HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 20.5s

Alternatives: 21

Speedup: 1.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. times-frac N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

6. Applied rewrites 98.7%

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

7. Final simplification 98.7%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. times-frac N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

6. Applied rewrites 98.7%

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

8. Applied rewrites 98.5%

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

9. Final simplification 98.5%

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

Alternative 3: 98.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. times-frac N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-/.f32 98.6

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-exp.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. rec-exp N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 98.6

      \[\leadsto \frac{cosTheta\_O}{v} \cdot \left(\frac{2}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}} \cdot \left(\frac{0.5}{v} \cdot cosTheta\_i\right)\right) \]

9. Applied rewrites 98.6%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. div-inv N/A

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

    4. lift-/.f32 N/A

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

    5. associate-*l* N/A

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

    6. lower-*.f32 N/A

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

    7. lower-*.f32 98.7

       \[\leadsto cosTheta\_O \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(\frac{2}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \left(\frac{0.5}{v} \cdot cosTheta\_i\right)\right)\right)} \]

11. Applied rewrites 98.6%

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

12. Final simplification 98.6%

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

Alternative 4: 98.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. times-frac N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-/.f32 98.6

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-exp.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. rec-exp N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 98.6

      \[\leadsto \frac{cosTheta\_O}{v} \cdot \left(\frac{2}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}} \cdot \left(\frac{0.5}{v} \cdot cosTheta\_i\right)\right) \]

9. Applied rewrites 98.6%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 98.5

       \[\leadsto \left(\frac{0.5}{v} \cdot cosTheta\_i\right) \cdot \color{blue}{\left(\frac{2}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{cosTheta\_O}{v}\right)} \]

11. Applied rewrites 98.4%

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

12. Final simplification 98.4%

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

Alternative 5: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. div-inv N/A

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

   6. associate-/r* N/A

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

   7. div-inv N/A

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

   8. lift-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 93.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.3

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

7. Applied rewrites 98.3%

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

9. Applied rewrites 98.3%

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

10. Final simplification 98.3%

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

Alternative 6: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Final simplification 98.3%

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

Alternative 7: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. times-frac N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l/ N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

6. Applied rewrites 98.5%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Applied rewrites 98.3%

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

10. Final simplification 98.3%

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

Alternative 8: 70.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((2.0e0 / ((((0.3333333333333333e0 + (0.016666666666666666e0 / (v * v))) / (v * v)) + 2.0e0) / v)) * ((0.5e0 / v) * costheta_i)) * (costheta_o / v)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((single(2.0) / ((((single(0.3333333333333333) + (single(0.016666666666666666) / (v * v))) / (v * v)) + single(2.0)) / v)) * ((single(0.5) / v) * cosTheta_i)) * (cosTheta_O / v);
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. times-frac N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-/.f32 98.6

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-exp.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. rec-exp N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 98.6

      \[\leadsto \frac{cosTheta\_O}{v} \cdot \left(\frac{2}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}} \cdot \left(\frac{0.5}{v} \cdot cosTheta\_i\right)\right) \]

9. Applied rewrites 98.6%

   \[\leadsto \frac{cosTheta\_O}{v} \cdot \left(\color{blue}{\frac{2}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \cdot \left(\frac{0.5}{v} \cdot cosTheta\_i\right)\right) \]

10. Taylor expanded in v around -inf

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

12. Applied rewrites 67.3%

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

13. Final simplification 67.3%

    \[\leadsto \left(\frac{2}{\frac{\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} + 2}{v}} \cdot \left(\frac{0.5}{v} \cdot cosTheta\_i\right)\right) \cdot \frac{cosTheta\_O}{v} \]
14. Add Preprocessing

Alternative 9: 64.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. exp-neg N/A

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

   10. un-div-inv N/A

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

   11. associate-/l/ N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 60.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 60.8

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

9. Applied rewrites 60.8%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. clear-num N/A

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

    4. un-div-inv N/A

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

    5. lower-/.f32 N/A

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

11. Applied rewrites 60.8%

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

Alternative 10: 70.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. div-inv N/A

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

   6. associate-/r* N/A

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

   7. div-inv N/A

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

   8. lift-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 93.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.3

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 67.3%

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

11. Final simplification 67.3%

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

Alternative 11: 64.2% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. exp-neg N/A

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

   10. un-div-inv N/A

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

   11. associate-/l/ N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 60.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 60.8

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

9. Applied rewrites 60.8%

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

10. Taylor expanded in v around -inf

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

12. Applied rewrites 60.8%

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

13. Final simplification 60.8%

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

Alternative 12: 64.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. exp-neg N/A

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

   10. un-div-inv N/A

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

   11. associate-/l/ N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 60.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 60.8

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

9. Applied rewrites 60.8%

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

10. Taylor expanded in sinTheta_i around 0

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

12. Applied rewrites 60.8%

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

13. Final simplification 60.8%

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

Alternative 13: 64.2% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. exp-neg N/A

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

   10. un-div-inv N/A

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

   11. associate-/l/ N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 60.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 60.8

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

9. Applied rewrites 60.8%

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

10. Taylor expanded in sinTheta_i around 0

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

12. Applied rewrites 60.8%

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

13. Final simplification 60.8%

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

Alternative 14: 64.2% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. exp-neg N/A

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

   10. un-div-inv N/A

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

   11. associate-/l/ N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 60.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 60.8

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

9. Applied rewrites 60.8%

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

10. Taylor expanded in sinTheta_i around 0

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

12. Applied rewrites 60.8%

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

13. Final simplification 60.8%

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

Alternative 15: 58.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 54.7

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

5. Applied rewrites 54.7%

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

7. Applied rewrites 55.1%

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

Alternative 16: 58.9% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 54.7

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

5. Applied rewrites 54.7%

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

7. Applied rewrites 55.1%

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

8. Final simplification 55.1%

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

Alternative 17: 58.9% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 54.7

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

5. Applied rewrites 54.7%

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

7. Applied rewrites 55.1%

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

Alternative 18: 58.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 54.7

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

5. Applied rewrites 54.7%

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

7. Applied rewrites 54.8%

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

9. Applied rewrites 54.8%

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

10. Final simplification 54.8%

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

Alternative 19: 58.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 54.7

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

5. Applied rewrites 54.7%

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

7. Applied rewrites 54.8%

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

8. Final simplification 54.8%

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

Alternative 20: 58.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 54.7

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

5. Applied rewrites 54.7%

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

7. Applied rewrites 54.8%

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

8. Final simplification 54.8%

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

Alternative 21: 58.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 54.7

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

5. Applied rewrites 54.7%

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

7. Applied rewrites 54.8%

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

9. Applied rewrites 54.8%

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

11. Applied rewrites 54.8%

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

Reproduce

herbie shell --seed 2024235 
(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)))