HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 18.8s

Alternatives: 15

Speedup: 1.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh \left(\frac{-1}{v}\right) \cdot 2\\ \frac{cosTheta\_i \cdot cosTheta\_O}{\frac{\left(-2 \cdot \sinh \left(\frac{-1}{v}\right)\right) \cdot t\_0}{\frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v} \cdot t\_0} \cdot v} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (* (cosh (/ -1.0 v)) 2.0)))
   (/
    (* cosTheta_i cosTheta_O)
    (*
     (/
      (* (* -2.0 (sinh (/ -1.0 v))) t_0)
      (* (/ (exp (* (/ (- sinTheta_i) v) sinTheta_O)) v) t_0))
     v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = coshf((-1.0f / v)) * 2.0f;
	return (cosTheta_i * cosTheta_O) / ((((-2.0f * sinhf((-1.0f / v))) * t_0) / ((expf(((-sinTheta_i / v) * sinTheta_O)) / v) * t_0)) * 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
    real(4) :: t_0
    t_0 = cosh(((-1.0e0) / v)) * 2.0e0
    code = (costheta_i * costheta_o) / (((((-2.0e0) * sinh(((-1.0e0) / v))) * t_0) / ((exp(((-sintheta_i / v) * sintheta_o)) / v) * t_0)) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(cosh(Float32(Float32(-1.0) / v)) * Float32(2.0))
	return Float32(Float32(cosTheta_i * cosTheta_O) / Float32(Float32(Float32(Float32(Float32(-2.0) * sinh(Float32(Float32(-1.0) / v))) * t_0) / Float32(Float32(exp(Float32(Float32(Float32(-sinTheta_i) / v) * sinTheta_O)) / v) * t_0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cosh((single(-1.0) / v)) * single(2.0);
	tmp = (cosTheta_i * cosTheta_O) / ((((single(-2.0) * sinh((single(-1.0) / v))) * t_0) / ((exp(((-sinTheta_i / v) * sinTheta_O)) / v) * t_0)) * v);
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. associate-/l/ N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

6. Applied rewrites 98.9%

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

7. Final simplification 98.9%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-/.f32 N/A

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

   14. div-inv N/A

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

   15. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. div-inv N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lower-/.f32 N/A

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

8. Applied rewrites 98.8%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lift-/.f32 N/A

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

   7. clear-num N/A

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

   8. lift-/.f32 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

10. Applied rewrites 98.8%

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

11. Final simplification 98.8%

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

Alternative 3: 98.7% accurate, 1.6× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-/.f32 N/A

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

   14. div-inv N/A

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

   15. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Final simplification 98.8%

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

Alternative 4: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-/.f32 N/A

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

   14. div-inv N/A

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

   15. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 5: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.6

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

5. Applied rewrites 98.6%

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

Alternative 6: 64.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \cdot \frac{1}{2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, \left(sinTheta\_i \cdot sinTheta\_O\right) \cdot -2\right)}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* (/ cosTheta_O v) cosTheta_i)
  (/
   1.0
   (-
    2.0
    (/
     (fma
      (/
       (fma
        (* (* (* sinTheta_O sinTheta_O) sinTheta_i) sinTheta_i)
        -0.5
        -0.16666666666666666)
       v)
      2.0
      (* (* sinTheta_i sinTheta_O) -2.0))
     v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O / v) * cosTheta_i) * (1.0f / (2.0f - (fmaf((fmaf((((sinTheta_O * sinTheta_O) * sinTheta_i) * sinTheta_i), -0.5f, -0.16666666666666666f) / v), 2.0f, ((sinTheta_i * sinTheta_O) * -2.0f)) / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_O / v) * cosTheta_i) * Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(fma(Float32(fma(Float32(Float32(Float32(sinTheta_O * sinTheta_O) * sinTheta_i) * sinTheta_i), Float32(-0.5), Float32(-0.16666666666666666)) / v), Float32(2.0), Float32(Float32(sinTheta_i * sinTheta_O) * Float32(-2.0))) / v))))
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. associate-/l/ N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 64.5%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\color{blue}{\left(2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, -2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}\right)} \cdot v} \]
8. Step-by-step derivation

9. Applied rewrites 64.5%

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

    1. lift-/.f32 N/A

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

    2. clear-num N/A

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

    3. inv-pow N/A

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

11. Applied rewrites 64.6%

    \[\leadsto \color{blue}{\frac{1}{2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, -2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}} \cdot \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)} \]

12. Final simplification 64.6%

    \[\leadsto \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \cdot \frac{1}{2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, \left(sinTheta\_i \cdot sinTheta\_O\right) \cdot -2\right)}{v}} \]
13. Add Preprocessing

