HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 18.1s

Alternatives: 21

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_O}{\frac{1}{{\left(\frac{1}{v}\right)}^{3}}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (* (/ (* cosTheta_i cosTheta_O) (/ 1.0 (pow (/ 1.0 v) 3.0))) (fma v v 0.0)))
  (* 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 (expf(((sinTheta_O * -sinTheta_i) / v)) * (((cosTheta_i * cosTheta_O) / (1.0f / powf((1.0f / v), 3.0f))) * fmaf(v, v, 0.0f))) / (v * (sinhf((1.0f / v)) * 2.0f));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(Float32(Float32(cosTheta_i * cosTheta_O) / Float32(Float32(1.0) / (Float32(Float32(1.0) / v) ^ Float32(3.0)))) * fma(v, v, Float32(0.0)))) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))))
end

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. frac-2neg N/A

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

   3. neg-sub0 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{0 - v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. flip3-- N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\frac{{0}^{3} - {v}^{3}}{0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. associate-/r/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-neg.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{0} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   10. sub0-neg N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\mathsf{neg}\left({v}^{3}\right)}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. lower-neg.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\mathsf{neg}\left({v}^{3}\right)}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. cube-mult N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \color{blue}{\left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   15. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \left(\color{blue}{0} + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   16. +-lft-identity N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \color{blue}{\left(v \cdot v + 0 \cdot v\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   17. mul0-lft N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \left(v \cdot v + \color{blue}{0}\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   18. lower-fma.f32 98.5

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-v \cdot \left(v \cdot v\right)} \cdot \color{blue}{\mathsf{fma}\left(v, v, 0\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.5%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-v \cdot \left(v \cdot v\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
5. Step-by-step derivation

   1. cube-unmult N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{{v}^{3}}\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. remove-double-div N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left({\color{blue}{\left(\frac{1}{\frac{1}{v}}\right)}}^{3}\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left({\left(\frac{1}{\color{blue}{\frac{1}{v}}}\right)}^{3}\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. cube-div N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{\frac{{1}^{3}}{{\left(\frac{1}{v}\right)}^{3}}}\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\frac{\color{blue}{1}}{{\left(\frac{1}{v}\right)}^{3}}\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{\frac{1}{{\left(\frac{1}{v}\right)}^{3}}}\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. lower-pow.f32 98.8

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-\frac{1}{\color{blue}{{\left(\frac{1}{v}\right)}^{3}}}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

6. Applied egg-rr 98.8%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-\color{blue}{\frac{1}{{\left(\frac{1}{v}\right)}^{3}}}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

7. Final simplification 98.8%

   \[\leadsto \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_O}{\frac{1}{{\left(\frac{1}{v}\right)}^{3}}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
8. Add Preprocessing

Alternative 2: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.7

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

4. Applied egg-rr 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. remove-double-div N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 98.8

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

6. Applied egg-rr 98.8%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-/l* N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. metadata-eval N/A

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

   9. associate-/r/ N/A

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

   10. clear-num N/A

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

   11. lift-/.f32 N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 98.8

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

8. Applied egg-rr 98.8%

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

9. Final simplification 98.8%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (* cosTheta_O (* cosTheta_i (/ 1.0 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 (expf(((sinTheta_O * -sinTheta_i) / v)) * (cosTheta_O * (cosTheta_i * (1.0f / 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 = (exp(((sintheta_o * -sintheta_i) / v)) * (costheta_o * (costheta_i * (1.0e0 / 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(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(cosTheta_O * Float32(cosTheta_i * Float32(Float32(1.0) / 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 = (exp(((sinTheta_O * -sinTheta_i) / v)) * (cosTheta_O * (cosTheta_i * (single(1.0) / v)))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.7

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

4. Applied egg-rr 98.7%

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

   1. frac-2neg N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. distribute-frac-neg2 N/A

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

   5. distribute-lft-neg-out N/A

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

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

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

   7. lift-*.f32 N/A

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

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

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-neg.f32 98.8

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (* (* cosTheta_i cosTheta_O) (/ 1.0 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 (expf(((sinTheta_O * -sinTheta_i) / v)) * ((cosTheta_i * cosTheta_O) * (1.0f / 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 = (exp(((sintheta_o * -sintheta_i) / v)) * ((costheta_i * costheta_o) * (1.0e0 / 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(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(1.0) / 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 = (exp(((sinTheta_O * -sinTheta_i) / v)) * ((cosTheta_i * cosTheta_O) * (single(1.0) / v))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.7

