/A

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

   8. clear-num N/A

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

   9. associate-*r/ N/A

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

   10. remove-double-div N/A

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

   11. lift-/.f32 N/A

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

   12. frac-times N/A

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

   13. *-rgt-identity N/A

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

   14. count-2 N/A

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

   15. lower-/.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f32 N/A

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

   20. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{4}}{\frac{1}{v} + \frac{1}{v}}} \]

   21. count-2 N/A

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

4. Applied egg-rr 98.7%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ 1.0 v) (* cosTheta_i cosTheta_O))
  (/ (* (sinh (/ 1.0 v)) 4.0) (/ 2.0 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)) / ((sinhf((1.0f / v)) * 4.0f) / (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 = ((1.0e0 / v) * (costheta_i * costheta_o)) / ((sinh((1.0e0 / v)) * 4.0e0) / (2.0e0 / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(4.0)) / Float32(Float32(2.0) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((single(1.0) / v) * (cosTheta_i * cosTheta_O)) / ((sinh((single(1.0) / v)) * single(4.0)) / (single(2.0) / v));
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. un-div-inv N/A

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

   8. clear-num N/A

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

   9. associate-*r/ N/A

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

   10. remove-double-div N/A

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

   11. lift-/.f32 N/A

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

   12. frac-times N/A

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

   13. *-rgt-identity N/A

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

   14. count-2 N/A

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

   15. lower-/.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f32 N/A

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

   20. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{4}}{\frac{1}{v} + \frac{1}{v}}} \]

   21. count-2 N/A

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

4. Applied egg-rr 98.7%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.8%

   \[\leadsto \frac{\color{blue}{1} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

10. Final simplification 98.8%

    \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]
11. Add Preprocessing

Alternative 3: 98.4% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* (/ 1.0 v) (* cosTheta_i cosTheta_O)) (/ v (/ 0.5 (sinh (/ 1.0 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)) / (v / (0.5f / sinhf((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 = ((1.0e0 / v) * (costheta_i * costheta_o)) / (v / (0.5e0 / sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. un-div-inv N/A

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

   8. clear-num N/A

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

   9. associate-*r/ N/A

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

   10. remove-double-div N/A

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

   11. lift-/.f32 N/A

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

   12. frac-times N/A

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

   13. *-rgt-identity N/A

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

   14. count-2 N/A

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

   15. lower-/.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f32 N/A

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

   20. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{4}}{\frac{1}{v} + \frac{1}{v}}} \]

   21. count-2 N/A

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

4. Applied egg-rr 98.7%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.8%

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

    1. lift-/.f32 N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 4}{\frac{2}{v}}} \]

    2. lift-sinh.f32 N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 4}{\frac{2}{v}}} \]

    3. lift-*.f32 N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 4}}{\frac{2}{v}}} \]

    4. associate-/r/ N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{2} \cdot v}} \]

    5. lift-*.f32 N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 4}}{2} \cdot v} \]

    6. associate-/l* N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \frac{4}{2}\right)} \cdot v} \]

    7. metadata-eval N/A

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

    8. lift-*.f32 N/A

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

    9. *-commutative N/A

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

    10. /-rgt-identity N/A

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

    11. clear-num N/A

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

    12. un-div-inv N/A

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

    13. lower-/.f32 N/A

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

    14. lift-*.f32 N/A

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

    15. *-commutative N/A

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

    16. associate-/r* N/A

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

    17. metadata-eval N/A

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

    18. lower-/.f32 98.7

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

11. Applied egg-rr 98.7%

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

12. Final simplification 98.7%

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

Alternative 4: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. un-div-inv N/A

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

   8. clear-num N/A

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

   9. associate-*r/ N/A

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

   10. remove-double-div N/A

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

   11. lift-/.f32 N/A

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

   12. frac-times N/A

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

   13. *-rgt-identity N/A

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

   14. count-2 N/A

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

   15. lower-/.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f32 N/A

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

   20. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{4}}{\frac{1}{v} + \frac{1}{v}}} \]

