HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.9%

Time: 23.1s

Alternatives: 20

Speedup: 1.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-*.f32 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. lift-*.f32 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. distribute-frac-neg2 N/A

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

   3. lift-/.f32 N/A

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

   4. neg-mul-1 N/A

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

   5. neg-mul-1 N/A

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

   6. lift-/.f32 N/A

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

   7. distribute-frac-neg2 N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/l/ N/A

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 99.0

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

8. Applied rewrites 99.0%

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

9. Final simplification 99.0%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-sinh.f32 N/A

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

   5. sinh-undef N/A

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

   6. flip-- N/A

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

   7. remove-double-div N/A

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

   8. lift-/.f32 N/A

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

   9. frac-times N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 99.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

8. Applied rewrites 99.0%

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

9. Final simplification 99.0%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-*.f32 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. lift-*.f32 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. distribute-frac-neg2 N/A

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

   3. lift-/.f32 N/A

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

   4. neg-mul-1 N/A

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

   5. neg-mul-1 N/A

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

   6. lift-/.f32 N/A

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

   7. distribute-frac-neg2 N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/l/ N/A

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

6. Applied rewrites 98.9%

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

7. Final simplification 98.9%

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

Alternative 4: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l/ N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. mul-1-neg N/A

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

   13. distribute-neg-frac2 N/A

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

   14. mul-1-neg N/A

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

   15. lower-/.f32 N/A

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

   16. mul-1-neg N/A

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

   17. lower-neg.f32 98.7

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

7. Applied rewrites 98.7%

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

9. Applied rewrites 98.9%

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

Alternative 5: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l/ N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. mul-1-neg N/A

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

   13. distribute-neg-frac2 N/A

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

   14. mul-1-neg N/A

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

   15. lower-/.f32 N/A

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

   16. mul-1-neg N/A

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

   17. lower-neg.f32 98.7

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

7. Applied rewrites 98.7%

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

9. Applied rewrites 98.9%

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

Alternative 6: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-sinh.f32 N/A

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

   5. sinh-undef N/A

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

   6. flip-- N/A

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

   7. remove-double-div N/A

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

   8. lift-/.f32 N/A

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

   9. frac-times N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 99.0

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

9. Applied rewrites 99.0%

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

10. Final simplification 99.0%

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

Alternative 7: 98.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-sinh.f32 N/A

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

   5. sinh-undef N/A

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

   6. flip-- N/A

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

   7. remove-double-div N/A

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

   8. lift-/.f32 N/A

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

   9. frac-times N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 98.9

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

9. Applied rewrites 98.9%

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

10. Final simplification 98.9%

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

Alternative 8: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-*.f32 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. lift-*.f32 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. distribute-frac-neg2 N/A

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

   3. lift-/.f32 N/A

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

   4. neg-mul-1 N/A

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

   5. neg-mul-1 N/A

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

   6. lift-/.f32 N/A

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

   7. distribute-frac-neg2 N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/l/ N/A

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Applied rewrites 98.7%

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

Alternative 9: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-sinh.f32 N/A

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

   5. sinh-undef N/A

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

   6. flip-- N/A

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

   7. remove-double-div N/A

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

   8. lift-/.f32 N/A

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

   9. frac-times N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.7

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 10: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-*.f32 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. lift-*.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 98.5

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

7. Applied rewrites 98.5%

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

Alternative 11: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l/ N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.4%

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

Alternative 12: 70.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(0.5, \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}, \frac{sinTheta\_O}{-v}\right), 1\right)}{v \cdot \left(\left(v \cdot 2\right) \cdot \frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} + 1}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i cosTheta_O)
  (/
   (fma
    sinTheta_i
    (fma
     0.5
     (/ (* sinTheta_i (* sinTheta_O sinTheta_O)) (* v v))
     (/ sinTheta_O (- v)))
    1.0)
   (*
    v
    (*
     (* v 2.0)
     (/
      (+
       (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
       1.0)
      v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * (fmaf(sinTheta_i, fmaf(0.5f, ((sinTheta_i * (sinTheta_O * sinTheta_O)) / (v * v)), (sinTheta_O / -v)), 1.0f) / (v * ((v * 2.0f) * ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) + 1.0f) / v))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * cosTheta_O) * Float32(fma(sinTheta_i, fma(Float32(0.5), Float32(Float32(sinTheta_i * Float32(sinTheta_O * sinTheta_O)) / Float32(v * v)), Float32(sinTheta_O / Float32(-v))), Float32(1.0)) / Float32(v * Float32(Float32(v * Float32(2.0)) * Float32(Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v)) + Float32(1.0)) / v)))))
end

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l/ N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. mul-1-neg N/A

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

   13. distribute-neg-frac2 N/A

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

   14. mul-1-neg N/A

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

   15. lower-/.f32 N/A

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

   16. mul-1-neg N/A

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

   17. lower-neg.f32 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 72.3%

    \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(0.5, \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}, \frac{sinTheta\_O}{-v}\right), 1\right)}{v \cdot \left(\color{blue}{\frac{\left(-\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right) + -1}{-v}} \cdot \left(v \cdot 2\right)\right)} \]