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

4. Applied egg-rr 98.7%

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

5. Final simplification 98.7%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (/ (* sinTheta_O (- sinTheta_i)) v)) (* cosTheta_O (/ cosTheta_i 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 (expf(((sinTheta_O * -sinTheta_i) / v)) * (cosTheta_O * (cosTheta_i / 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 = (exp(((sintheta_o * -sintheta_i) / v)) * (costheta_o * (costheta_i / 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(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(cosTheta_O * Float32(cosTheta_i / 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 = (exp(((sinTheta_O * -sinTheta_i) / v)) * (cosTheta_O * (cosTheta_i / v))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.6

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 6: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (/ (* sinTheta_O (- sinTheta_i)) v)) (* 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 (expf(((sinTheta_O * -sinTheta_i) / v)) * (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 = (exp(((sintheta_o * -sintheta_i) / v)) * (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(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(cosTheta_i * Float32(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 = (exp(((sinTheta_O * -sinTheta_i) / v)) * (cosTheta_i * (cosTheta_O / v))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 98.6

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 7: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\frac{2 \cdot \frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v}}{\frac{1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (* (* cosTheta_i cosTheta_O) (/ 1.0 v)))
  (/
   (*
    2.0
    (/
     (+
      (/
       (+ 0.008333333333333333 (/ 0.0001984126984126984 (* v v)))
       (* (* v v) (* v v)))
      (+ 1.0 (/ 0.16666666666666666 (* v v))))
     v))
   (/ 1.0 v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(((sinTheta_O * -sinTheta_i) / v)) * ((cosTheta_i * cosTheta_O) * (1.0f / v))) / ((2.0f * ((((0.008333333333333333f + (0.0001984126984126984f / (v * v))) / ((v * v) * (v * v))) + (1.0f + (0.16666666666666666f / (v * v)))) / v)) / (1.0f / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(((sintheta_o * -sintheta_i) / v)) * ((costheta_i * costheta_o) * (1.0e0 / v))) / ((2.0e0 * ((((0.008333333333333333e0 + (0.0001984126984126984e0 / (v * v))) / ((v * v) * (v * v))) + (1.0e0 + (0.16666666666666666e0 / (v * v)))) / v)) / (1.0e0 / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(1.0) / v))) / Float32(Float32(Float32(2.0) * Float32(Float32(Float32(Float32(Float32(0.008333333333333333) + Float32(Float32(0.0001984126984126984) / Float32(v * v))) / Float32(Float32(v * v) * Float32(v * v))) + Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) / Float32(v * v)))) / v)) / Float32(Float32(1.0) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((sinTheta_O * -sinTheta_i) / v)) * ((cosTheta_i * cosTheta_O) * (single(1.0) / v))) / ((single(2.0) * ((((single(0.008333333333333333) + (single(0.0001984126984126984) / (v * v))) / ((v * v) * (v * v))) + (single(1.0) + (single(0.16666666666666666) / (v * v)))) / v)) / (single(1.0) / v));
end

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.7

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

4. Applied egg-rr 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. remove-double-div N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 98.8

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in v around -inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)} \cdot 2}{\frac{1}{v}}} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)\right)} \cdot 2}{\frac{1}{v}}} \]

   2. distribute-frac-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\color{blue}{\frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}{v}} \cdot 2}{\frac{1}{v}}} \]

   3. sub-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v} \cdot 2}{\frac{1}{v}}} \]

   4. mul-1-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right)\right)} + \left(\mathsf{neg}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)}{v} \cdot 2}{\frac{1}{v}}} \]

   5. distribute-neg-out N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v} \cdot 2}{\frac{1}{v}}} \]

   6. remove-double-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\frac{\color{blue}{\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}}{v} \cdot 2}{\frac{1}{v}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\color{blue}{\frac{\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}} \cdot 2}{\frac{1}{v}}} \]

9. Simplified 77.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\frac{\color{blue}{\frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v}} \cdot 2}{\frac{1}{v}}} \]