   21. count-2 N/A

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

4. Applied egg-rr 98.7%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.8%

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

    1. lift-*.f32 N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    2. associate-/r/ N/A

       \[\leadsto \frac{1 \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    3. lift-/.f32 N/A

       \[\leadsto \frac{1 \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    4. div-inv N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    5. lift-/.f32 N/A

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    6. associate-/r/ N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    7. lift-/.f32 N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    8. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    9. lift-/.f32 N/A

       \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 4}{\frac{2}{v}}} \]

    10. lift-sinh.f32 N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 4}{\frac{2}{v}}} \]

    11. lift-*.f32 N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 4}}{\frac{2}{v}}} \]

    12. lift-/.f32 N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\color{blue}{\frac{2}{v}}}} \]

    13. div-inv N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 4\right) \cdot \frac{1}{\frac{2}{v}}}} \]

    14. div-inv N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}}} \]

    15. lift-/.f32 N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\color{blue}{\frac{2}{v}}}} \]

    16. associate-/r/ N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{2} \cdot v}} \]

11. Applied egg-rr 98.7%

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

Alternative 5: 98.3% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* cosTheta_i cosTheta_O) (* v (* (sinh (/ 1.0 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 * (sinhf((1.0f / 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 * (sinh((1.0e0 / v)) * (v * 2.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. un-div-inv N/A

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

   8. clear-num N/A

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

   9. associate-*r/ N/A

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

   10. remove-double-div N/A

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

   11. lift-/.f32 N/A

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

   12. frac-times N/A

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

   13. *-rgt-identity N/A

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

   14. count-2 N/A

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

   15. lower-/.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f32 N/A

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

   20. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{4}}{\frac{1}{v} + \frac{1}{v}}} \]

   21. count-2 N/A

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

4. Applied egg-rr 98.7%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.8%

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

    1. lift-*.f32 N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    2. associate-/r/ N/A

       \[\leadsto \frac{1 \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    3. lift-/.f32 N/A

       \[\leadsto \frac{1 \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    4. div-inv N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    5. lift-/.f32 N/A

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    6. associate-/r/ N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    7. lift-/.f32 N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    8. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

    9. lift-/.f32 N/A

       \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 4}{\frac{2}{v}}} \]

    10. lift-sinh.f32 N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 4}{\frac{2}{v}}} \]

    11. lift-*.f32 N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 4}}{\frac{2}{v}}} \]

    12. lift-/.f32 N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\color{blue}{\frac{2}{v}}}} \]

    13. div-inv N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 4\right) \cdot \frac{1}{\frac{2}{v}}}} \]

    14. div-inv N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}}} \]

    15. lift-/.f32 N/A

        \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}}} \]

    16. associate-/l* N/A

        \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\frac{1}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}}} \]

    17. clear-num N/A

        \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{1}{\frac{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}}{\frac{1}{v}}}} \]

11. Applied egg-rr 98.5%

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

Alternative 6: 70.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{4 \cdot \frac{1 + \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{v}}{\frac{2}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ 1.0 v) (* cosTheta_i cosTheta_O))
  (/
   (*
    4.0
    (/
     (+
      1.0
      (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v)))
     v))
   (/ 2.0 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)) / ((4.0f * ((1.0f + ((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v))) / 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 = ((1.0e0 / v) * (costheta_i * costheta_o)) / ((4.0e0 * ((1.0e0 + ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v))) / v)) / (2.0e0 / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O)) / Float32(Float32(Float32(4.0) * Float32(Float32(Float32(1.0) + Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v))) / v)) / Float32(Float32(2.0) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((single(1.0) / v) * (cosTheta_i * cosTheta_O)) / ((single(4.0) * ((single(1.0) + ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v))) / v)) / (single(2.0) / v));
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. un-div-inv N/A

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

   8. clear-num N/A

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

   9. associate-*r/ N/A

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

   10. remove-double-div N/A

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

   11. lift-/.f32 N/A

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

   12. frac-times N/A

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

   13. *-rgt-identity N/A

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

   14. count-2 N/A

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

   15. lower-/.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f32 N/A

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

   20. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{4}}{\frac{1}{v} + \frac{1}{v}}} \]

