HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 22.7s

Alternatives: 26

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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 98.9

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

6. Applied rewrites 98.9%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

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)) / (0.5f / v)) * expf(((sinTheta_i * sinTheta_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 * costheta_o) / (v * ((sinh((1.0e0 / v)) / (0.5e0 / v)) * exp(((sintheta_i * sintheta_o) / v))))
end function

Julia

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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. associate-/l/ N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 98.7

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

6. Applied rewrites 98.7%

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

7. Final simplification 98.7%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

6. Applied rewrites 98.7%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (/
   (/ cosTheta_i v)
   (* (exp (/ (* sinTheta_i sinTheta_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_O * ((cosTheta_i / v) / (expf(((sinTheta_i * sinTheta_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_o * ((costheta_i / v) / (exp(((sintheta_i * sintheta_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(cosTheta_O * Float32(Float32(cosTheta_i / v) / Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / 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_O * ((cosTheta_i / v) / (exp(((sinTheta_i * sinTheta_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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

5. Final simplification 98.6%

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

Alternative 6: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 98.9

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

6. Applied rewrites 98.9%

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

8. Applied rewrites 98.6%

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

9. Final simplification 98.6%

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

Alternative 7: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i
  (/
   cosTheta_O
   (*
    (* v (exp (* sinTheta_i (/ sinTheta_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 * expf((sinTheta_i * (sinTheta_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 * exp((sintheta_i * (sintheta_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(cosTheta_i * Float32(cosTheta_O / Float32(Float32(v * exp(Float32(sinTheta_i * Float32(sinTheta_O / 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 * exp((sinTheta_i * (sinTheta_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

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

6. Applied rewrites 98.7%

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

7. Final simplification 98.7%

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

Alternative 8: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. associate-/r/ N/A

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

   6. lift-/.f32 N/A

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

   7. remove-double-div N/A

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

   8. lift-/.f32 N/A

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

   9. frac-2neg N/A

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

   10. metadata-eval N/A

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

   11. associate-/l/ N/A

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

   12. lower-/.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac2 N/A

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

   15. div-inv N/A

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

   16. lift-/.f32 N/A

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

   17. frac-2neg N/A

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

   18. distribute-frac-neg2 N/A

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

   19. distribute-neg-frac N/A

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

   20. lift-/.f32 N/A

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

9. Applied rewrites 98.6%

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

Alternative 9: 98.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i cosTheta_O)
  (/
   (/ (fma (- sinTheta_O) (/ sinTheta_i v) 1.0) 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) * ((fmaf(-sinTheta_O, (sinTheta_i / v), 1.0f) / v) / (sinhf((1.0f / v)) * (v * 2.0f)));
}

Julia

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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f32 N/A

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

   8. lower-/.f32 98.6

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

7. Applied rewrites 98.6%

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

Alternative 10: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i / (v / costheta_o)) / (sinh((1.0e0 / v)) / (0.5e0 / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. associate-/r/ N/A

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

   6. lift-/.f32 N/A

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

   7. lower-/.f32 98.5

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

9. Applied rewrites 98.5%

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

Alternative 11: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i * costheta_o) * ((0.5e0 / (v * sinh((1.0e0 / v)))) * (1.0e0 / v))
end function

Julia

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

MATLAB

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

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.5

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

7. Applied rewrites 98.5%

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

8. Final simplification 98.5%

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

Alternative 12: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 13: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 14: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. times-frac N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-/.f32 N/A

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 98.3

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

7. Applied rewrites 98.3%

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

Alternative 15: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / (v * sinh((single(1.0) / v)))) * ((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 sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. associate-/r/ N/A

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

   9. lift-/.f32 N/A

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

   10. clear-num N/A

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

   11. lift-/.f32 N/A

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

   12. lower-*.f32 98.5

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

7. Applied rewrites 98.2%

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

Alternative 16: 70.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}{v \cdot \frac{\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ cosTheta_i (/ v cosTheta_O))
  (*
   v
   (/
    (-
     (/ (+ 0.3333333333333333 (/ 0.016666666666666666 (* 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 (cosTheta_i / (v / cosTheta_O)) / (v * ((((0.3333333333333333f + (0.016666666666666666f / (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 = (costheta_i / (v / costheta_o)) / (v * ((((0.3333333333333333e0 + (0.016666666666666666e0 / (v * v))) / (v * v)) - (-2.0e0)) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i / Float32(v / cosTheta_O)) / Float32(v * Float32(Float32(Float32(Float32(Float32(0.3333333333333333) + Float32(Float32(0.016666666666666666) / Float32(v * v))) / Float32(v * v)) - Float32(-2.0)) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / (v / cosTheta_O)) / (v * ((((single(0.3333333333333333) + (single(0.016666666666666666) / (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. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. neg-mul-1 N/A

      \[\leadsto \frac{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}{\frac{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2}{\color{blue}{-1 \cdot v}} \cdot v} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}{\color{blue}{\frac{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2}{-1 \cdot v}} \cdot v} \]

10. Applied rewrites 71.2%

    \[\leadsto \frac{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}{\color{blue}{\frac{\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{-v \cdot v} + -2}{-v}} \cdot v} \]

11. Final simplification 71.2%

    \[\leadsto \frac{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}{v \cdot \frac{\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2}{v}} \]
12. Add Preprocessing

Alternative 17: 70.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) / v) / (v * (2.0f * ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) + 1.0f) / v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_i * costheta_o) / v) / (v * (2.0e0 * ((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) + 1.0e0) / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-/.f32 N/A

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

8. Applied rewrites 71.1%

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

9. Final simplification 71.1%

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

Alternative 18: 64.0% accurate, 5.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (/ cosTheta_i (/ v 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 (cosTheta_i / (v / 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 = (costheta_i / (v / 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(cosTheta_i / Float32(v / 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 = (cosTheta_i / (v / 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. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around inf

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

   1. lower-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 65.2

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

10. Applied rewrites 65.2%

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

Alternative 19: 64.0% 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. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in v around inf

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

   1. lower-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 65.1

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

8. Applied rewrites 65.1%

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

9. Final simplification 65.1%

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

Alternative 20: 58.8% 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

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 59.6

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

5. Applied rewrites 59.6%

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

7. Applied rewrites 60.1%

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

8. Final simplification 60.1%

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

Alternative 21: 58.8% 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

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 59.6

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

5. Applied rewrites 59.6%

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

7. Applied rewrites 60.1%

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

Alternative 22: 58.3% accurate, 12.4× speedup

Math

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

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(Float32(cosTheta_i * 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

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 59.6

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

5. Applied rewrites 59.6%

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

6. Final simplification 59.6%

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

Alternative 23: 58.3% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 59.6

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

5. Applied rewrites 59.6%

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

7. Applied rewrites 59.6%

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

8. Final simplification 59.6%

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

Alternative 24: 58.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 0.5e0 * ((costheta_i * costheta_o) / v)
end function

Julia

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

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 59.6

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

5. Applied rewrites 59.6%

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

7. Applied rewrites 59.6%

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

8. Final simplification 59.6%

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

Alternative 25: 58.2% accurate, 12.4× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 59.6

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

5. Applied rewrites 59.6%

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

7. Applied rewrites 59.6%

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

8. Final simplification 59.6%

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

Alternative 26: 58.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 59.6

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

5. Applied rewrites 59.6%

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

7. Applied rewrites 59.6%

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

9. Applied rewrites 59.6%

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

10. Final simplification 59.6%

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

Reproduce

herbie shell --seed 2024219 
(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)))