10. Final simplification 77.6%

    \[\leadsto \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\frac{2 \cdot \frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v}}{\frac{1}{v}}} \]
11. Add Preprocessing

Alternative 8: 70.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \left(\mathsf{fma}\left(v, v, 0\right) \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v}\right)}{v \cdot \left(2 \cdot \frac{\frac{\mathsf{fma}\left(\frac{1}{v}, \frac{0.008333333333333333}{v}, 0.16666666666666666\right)}{v \cdot v} - -1}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (* (fma v v 0.0) (/ (* cosTheta_i (/ cosTheta_O (* v v))) v)))
  (*
   v
   (*
    2.0
    (/
     (-
      (/
       (fma (/ 1.0 v) (/ 0.008333333333333333 v) 0.16666666666666666)
       (* v v))
      -1.0)
     v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(((sinTheta_O * -sinTheta_i) / v)) * (fmaf(v, v, 0.0f) * ((cosTheta_i * (cosTheta_O / (v * v))) / v))) / (v * (2.0f * (((fmaf((1.0f / v), (0.008333333333333333f / v), 0.16666666666666666f) / (v * v)) - -1.0f) / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(fma(v, v, Float32(0.0)) * Float32(Float32(cosTheta_i * Float32(cosTheta_O / Float32(v * v))) / v))) / Float32(v * Float32(Float32(2.0) * Float32(Float32(Float32(fma(Float32(Float32(1.0) / v), Float32(Float32(0.008333333333333333) / v), Float32(0.16666666666666666)) / Float32(v * v)) - Float32(-1.0)) / v))))
end

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. frac-2neg N/A

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

   3. neg-sub0 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{0 - v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. flip3-- N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\frac{{0}^{3} - {v}^{3}}{0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. associate-/r/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-neg.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{0} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   10. sub0-neg N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\mathsf{neg}\left({v}^{3}\right)}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. lower-neg.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\mathsf{neg}\left({v}^{3}\right)}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. cube-mult N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \color{blue}{\left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   15. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \left(\color{blue}{0} + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   16. +-lft-identity N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \color{blue}{\left(v \cdot v + 0 \cdot v\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   17. mul0-lft N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \left(v \cdot v + \color{blue}{0}\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   18. lower-fma.f32 98.5

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-v \cdot \left(v \cdot v\right)} \cdot \color{blue}{\mathsf{fma}\left(v, v, 0\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.5%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-v \cdot \left(v \cdot v\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(\color{blue}{cosTheta\_i \cdot cosTheta\_O}\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \color{blue}{\left(v \cdot v\right)}\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. frac-2neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(v \cdot v\right)}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(v \cdot v\right)}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. associate-/r* N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. associate-/l/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(v \cdot v\right) \cdot v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. associate-/r* N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. associate-/l* N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. lower-/.f32 98.4

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{v \cdot v}}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

6. Applied egg-rr 98.4%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

7. Taylor expanded in v around -inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)} \cdot 2\right) \cdot v} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right)} \cdot 2\right) \cdot v} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}} \cdot 2\right) \cdot v} \]

   3. neg-mul-1 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\color{blue}{-1 \cdot v}} \cdot 2\right) \cdot v} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{-1 \cdot v}} \cdot 2\right) \cdot v} \]

9. Simplified 71.2%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\frac{-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{-v}} \cdot 2\right) \cdot v} \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\frac{1}{6} + \frac{\frac{1}{120}}{\color{blue}{v \cdot v}}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    2. lift-/.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\frac{1}{6} + \color{blue}{\frac{\frac{1}{120}}{v \cdot v}}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    3. +-commutative N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\color{blue}{\frac{\frac{1}{120}}{v \cdot v} + \frac{1}{6}}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    4. lift-/.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\color{blue}{\frac{\frac{1}{120}}{v \cdot v}} + \frac{1}{6}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    5. clear-num N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\color{blue}{\frac{1}{\frac{v \cdot v}{\frac{1}{120}}}} + \frac{1}{6}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    6. inv-pow N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\color{blue}{{\left(\frac{v \cdot v}{\frac{1}{120}}\right)}^{-1}} + \frac{1}{6}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    7. lift-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{{\left(\frac{\color{blue}{v \cdot v}}{\frac{1}{120}}\right)}^{-1} + \frac{1}{6}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    8. associate-/l* N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{{\color{blue}{\left(v \cdot \frac{v}{\frac{1}{120}}\right)}}^{-1} + \frac{1}{6}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    9. unpow-prod-down N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\color{blue}{{v}^{-1} \cdot {\left(\frac{v}{\frac{1}{120}}\right)}^{-1}} + \frac{1}{6}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    10. inv-pow N/A