   21. count-2 N/A

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

4. Applied egg-rr 98.7%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.8%

   \[\leadsto \frac{\color{blue}{1} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

10. Taylor expanded in v around -inf

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

    1. associate-*r/ N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)}{v}} \cdot 4}{\frac{2}{v}}} \]

    2. mul-1-neg N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)\right)}}{v} \cdot 4}{\frac{2}{v}}} \]

    3. sub-neg N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{v} \cdot 4}{\frac{2}{v}}} \]

    4. metadata-eval N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + \color{blue}{-1}\right)\right)}{v} \cdot 4}{\frac{2}{v}}} \]

    5. distribute-neg-in N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)}}{v} \cdot 4}{\frac{2}{v}}} \]

    6. mul-1-neg N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)}{v} \cdot 4}{\frac{2}{v}}} \]

    7. remove-double-neg N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{\color{blue}{\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}} + \left(\mathsf{neg}\left(-1\right)\right)}{v} \cdot 4}{\frac{2}{v}}} \]

    8. metadata-eval N/A

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

    9. lower-/.f32 N/A

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

12. Simplified 70.0%

    \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\color{blue}{\frac{1 + \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{v}} \cdot 4}{\frac{2}{v}}} \]

13. Final simplification 70.0%

    \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{4 \cdot \frac{1 + \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{v}}{\frac{2}{v}}} \]
14. Add Preprocessing

Alternative 7: 63.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{4 \cdot \frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}}{\frac{2}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ 1.0 v) (* cosTheta_i cosTheta_O))
  (/ (* 4.0 (/ (+ 1.0 (/ 0.16666666666666666 (* v v))) v)) (/ 2.0 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)) / ((4.0f * ((1.0f + (0.16666666666666666f / (v * v))) / 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 = ((1.0e0 / v) * (costheta_i * costheta_o)) / ((4.0e0 * ((1.0e0 + (0.16666666666666666e0 / (v * v))) / v)) / (2.0e0 / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O)) / Float32(Float32(Float32(4.0) * Float32(Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) / Float32(v * v))) / v)) / Float32(Float32(2.0) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((single(1.0) / v) * (cosTheta_i * cosTheta_O)) / ((single(4.0) * ((single(1.0) + (single(0.16666666666666666) / (v * v))) / v)) / (single(2.0) / v));
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. un-div-inv N/A

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

   8. clear-num N/A

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

   9. associate-*r/ N/A

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

   10. remove-double-div N/A

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

   11. lift-/.f32 N/A

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

   12. frac-times N/A

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

   13. *-rgt-identity N/A

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

   14. count-2 N/A

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

   15. lower-/.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f32 N/A

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

   20. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{4}}{\frac{1}{v} + \frac{1}{v}}} \]

   21. count-2 N/A

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

4. Applied egg-rr 98.7%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.8%

   \[\leadsto \frac{\color{blue}{1} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

10. Taylor expanded in v around inf

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

    1. lower-/.f32 N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\color{blue}{\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}} \cdot 4}{\frac{2}{v}}} \]

    2. lower-+.f32 N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{\color{blue}{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}}{v} \cdot 4}{\frac{2}{v}}} \]

    3. associate-*r/ N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{1 + \color{blue}{\frac{\frac{1}{6} \cdot 1}{{v}^{2}}}}{v} \cdot 4}{\frac{2}{v}}} \]

    4. metadata-eval N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{1 + \frac{\color{blue}{\frac{1}{6}}}{{v}^{2}}}{v} \cdot 4}{\frac{2}{v}}} \]

    5. lower-/.f32 N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{1 + \color{blue}{\frac{\frac{1}{6}}{{v}^{2}}}}{v} \cdot 4}{\frac{2}{v}}} \]

    6. unpow2 N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{1 + \frac{\frac{1}{6}}{\color{blue}{v \cdot v}}}{v} \cdot 4}{\frac{2}{v}}} \]

    7. lower-*.f32 64.0

       \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{1 + \frac{0.16666666666666666}{\color{blue}{v \cdot v}}}{v} \cdot 4}{\frac{2}{v}}} \]

12. Simplified 64.0%

    \[\leadsto \frac{1 \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\color{blue}{\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}} \cdot 4}{\frac{2}{v}}} \]

