HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 18.5s

Alternatives: 15

Speedup: 1.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   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 98.8

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

6. Applied rewrites 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   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 98.8

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Applied rewrites 98.8%

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

10. Final simplification 98.8%

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

Alternative 3: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l/ N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.4%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f32 N/A

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

   7. div-inv N/A

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

   8. div-inv N/A

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

   9. lift-/.f32 N/A

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

   10. clear-num N/A

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

   11. associate-*r* N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*r* N/A

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

   16. lift-*.f32 N/A

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

   17. lift-*.f32 N/A

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 4: 98.5% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O (/ (* cosTheta_i (/ 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_O * ((cosTheta_i * (1.0f / 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 * (1.0e0 / 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 * Float32(Float32(1.0) / 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 * (single(1.0) / v)) / (sinh((single(1.0) / v)) * (v * single(2.0))));
end

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   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 98.8

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

6. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-approx N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-sinh.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*r* N/A

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

   9. lift-/.f32 N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

8. Applied rewrites 98.7%

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

Alternative 5: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   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 98.8

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Applied rewrites 98.8%

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

11. Applied rewrites 98.6%

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

12. Final simplification 98.6%

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

Alternative 6: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l/ N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.4%

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

8. Final simplification 98.4%

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

Alternative 7: 98.4% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i (* cosTheta_O (/ 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 * (1.0f / (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 * (1.0e0 / (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(1.0) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(2.0)))))))
end

MATLAB

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

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   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 98.8

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

6. Applied rewrites 98.8%

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

8. Applied rewrites 98.5%

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

9. Final simplification 98.5%

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

Alternative 8: 71.0% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_O 1.0)
  (/
   cosTheta_i
   (*
    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_O * 1.0f) * (cosTheta_i / (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_o * 1.0e0) * (costheta_i / (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(cosTheta_O * Float32(1.0)) * Float32(cosTheta_i / Float32(v * Float32(Float32(v * 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_O * single(1.0)) * (cosTheta_i / (v * ((v * single(2.0)) * ((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)) + single(1.0)) / v))));
end

Derivation

1. Initial program 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l/ N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 70.7%

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

11. Final simplification 70.7%

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

Alternative 9: 64.8% accurate, 4.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * 1.0f) * (cosTheta_i / (v * ((v * 2.0f) * (((0.16666666666666666f / (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_o * 1.0e0) * (costheta_i / (v * ((v * 2.0e0) * (((0.16666666666666666e0 / (v * v)) + 1.0e0) / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l/ N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 64.9

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

10. Applied rewrites 64.9%

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

11. Final simplification 64.9%

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

Alternative 10: 71.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_O \cdot 1\right) \cdot \frac{cosTheta\_i}{v \cdot \left(\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_O 1.0)
  (/
   cosTheta_i
   (*
    v
    (-
     (/ (+ 0.3333333333333333 (/ 0.016666666666666666 (* v v))) (* v v))
     -2.0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * 1.0f) * (cosTheta_i / (v * (((0.3333333333333333f + (0.016666666666666666f / (v * v))) / (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 * 1.0e0) * (costheta_i / (v * (((0.3333333333333333e0 + (0.016666666666666666e0 / (v * v))) / (v * v)) - (-2.0e0))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * Float32(1.0)) * Float32(cosTheta_i / Float32(v * Float32(Float32(Float32(Float32(0.3333333333333333) + Float32(Float32(0.016666666666666666) / Float32(v * v))) / Float32(v * v)) - Float32(-2.0)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * single(1.0)) * (cosTheta_i / (v * (((single(0.3333333333333333) + (single(0.016666666666666666) / (v * v))) / (v * v)) - single(-2.0))));
end

Derivation

1. Initial program 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l/ N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f32 N/A

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

10. Applied rewrites 70.7%

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

11. Final simplification 70.7%

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

Alternative 11: 64.8% accurate, 5.9× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l/ N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around inf

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

   1. lower-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 64.9

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

10. Applied rewrites 64.9%

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

11. Final simplification 64.9%

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

Alternative 12: 59.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-*.f32 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. lift-sinh.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-/l/ N/A

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

4. Applied rewrites 92.3%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 92.3%

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

8. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 60.1

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

10. Applied rewrites 60.1%

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

11. Final simplification 60.1%

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

Alternative 13: 59.1% 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 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-*.f32 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. lift-sinh.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-/l/ N/A

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

4. Applied rewrites 92.3%

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

5. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   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. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 59.6

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

7. Applied rewrites 59.6%

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

Alternative 14: 59.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) * 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 98.4%

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

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

Alternative 15: 59.1% 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 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-*.f32 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. lift-sinh.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-/l/ N/A

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

4. Applied rewrites 92.3%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 92.3%

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

   1. lift-approx N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-/.f32 92.1

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

9. Applied rewrites 92.1%

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

10. Taylor expanded in v around inf

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

    1. lower-*.f32 N/A

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

    2. lower-/.f32 N/A

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

    3. lower-*.f32 59.5

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

12. Applied rewrites 59.5%

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

13. Final simplification 59.5%

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

Reproduce

herbie shell --seed 2024212 
(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)))