Alternative 7: 64.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, \left(sinTheta\_i \cdot sinTheta\_O\right) \cdot -2\right)}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ cosTheta_O v) cosTheta_i)
  (-
   2.0
   (/
    (fma
     (/
      (fma
       (* (* (* sinTheta_O sinTheta_O) sinTheta_i) sinTheta_i)
       -0.5
       -0.16666666666666666)
      v)
     2.0
     (* (* sinTheta_i sinTheta_O) -2.0))
    v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O / v) * cosTheta_i) / (2.0f - (fmaf((fmaf((((sinTheta_O * sinTheta_O) * sinTheta_i) * sinTheta_i), -0.5f, -0.16666666666666666f) / v), 2.0f, ((sinTheta_i * sinTheta_O) * -2.0f)) / v));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_O / v) * cosTheta_i) / Float32(Float32(2.0) - Float32(fma(Float32(fma(Float32(Float32(Float32(sinTheta_O * sinTheta_O) * sinTheta_i) * sinTheta_i), Float32(-0.5), Float32(-0.16666666666666666)) / v), Float32(2.0), Float32(Float32(sinTheta_i * sinTheta_O) * Float32(-2.0))) / v)))
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. associate-/l/ N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 64.5%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\color{blue}{\left(2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, -2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}\right)} \cdot v} \]
8. Step-by-step derivation

9. Applied rewrites 64.5%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. associate-/r* N/A

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

    5. lift-*.f32 N/A

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

    6. *-commutative N/A

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

    7. lower-/.f32 N/A

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

11. Applied rewrites 64.6%

    \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, -2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}}} \]

12. Final simplification 64.6%

    \[\leadsto \frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, \left(sinTheta\_i \cdot sinTheta\_O\right) \cdot -2\right)}{v}} \]
13. Add Preprocessing

Alternative 8: 64.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O}{\left(2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.16666666666666666\right)}{v}, 2, \left(sinTheta\_i \cdot sinTheta\_O\right) \cdot -2\right)}{v}\right) \cdot v} \cdot cosTheta\_i \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/
   cosTheta_O
   (*
    (-
     2.0
     (/
      (fma
       (/
        (fma
         -0.5
         (* (* (* sinTheta_O sinTheta_O) sinTheta_i) sinTheta_i)
         -0.16666666666666666)
        v)
       2.0
       (* (* sinTheta_i sinTheta_O) -2.0))
      v))
    v))
  cosTheta_i))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O / ((2.0f - (fmaf((fmaf(-0.5f, (((sinTheta_O * sinTheta_O) * sinTheta_i) * sinTheta_i), -0.16666666666666666f) / v), 2.0f, ((sinTheta_i * sinTheta_O) * -2.0f)) / v)) * v)) * cosTheta_i;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O / Float32(Float32(Float32(2.0) - Float32(fma(Float32(fma(Float32(-0.5), Float32(Float32(Float32(sinTheta_O * sinTheta_O) * sinTheta_i) * sinTheta_i), Float32(-0.16666666666666666)) / v), Float32(2.0), Float32(Float32(sinTheta_i * sinTheta_O) * Float32(-2.0))) / v)) * v)) * cosTheta_i)
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. associate-/l/ N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 64.5%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\color{blue}{\left(2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, -2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}\right)} \cdot v} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 64.6

      \[\leadsto cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{\left(2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, -2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}\right) \cdot v}} \]

9. Applied rewrites 64.6%

   \[\leadsto \color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{\left(2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.16666666666666666\right)}{v}, 2, \left(sinTheta\_i \cdot sinTheta\_O\right) \cdot -2\right)}{v}\right) \cdot v}} \]

10. Final simplification 64.6%

    \[\leadsto \frac{cosTheta\_O}{\left(2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.16666666666666666\right)}{v}, 2, \left(sinTheta\_i \cdot sinTheta\_O\right) \cdot -2\right)}{v}\right) \cdot v} \cdot cosTheta\_i \]
11. Add Preprocessing