        \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\color{blue}{\frac{1}{v}} \cdot {\left(\frac{v}{\frac{1}{120}}\right)}^{-1} + \frac{1}{6}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    11. lift-/.f32 N/A

        \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\color{blue}{\frac{1}{v}} \cdot {\left(\frac{v}{\frac{1}{120}}\right)}^{-1} + \frac{1}{6}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    12. inv-pow N/A

        \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\frac{1}{v} \cdot \color{blue}{\frac{1}{\frac{v}{\frac{1}{120}}}} + \frac{1}{6}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    13. clear-num N/A

        \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\frac{1}{v} \cdot \color{blue}{\frac{\frac{1}{120}}{v}} + \frac{1}{6}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    14. lower-fma.f32 N/A

        \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{v}, \frac{\frac{1}{120}}{v}, \frac{1}{6}\right)}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    15. lower-/.f32 71.2

        \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\mathsf{fma}\left(\frac{1}{v}, \color{blue}{\frac{0.008333333333333333}{v}}, 0.16666666666666666\right)}{v \cdot v}}{-v} \cdot 2\right) \cdot v} \]

11. Applied egg-rr 71.2%

    \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{v}, \frac{0.008333333333333333}{v}, 0.16666666666666666\right)}}{v \cdot v}}{-v} \cdot 2\right) \cdot v} \]

12. Final simplification 71.2%

    \[\leadsto \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \left(\mathsf{fma}\left(v, v, 0\right) \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v}\right)}{v \cdot \left(2 \cdot \frac{\frac{\mathsf{fma}\left(\frac{1}{v}, \frac{0.008333333333333333}{v}, 0.16666666666666666\right)}{v \cdot v} - -1}{v}\right)} \]
13. Add Preprocessing

Alternative 9: 70.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \left(\mathsf{fma}\left(v, v, 0\right) \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v}\right)}{v \cdot \left(2 \cdot \frac{\frac{0.16666666666666666 + 0.008333333333333333 \cdot \frac{1}{v \cdot v}}{v \cdot v} - -1}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (* (fma v v 0.0) (/ (* cosTheta_i (/ cosTheta_O (* v v))) v)))
  (*
   v
   (*
    2.0
    (/
     (-
      (/
       (+ 0.16666666666666666 (* 0.008333333333333333 (/ 1.0 (* 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 (expf(((sinTheta_O * -sinTheta_i) / v)) * (fmaf(v, v, 0.0f) * ((cosTheta_i * (cosTheta_O / (v * v))) / v))) / (v * (2.0f * ((((0.16666666666666666f + (0.008333333333333333f * (1.0f / (v * v)))) / (v * v)) - -1.0f) / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(fma(v, v, Float32(0.0)) * Float32(Float32(cosTheta_i * Float32(cosTheta_O / Float32(v * v))) / v))) / Float32(v * Float32(Float32(2.0) * Float32(Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) * Float32(Float32(1.0) / Float32(v * v)))) / Float32(v * v)) - Float32(-1.0)) / v))))
end

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. frac-2neg N/A

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

   3. neg-sub0 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{0 - v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. flip3-- N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\frac{{0}^{3} - {v}^{3}}{0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. associate-/r/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-neg.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{0} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   10. sub0-neg N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\mathsf{neg}\left({v}^{3}\right)}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. lower-neg.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\mathsf{neg}\left({v}^{3}\right)}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. cube-mult N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \color{blue}{\left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   15. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \left(\color{blue}{0} + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   16. +-lft-identity N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \color{blue}{\left(v \cdot v + 0 \cdot v\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   17. mul0-lft N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \left(v \cdot v + \color{blue}{0}\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   18. lower-fma.f32 98.5