11. Final simplification 72.3%

    \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(0.5, \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}, \frac{sinTheta\_O}{-v}\right), 1\right)}{v \cdot \left(\left(v \cdot 2\right) \cdot \frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} + 1}{v}\right)} \]
12. Add Preprocessing

Alternative 13: 70.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(0.5, \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}, \frac{sinTheta\_O}{-v}\right), 1\right)}{v \cdot \left(\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i cosTheta_O)
  (/
   (fma
    sinTheta_i
    (fma
     0.5
     (/ (* sinTheta_i (* sinTheta_O sinTheta_O)) (* v v))
     (/ sinTheta_O (- v)))
    1.0)
   (*
    v
    (-
     (/ (+ 0.3333333333333333 (/ 0.016666666666666666 (* v v))) (* v v))
     -2.0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * (fmaf(sinTheta_i, fmaf(0.5f, ((sinTheta_i * (sinTheta_O * sinTheta_O)) / (v * v)), (sinTheta_O / -v)), 1.0f) / (v * (((0.3333333333333333f + (0.016666666666666666f / (v * v))) / (v * v)) - -2.0f)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * cosTheta_O) * Float32(fma(sinTheta_i, fma(Float32(0.5), Float32(Float32(sinTheta_i * Float32(sinTheta_O * sinTheta_O)) / Float32(v * v)), Float32(sinTheta_O / Float32(-v))), Float32(1.0)) / Float32(v * Float32(Float32(Float32(Float32(0.3333333333333333) + Float32(Float32(0.016666666666666666) / Float32(v * v))) / Float32(v * v)) - Float32(-2.0)))))
end

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l/ N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. mul-1-neg N/A

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

   13. distribute-neg-frac2 N/A

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

   14. mul-1-neg N/A

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

   15. lower-/.f32 N/A

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

   16. mul-1-neg N/A

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

   17. lower-neg.f32 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

   5. lower-*.f32 N/A

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

10. Applied rewrites 72.3%

    \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(0.5, \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}, \frac{sinTheta\_O}{-v}\right), 1\right)}{\color{blue}{\left(\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{-v \cdot v} + -2\right) \cdot \left(-v\right)}} \]

11. Final simplification 72.3%

    \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(0.5, \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}, \frac{sinTheta\_O}{-v}\right), 1\right)}{v \cdot \left(\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2\right)} \]
12. Add Preprocessing

Alternative 14: 64.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(v \cdot v\right)}{\mathsf{fma}\left(v, \mathsf{fma}\left(v, 2 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right), \mathsf{fma}\left(-2, sinTheta\_i \cdot \left(sinTheta\_i \cdot \left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot -0.5\right)\right), 0.3333333333333333\right)\right), 0.3333333333333333 \cdot \left(sinTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_O, sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_i\right), sinTheta\_i\right)\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (* cosTheta_i cosTheta_O) (* v v))
  (fma
   v
   (fma
    v
    (* 2.0 (fma sinTheta_O sinTheta_i v))
    (fma
     -2.0
     (* sinTheta_i (* sinTheta_i (* (* sinTheta_O sinTheta_O) -0.5)))
     0.3333333333333333))
   (*
    0.3333333333333333
    (*
     sinTheta_O
     (fma
      (* sinTheta_O sinTheta_O)
      (* sinTheta_i (* sinTheta_i sinTheta_i))
      sinTheta_i))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) * (v * v)) / fmaf(v, fmaf(v, (2.0f * fmaf(sinTheta_O, sinTheta_i, v)), fmaf(-2.0f, (sinTheta_i * (sinTheta_i * ((sinTheta_O * sinTheta_O) * -0.5f))), 0.3333333333333333f)), (0.3333333333333333f * (sinTheta_O * fmaf((sinTheta_O * sinTheta_O), (sinTheta_i * (sinTheta_i * sinTheta_i)), sinTheta_i))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) * Float32(v * v)) / fma(v, fma(v, Float32(Float32(2.0) * fma(sinTheta_O, sinTheta_i, v)), fma(Float32(-2.0), Float32(sinTheta_i * Float32(sinTheta_i * Float32(Float32(sinTheta_O * sinTheta_O) * Float32(-0.5)))), Float32(0.3333333333333333))), Float32(Float32(0.3333333333333333) * Float32(sinTheta_O * fma(Float32(sinTheta_O * sinTheta_O), Float32(sinTheta_i * Float32(sinTheta_i * sinTheta_i)), sinTheta_i)))))
end

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-*.f32 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. times-frac N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around -inf

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{-1 \cdot \left(v \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-2 \cdot \frac{\frac{1}{6} \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + \frac{1}{6} \cdot \left({sinTheta\_O}^{3} \cdot {sinTheta\_i}^{3}\right)}{v} + 2 \cdot \left(\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}\right)}{v} + 2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v} - 2\right)\right)}} \]

7. Applied rewrites 66.2%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(\frac{\mathsf{fma}\left(2, sinTheta\_O \cdot sinTheta\_i, \frac{\mathsf{fma}\left(2, \mathsf{fma}\left(-0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right), sinTheta\_i \cdot sinTheta\_i, -0.16666666666666666\right), \frac{-0.3333333333333333 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right)\right)}{v}\right)}{-v}\right)}{-v} + -2\right) \cdot \left(-v\right)}} \]

8. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\frac{\frac{1}{3} \cdot \left(sinTheta\_O \cdot sinTheta\_i + {sinTheta\_O}^{3} \cdot {sinTheta\_i}^{3}\right) + v \cdot \left(-2 \cdot \left(\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}\right) + v \cdot \left(2 \cdot v + 2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right)}{{v}^{2}}}} \]

10. Applied rewrites 66.2%

    \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\frac{\mathsf{fma}\left(v, \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right), sinTheta\_i \cdot sinTheta\_i, -0.16666666666666666\right), -2, v \cdot \left(2 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right)\right)\right), 0.3333333333333333 \cdot \mathsf{fma}\left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right), sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_i\right), sinTheta\_O \cdot sinTheta\_i\right)\right)}{v \cdot v}}} \]

12. Applied rewrites 66.3%

    \[\leadsto \color{blue}{\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(v \cdot v\right)}{\mathsf{fma}\left(v, \mathsf{fma}\left(v, 2 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right), \mathsf{fma}\left(-2, sinTheta\_i \cdot \left(sinTheta\_i \cdot \left(-0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right), 0.3333333333333333\right)\right), 0.3333333333333333 \cdot \left(sinTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_O, sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_i\right), sinTheta\_i\right)\right)\right)}} \]

13. Final simplification 66.3%

    \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(v \cdot v\right)}{\mathsf{fma}\left(v, \mathsf{fma}\left(v, 2 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right), \mathsf{fma}\left(-2, sinTheta\_i \cdot \left(sinTheta\_i \cdot \left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot -0.5\right)\right), 0.3333333333333333\right)\right), 0.3333333333333333 \cdot \left(sinTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_O, sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_i\right), sinTheta\_i\right)\right)\right)} \]
14. Add Preprocessing

Alternative 15: 64.4% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-*.f32 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. times-frac N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around -inf

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{-1 \cdot \left(v \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-2 \cdot \frac{\frac{1}{6} \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + \frac{1}{6} \cdot \left({sinTheta\_O}^{3} \cdot {sinTheta\_i}^{3}\right)}{v} + 2 \cdot \left(\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}\right)}{v} + 2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v} - 2\right)\right)}} \]

7. Applied rewrites 66.2%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(\frac{\mathsf{fma}\left(2, sinTheta\_O \cdot sinTheta\_i, \frac{\mathsf{fma}\left(2, \mathsf{fma}\left(-0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right), sinTheta\_i \cdot sinTheta\_i, -0.16666666666666666\right), \frac{-0.3333333333333333 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right)\right)}{v}\right)}{-v}\right)}{-v} + -2\right) \cdot \left(-v\right)}} \]

8. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\frac{\frac{1}{3} \cdot \left(sinTheta\_O \cdot sinTheta\_i + {sinTheta\_O}^{3} \cdot {sinTheta\_i}^{3}\right) + v \cdot \left(-2 \cdot \left(\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}\right) + v \cdot \left(2 \cdot v + 2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right)}{{v}^{2}}}} \]

10. Applied rewrites 66.2%

    \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\frac{\mathsf{fma}\left(v, \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right), sinTheta\_i \cdot sinTheta\_i, -0.16666666666666666\right), -2, v \cdot \left(2 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right)\right)\right), 0.3333333333333333 \cdot \mathsf{fma}\left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right), sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_i\right), sinTheta\_O \cdot sinTheta\_i\right)\right)}{v \cdot v}}} \]

11. Taylor expanded in sinTheta_O around 0

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

    1. lower-/.f32 N/A

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

    2. associate-*r* N/A

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

    3. lower-*.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. +-commutative N/A

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

    6. lower-fma.f32 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f32 66.2

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

13. Applied rewrites 66.2%

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

14. Final simplification 66.2%

    \[\leadsto \frac{v \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{fma}\left(2, v \cdot v, 0.3333333333333333\right)} \]
15. Add Preprocessing

Alternative 16: 59.2% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.7

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

5. Applied rewrites 60.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. clear-num N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 61.5

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 61.5

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

7. Applied rewrites 61.5%

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

Alternative 17: 59.1% 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 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.7

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

5. Applied rewrites 60.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 61.4

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

7. Applied rewrites 61.4%

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

Alternative 18: 58.7% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.7

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

5. Applied rewrites 60.7%

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

6. Final simplification 60.7%

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

Alternative 19: 58.7% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.7

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

5. Applied rewrites 60.7%

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

   1. lift-*.f32 N/A

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

   2. associate-/l* N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. clear-num N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. lift-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. clear-num N/A

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

   16. lower-/.f32 60.6

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

7. Applied rewrites 60.6%

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

8. Final simplification 60.6%

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

Alternative 20: 58.7% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.7

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

5. Applied rewrites 60.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. lift-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lower-*.f32 60.6

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. associate-*r/ N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. lift-/.f32 60.6

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

7. Applied rewrites 60.6%

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

Reproduce

herbie shell --seed 2024214 
(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)))