Alternative 9: 64.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i}{\mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{\mathsf{fma}\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot 0.5, sinTheta\_i \cdot sinTheta\_i, 0.16666666666666666\right)}{v \cdot v}\right), 2\right) \cdot v} \cdot cosTheta\_O \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/
   cosTheta_i
   (*
    (fma
     2.0
     (fma
      sinTheta_O
      (/ sinTheta_i v)
      (/
       (fma
        (* (* sinTheta_O sinTheta_O) 0.5)
        (* sinTheta_i sinTheta_i)
        0.16666666666666666)
       (* v v)))
     2.0)
    v))
  cosTheta_O))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i / (fmaf(2.0f, fmaf(sinTheta_O, (sinTheta_i / v), (fmaf(((sinTheta_O * sinTheta_O) * 0.5f), (sinTheta_i * sinTheta_i), 0.16666666666666666f) / (v * v))), 2.0f) * v)) * cosTheta_O;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i / Float32(fma(Float32(2.0), fma(sinTheta_O, Float32(sinTheta_i / v), Float32(fma(Float32(Float32(sinTheta_O * sinTheta_O) * Float32(0.5)), Float32(sinTheta_i * sinTheta_i), Float32(0.16666666666666666)) / Float32(v * v))), Float32(2.0)) * v)) * cosTheta_O)
end

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-/.f32 N/A

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

   14. div-inv N/A

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

   15. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.7%

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

7. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

9. Applied rewrites 64.6%

   \[\leadsto \frac{cosTheta\_i}{\color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{\mathsf{fma}\left(0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right), sinTheta\_i \cdot sinTheta\_i, 0.16666666666666666\right)}{v \cdot v}\right), 2\right) \cdot v}} \cdot cosTheta\_O \]

10. Final simplification 64.6%

    \[\leadsto \frac{cosTheta\_i}{\mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{\mathsf{fma}\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot 0.5, sinTheta\_i \cdot sinTheta\_i, 0.16666666666666666\right)}{v \cdot v}\right), 2\right) \cdot v} \cdot cosTheta\_O \]
11. Add Preprocessing

Alternative 10: 64.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 - \frac{\mathsf{fma}\left(\frac{-0.16666666666666666}{v}, 2, \left(sinTheta\_i \cdot sinTheta\_O\right) \cdot -2\right)}{v}\right) \cdot v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) / ((2.0f - (fmaf((-0.16666666666666666f / v), 2.0f, ((sinTheta_i * sinTheta_O) * -2.0f)) / v)) * v);
}

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. associate-/l/ N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 64.5%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\color{blue}{\left(2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, -2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}\right)} \cdot v} \]

8. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\left(2 - \frac{\mathsf{fma}\left(\frac{\frac{-1}{6}}{v}, 2, -2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}\right) \cdot v} \]
9. Step-by-step derivation

10. Applied rewrites 64.5%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\left(2 - \frac{\mathsf{fma}\left(\frac{-0.16666666666666666}{v}, 2, -2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}\right) \cdot v} \]

11. Final simplification 64.5%

    \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 - \frac{\mathsf{fma}\left(\frac{-0.16666666666666666}{v}, 2, \left(sinTheta\_i \cdot sinTheta\_O\right) \cdot -2\right)}{v}\right) \cdot v} \]
12. Add Preprocessing

Alternative 11: 64.9% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i * costheta_o) / ((2.0e0 - (((-0.3333333333333333e0) / v) * (1.0e0 / v))) * v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. associate-/l/ N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 64.5%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\color{blue}{\left(2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, -2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}\right)} \cdot v} \]
8. Step-by-step derivation

9. Applied rewrites 64.5%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\left(2 - \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.16666666666666666\right)}{v}, 2, \left(sinTheta\_i \cdot sinTheta\_O\right) \cdot -2\right) \cdot \color{blue}{\frac{1}{v}}\right) \cdot v} \]

10. Taylor expanded in sinTheta_i around 0

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

12. Applied rewrites 64.5%

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

13. Final simplification 64.5%

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

Alternative 12: 64.9% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i * costheta_o) / ((2.0e0 - ((-0.3333333333333333e0) / (v * v))) * v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. associate-/l/ N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 64.5%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\color{blue}{\left(2 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i\right) \cdot sinTheta\_i, -0.5, -0.16666666666666666\right)}{v}, 2, -2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}\right)} \cdot v} \]

8. Taylor expanded in sinTheta_i around 0

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

10. Applied rewrites 64.5%

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

11. Final simplification 64.5%

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

Alternative 13: 59.6% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 59.0

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

5. Applied rewrites 59.0%

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

7. Applied rewrites 59.4%

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

Alternative 14: 59.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 59.0

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

5. Applied rewrites 59.0%

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

7. Applied rewrites 59.0%

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

9. Applied rewrites 59.0%

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

10. Final simplification 59.0%

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

Alternative 15: 59.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 59.0

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

5. Applied rewrites 59.0%

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

7. Applied rewrites 59.0%

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

9. Applied rewrites 59.0%

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

Reproduce

herbie shell --seed 2024240 
(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)))