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-v \cdot \left(v \cdot v\right)} \cdot \color{blue}{\mathsf{fma}\left(v, v, 0\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.5%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-v \cdot \left(v \cdot v\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(\color{blue}{cosTheta\_i \cdot cosTheta\_O}\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \color{blue}{\left(v \cdot v\right)}\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. frac-2neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(v \cdot v\right)}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(v \cdot v\right)}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. associate-/r* N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. associate-/l/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(v \cdot v\right) \cdot v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. associate-/r* N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. associate-/l* N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. lower-/.f32 98.4

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{v \cdot v}}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

6. Applied egg-rr 98.4%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

7. Taylor expanded in v around -inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)} \cdot 2\right) \cdot v} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right)} \cdot 2\right) \cdot v} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}} \cdot 2\right) \cdot v} \]

   3. neg-mul-1 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\color{blue}{-1 \cdot v}} \cdot 2\right) \cdot v} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{-1 \cdot v}} \cdot 2\right) \cdot v} \]

9. Simplified 71.2%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\frac{-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{-v}} \cdot 2\right) \cdot v} \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\frac{1}{6} + \frac{\frac{1}{120}}{\color{blue}{v \cdot v}}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    2. clear-num N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\frac{1}{6} + \color{blue}{\frac{1}{\frac{v \cdot v}{\frac{1}{120}}}}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    3. associate-/r/ N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\frac{1}{6} + \color{blue}{\frac{1}{v \cdot v} \cdot \frac{1}{120}}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    4. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{\frac{1}{6} + \color{blue}{\frac{1}{v \cdot v} \cdot \frac{1}{120}}}{v \cdot v}}{\mathsf{neg}\left(v\right)} \cdot 2\right) \cdot v} \]

    5. lower-/.f32 71.2

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{0.16666666666666666 + \color{blue}{\frac{1}{v \cdot v}} \cdot 0.008333333333333333}{v \cdot v}}{-v} \cdot 2\right) \cdot v} \]

11. Applied egg-rr 71.2%

    \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 - \frac{0.16666666666666666 + \color{blue}{\frac{1}{v \cdot v} \cdot 0.008333333333333333}}{v \cdot v}}{-v} \cdot 2\right) \cdot v} \]

12. Final simplification 71.2%

    \[\leadsto \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \left(\mathsf{fma}\left(v, v, 0\right) \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v}\right)}{v \cdot \left(2 \cdot \frac{\frac{0.16666666666666666 + 0.008333333333333333 \cdot \frac{1}{v \cdot v}}{v \cdot v} - -1}{v}\right)} \]
13. Add Preprocessing

Alternative 10: 70.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \left(\mathsf{fma}\left(v, v, 0\right) \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v}\right)}{v \cdot \left(2 \cdot \frac{1 + \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (* (fma v v 0.0) (/ (* cosTheta_i (/ cosTheta_O (* v v))) v)))
  (*
   v
   (*
    2.0
    (/
     (+
      1.0
      (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v)))
     v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(((sinTheta_O * -sinTheta_i) / v)) * (fmaf(v, v, 0.0f) * ((cosTheta_i * (cosTheta_O / (v * v))) / v))) / (v * (2.0f * ((1.0f + ((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v))) / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(fma(v, v, Float32(0.0)) * Float32(Float32(cosTheta_i * Float32(cosTheta_O / Float32(v * v))) / v))) / Float32(v * Float32(Float32(2.0) * Float32(Float32(Float32(1.0) + Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v))) / v))))
end

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. frac-2neg N/A

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

   3. neg-sub0 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{0 - v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. flip3-- N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\frac{{0}^{3} - {v}^{3}}{0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. associate-/r/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-neg.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{0} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   10. sub0-neg N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\mathsf{neg}\left({v}^{3}\right)}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. lower-neg.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\mathsf{neg}\left({v}^{3}\right)}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. cube-mult N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \color{blue}{\left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   15. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \left(\color{blue}{0} + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   16. +-lft-identity N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \color{blue}{\left(v \cdot v + 0 \cdot v\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   17. mul0-lft N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \left(v \cdot v + \color{blue}{0}\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   18. lower-fma.f32 98.5

