HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 20.2s

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, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-/.f32 N/A

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

   13. lift-sinh.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

4. Applied egg-rr 98.9%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f32 N/A

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

   6. /-rgt-identity N/A

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

   7. clear-num N/A

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

   8. lift-/.f32 N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f32 99.0

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

4. Applied egg-rr 98.8%

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

   1. associate-*r/ N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 98.9

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

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. lift-sinh.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/l* N/A

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

4. Applied egg-rr 98.7%

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

   1. lift-neg.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-sinh.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/l/ N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 98.9

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

   11. lift-/.f32 N/A

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

   12. lift-neg.f32 N/A

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

   13. distribute-frac-neg2 N/A

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

   14. lower-neg.f32 N/A

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

   15. lower-/.f32 98.9

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

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-/.f32 N/A

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

   13. lift-sinh.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 5: 98.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-/.f32 N/A

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

   13. lift-sinh.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-/l* N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f32 98.5

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

9. Simplified 98.5%

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

11. Applied egg-rr 98.8%

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

12. Final simplification 98.8%

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

Alternative 6: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-/.f32 N/A

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

   13. lift-sinh.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.7

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

7. Simplified 98.7%

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

Alternative 7: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. lift-sinh.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/l* N/A

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Simplified 98.5%

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

Alternative 8: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.5

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

5. Simplified 98.5%

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

6. Final simplification 98.5%

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

Alternative 9: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-/.f32 N/A

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

   13. lift-sinh.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

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

Alternative 10: 69.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-/.f32 N/A

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

   13. lift-sinh.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-/l* N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f32 98.5

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

9. Simplified 98.5%

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

10. Taylor expanded in v around -inf

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

    1. mul-1-neg N/A

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

    2. distribute-neg-frac2 N/A

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

    3. neg-mul-1 N/A

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

    4. lower-/.f32 N/A

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

12. Simplified 69.6%

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

13. Final simplification 69.6%

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

Alternative 11: 63.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-/.f32 N/A

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

   13. lift-sinh.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-/l* N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f32 98.5

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

9. Simplified 98.5%

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

10. Taylor expanded in v around inf

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

    1. lower-/.f32 N/A

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

    2. lower-+.f32 N/A

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

    5. lower-/.f32 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f32 63.9

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

12. Simplified 63.9%

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

13. Final simplification 63.9%

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

Alternative 12: 69.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-/.f32 N/A

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

   13. lift-sinh.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-/l* N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f32 98.5

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

9. Simplified 98.5%

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

10. Taylor expanded in v around -inf

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

    1. mul-1-neg N/A

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

    2. *-commutative N/A

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

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

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

    4. neg-mul-1 N/A

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

    5. lower-*.f32 N/A

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

12. Simplified 69.6%

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

13. Final simplification 69.6%

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

Alternative 13: 63.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-/.f32 N/A

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

   13. lift-sinh.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-/l* N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f32 98.5

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

9. Simplified 98.5%

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

10. Taylor expanded in v around inf

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

    1. lower-*.f32 N/A

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

    2. lower-+.f32 N/A

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

    5. lower-/.f32 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f32 63.9

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

12. Simplified 63.9%

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

13. Final simplification 63.9%

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

Alternative 14: 58.5% accurate, 7.0× speedup

Math

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

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(Float32(1.0) / Float32(Float32(2.0) / 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 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 58.6

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

5. Simplified 58.6%

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

   1. *-commutative N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. associate-/l* N/A

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

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

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. clear-num N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. lower-/.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 N/A

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

   19. metadata-eval N/A

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

   20. div-inv N/A

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

   21. clear-num N/A

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

   22. lower-/.f32 59.4

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

7. Applied egg-rr 59.4%

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

Alternative 15: 58.4% 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 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 58.6

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

5. Simplified 58.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. clear-num N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/l/ N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. clear-num N/A

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

   18. lower-/.f32 58.4

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

7. Applied egg-rr 58.4%

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

   1. lift-/.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 58.4

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

9. Applied egg-rr 58.4%

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

    1. associate-*l/ N/A

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

    2. associate-*l/ N/A

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

    3. associate-*r* N/A

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

    4. lift-*.f32 N/A

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

    5. clear-num N/A

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

    6. lower-/.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. lower-*.f32 59.1

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

11. Applied egg-rr 59.1%

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

12. Final simplification 59.1%

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

Alternative 16: 58.3% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 58.6

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

5. Simplified 58.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. clear-num N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/l/ N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. clear-num N/A

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

   18. lower-/.f32 58.4

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

7. Applied egg-rr 58.4%

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

   1. lift-/.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 58.4

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

9. Applied egg-rr 58.4%

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

    1. associate-*l/ N/A

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

    2. associate-*l/ N/A

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

    3. associate-*r* N/A

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

    4. lift-*.f32 N/A

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

    5. associate-*r/ N/A

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

    6. clear-num N/A

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

    7. un-div-inv N/A

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

    8. lower-/.f32 N/A

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

    9. lower-/.f32 59.1

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

11. Applied egg-rr 59.1%

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

12. Final simplification 59.1%

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

Alternative 17: 57.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 58.6

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

5. Simplified 58.6%

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

6. Final simplification 58.6%

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

Alternative 18: 57.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 58.6

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

5. Simplified 58.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. lift-/.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. associate-*r/ N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. lower-/.f32 58.5

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

7. Applied egg-rr 58.5%

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

8. Final simplification 58.5%

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

Alternative 19: 57.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 58.6

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

5. Simplified 58.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. clear-num N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/l/ N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. clear-num N/A

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

   18. lower-/.f32 58.4

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

7. Applied egg-rr 58.4%

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

8. Final simplification 58.4%

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

Alternative 20: 57.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 58.6

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

5. Simplified 58.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. clear-num N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/l/ N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. clear-num N/A

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

   18. lower-/.f32 58.4

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

7. Applied egg-rr 58.4%

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

   1. lift-/.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 58.4

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

9. Applied egg-rr 58.4%

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

    1. lift-/.f32 N/A

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

    2. associate-*l* N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. lower-*.f32 N/A

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

    6. lower-*.f32 58.4

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

11. Applied egg-rr 58.4%

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

12. Final simplification 58.4%

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

Alternative 21: 57.9% accurate, 12.4× speedup

Math

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

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(cosTheta_i * Float32(cosTheta_O * Float32(Float32(0.5) / v)))
end

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 58.6

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

5. Simplified 58.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. clear-num N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/l/ N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. clear-num N/A

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

   18. lower-/.f32 58.4

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

7. Applied egg-rr 58.4%

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

   1. lift-/.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 58.4

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

9. Applied egg-rr 58.4%

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

10. Final simplification 58.4%

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

Reproduce

herbie shell --seed 2024208 
(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)))