13. Final simplification 64.0%

    \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{4 \cdot \frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}}{\frac{2}{v}}} \]
14. Add Preprocessing

Alternative 8: 63.8% accurate, 5.2× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ 1.0 v) (* cosTheta_i cosTheta_O))
  (+ 2.0 (/ 0.3333333333333333 (* 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)) / (2.0f + (0.3333333333333333f / (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)) / (2.0e0 + (0.3333333333333333e0 / (v * v)))
end function

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. un-div-inv N/A

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

   8. clear-num N/A

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

   9. associate-*r/ N/A

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

   10. remove-double-div N/A

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

   11. lift-/.f32 N/A

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

   12. frac-times N/A

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

   13. *-rgt-identity N/A

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

   14. count-2 N/A

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

   15. lower-/.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f32 N/A

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

   20. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{4}}{\frac{1}{v} + \frac{1}{v}}} \]

   21. count-2 N/A

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

4. Applied egg-rr 98.7%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.8%

   \[\leadsto \frac{\color{blue}{1} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 4}{\frac{2}{v}}} \]

10. Taylor expanded in v around inf

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

    1. lower-+.f32 N/A

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

    2. associate-*r/ N/A

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

    3. metadata-eval N/A

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

    4. lower-/.f32 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f32 64.0

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

12. Simplified 64.0%

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

13. Final simplification 64.0%

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

Alternative 9: 63.8% accurate, 5.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) / v) / (2.0f + (0.3333333333333333f / (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 = ((costheta_i * costheta_o) / v) / (2.0e0 + (0.3333333333333333e0 / (v * v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. remove-double-div N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-sinh.f32 N/A

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

   10. sinh-undef N/A

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

   11. flip-- N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f32 N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Simplified 98.6%

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

8. Taylor expanded in v around inf

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

   1. lower-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 64.0

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

10. Simplified 64.0%

    \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

11. Final simplification 64.0%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]
12. Add Preprocessing

Alternative 10: 58.7% accurate, 6.2× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * ((1.0f / v) / (1.0f / (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 * ((1.0e0 / v) / (1.0e0 / (costheta_i * costheta_o)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

5. Simplified 58.3%

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

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.3

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

8. Simplified 58.3%

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

   1. associate-*l/ N/A

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

   2. lift-/.f32 N/A

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

   3. lower-*.f32 58.3

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

10. Applied egg-rr 58.3%

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

    1. associate-*l/ N/A

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

    2. *-commutative N/A

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

    3. lift-*.f32 N/A

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

    4. clear-num N/A

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

    5. div-inv N/A

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

    6. associate-/r* N/A

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

    7. lift-/.f32 N/A

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

    8. lower-/.f32 N/A

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

    9. lower-/.f32 58.7

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

12. Applied egg-rr 58.7%

    \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{1}{v}}{\frac{1}{cosTheta\_i \cdot cosTheta\_O}}} \]
13. Add Preprocessing

Alternative 11: 58.6% accurate, 7.0× speedup

Math

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

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(Float32(v / cosTheta_i) / Float32(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 98.5%

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

3. Taylor expanded in v around inf

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

5. Simplified 58.3%

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

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.3

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

8. Simplified 58.3%

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

   1. associate-*l/ N/A

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

   2. lift-/.f32 N/A

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

   3. lower-*.f32 58.3

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

10. Applied egg-rr 58.3%

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

    1. associate-*l/ N/A

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

    2. associate-*r/ N/A

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

    3. lift-/.f32 N/A

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

    4. associate-*r* N/A

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

    5. *-lft-identity N/A

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

    6. metadata-eval N/A

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

    7. associate-/r/ N/A

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

    8. lift-/.f32 N/A

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

    9. lift-/.f32 N/A

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

    10. lift-/.f32 N/A

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

    11. clear-num N/A

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

    12. un-div-inv N/A

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

    13. clear-num N/A

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

    14. lower-/.f32 N/A

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

    15. lower-/.f32 N/A

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

    16. lower-/.f32 N/A

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

    17. lift-/.f32 N/A

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

    18. lift-/.f32 N/A

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

    19. associate-/r/ N/A

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

    20. metadata-eval N/A

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

    21. *-lft-identity N/A

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

    22. *-commutative N/A

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

    23. lower-*.f32 58.7

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

12. Applied egg-rr 58.7%

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta\_i}}{cosTheta\_O \cdot 0.5}}} \]
13. Add Preprocessing

Alternative 12: 58.6% 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 98.5%

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

3. Taylor expanded in v around inf

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

5. Simplified 58.3%

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

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.3

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

8. Simplified 58.3%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lift-/.f32 N/A

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

   7. div-inv N/A

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

   8. clear-num N/A

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

   9. lift-/.f32 N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. div-inv N/A

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

   13. clear-num N/A

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

   14. lower-/.f32 N/A

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

   15. clear-num N/A

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

   16. div-inv N/A

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

   19. lift-/.f32 N/A

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

   20. clear-num N/A

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

   21. div-inv N/A

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

   22. lift-/.f32 N/A

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

   23. associate-/r* N/A

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

   24. lift-*.f32 N/A

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

   25. un-div-inv N/A

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

   26. clear-num N/A

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

   27. associate-*r/ N/A

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

10. Applied egg-rr 58.7%

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

Alternative 13: 58.5% 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 98.5%

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

3. Taylor expanded in v around inf

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

5. Simplified 58.3%

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

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.3

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

8. Simplified 58.3%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lift-/.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. associate-/r* N/A

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

   10. lift-*.f32 N/A

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

   11. lower-/.f32 58.5

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 58.5

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

10. Applied egg-rr 58.5%

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

Alternative 14: 58.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

5. Simplified 58.3%

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

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.3

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

8. Simplified 58.3%

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

   1. associate-*l/ N/A

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

   2. lift-/.f32 N/A

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

   3. lower-*.f32 58.3

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

10. Applied egg-rr 58.3%

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

    1. associate-*l/ N/A

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

    2. associate-*r/ N/A

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

    3. lift-/.f32 N/A

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

    4. associate-*r* N/A

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

    5. *-lft-identity N/A

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

    6. metadata-eval N/A

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

    7. associate-/r/ N/A

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

    8. lift-/.f32 N/A

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

    9. lift-/.f32 N/A

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

    10. lift-/.f32 N/A

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

    11. associate-*r/ N/A

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

    12. lower-/.f32 N/A

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

    13. *-commutative N/A

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

    14. lower-*.f32 N/A

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

    15. lift-/.f32 N/A

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

    16. lift-/.f32 N/A

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

    17. associate-/r/ N/A

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

    18. metadata-eval N/A

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

    19. *-lft-identity N/A

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

    20. *-commutative N/A

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

    21. lower-*.f32 58.4

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

12. Applied egg-rr 58.4%

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

Alternative 15: 58.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

5. Simplified 58.3%

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

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.3

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

8. Simplified 58.3%

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

   1. lift-*.f32 N/A

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

   2. div-inv N/A

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

   3. lift-/.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-/.f32 N/A

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

   7. un-div-inv N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 58.4

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 58.4

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

10. Applied egg-rr 58.4%

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

11. Final simplification 58.4%

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

Alternative 16: 58.0% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

5. Simplified 58.3%

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

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.3

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

8. Simplified 58.3%

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

   1. lift-*.f32 N/A

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

   2. div-inv N/A

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

   3. lift-/.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-/.f32 N/A

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

   7. un-div-inv N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 58.3

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

10. Applied egg-rr 58.3%

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

11. Final simplification 58.3%

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

Alternative 17: 58.1% accurate, 12.4× speedup

Math

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

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(0.5) * Float32(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 98.5%

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

3. Taylor expanded in v around inf

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

5. Simplified 58.3%

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

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.3

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

8. Simplified 58.3%

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

   1. associate-*l/ N/A

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

   2. lift-/.f32 N/A

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

   3. lower-*.f32 58.3

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

10. Applied egg-rr 58.3%

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

11. Final simplification 58.3%

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

Reproduce

herbie shell --seed 2024215 
(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)))