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-v \cdot \left(v \cdot v\right)} \cdot \color{blue}{\mathsf{fma}\left(v, v, 0\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.5%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-v \cdot \left(v \cdot v\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(\color{blue}{cosTheta\_i \cdot cosTheta\_O}\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \color{blue}{\left(v \cdot v\right)}\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. frac-2neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(v \cdot v\right)}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(v \cdot v\right)}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. associate-/r* N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. associate-/l/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(v \cdot v\right) \cdot v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. associate-/r* N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. associate-/l* N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. lower-/.f32 98.4

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{v \cdot v}}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

6. Applied egg-rr 98.4%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v}} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

7. Taylor expanded in v around -inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)} \cdot 2\right) \cdot v} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right)} \cdot 2\right) \cdot v} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}} \cdot 2\right) \cdot v} \]

   3. neg-mul-1 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\color{blue}{-1 \cdot v}} \cdot 2\right) \cdot v} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{-1 \cdot v}} \cdot 2\right) \cdot v} \]

9. Simplified 71.2%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\frac{-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{-v}} \cdot 2\right) \cdot v} \]

10. Final simplification 71.2%

    \[\leadsto \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \left(\mathsf{fma}\left(v, v, 0\right) \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot v}}{v}\right)}{v \cdot \left(2 \cdot \frac{1 + \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{v}\right)} \]
11. Add Preprocessing

Alternative 11: 70.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \left(\mathsf{fma}\left(v, v, 0\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(v \cdot v\right)}\right)}{v \cdot \left(2 \cdot \frac{1 - \frac{-0.16666666666666666 + \frac{-0.008333333333333333}{v \cdot v}}{v \cdot v}}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (* (fma v v 0.0) (/ (* cosTheta_i cosTheta_O) (* v (* v v)))))
  (*
   v
   (*
    2.0
    (/
     (-
      1.0
      (/ (+ -0.16666666666666666 (/ -0.008333333333333333 (* v v))) (* v v)))
     v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(((sinTheta_O * -sinTheta_i) / v)) * (fmaf(v, v, 0.0f) * ((cosTheta_i * cosTheta_O) / (v * (v * v))))) / (v * (2.0f * ((1.0f - ((-0.16666666666666666f + (-0.008333333333333333f / (v * v))) / (v * v))) / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(fma(v, v, Float32(0.0)) * Float32(Float32(cosTheta_i * cosTheta_O) / Float32(v * Float32(v * v))))) / Float32(v * Float32(Float32(2.0) * Float32(Float32(Float32(1.0) - Float32(Float32(Float32(-0.16666666666666666) + Float32(Float32(-0.008333333333333333) / Float32(v * v))) / Float32(v * v))) / v))))
end

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. frac-2neg N/A

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

   3. neg-sub0 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{0 - v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. flip3-- N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\frac{{0}^{3} - {v}^{3}}{0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. associate-/r/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{{0}^{3} - {v}^{3}}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-neg.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}}{{0}^{3} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{0} - {v}^{3}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   10. sub0-neg N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\mathsf{neg}\left({v}^{3}\right)}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. lower-neg.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{\mathsf{neg}\left({v}^{3}\right)}} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. cube-mult N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(\color{blue}{v \cdot \left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \color{blue}{\left(v \cdot v\right)}\right)} \cdot \left(0 \cdot 0 + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   15. metadata-eval N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \left(\color{blue}{0} + \left(v \cdot v + 0 \cdot v\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   16. +-lft-identity N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \color{blue}{\left(v \cdot v + 0 \cdot v\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   17. mul0-lft N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \left(v \cdot v + \color{blue}{0}\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   18. lower-fma.f32 98.5

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-v \cdot \left(v \cdot v\right)} \cdot \color{blue}{\mathsf{fma}\left(v, v, 0\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.5%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-v \cdot \left(v \cdot v\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Taylor expanded in v around -inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)} \cdot 2\right) \cdot v} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)}{v}} \cdot 2\right) \cdot v} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)\right)}}{v} \cdot 2\right) \cdot v} \]

   3. sub-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{v} \cdot 2\right) \cdot v} \]

   4. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + \color{blue}{-1}\right)\right)}{v} \cdot 2\right) \cdot v} \]

   5. distribute-neg-in N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)}}{v} \cdot 2\right) \cdot v} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)}{v} \cdot 2\right) \cdot v} \]

   7. remove-double-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{\color{blue}{\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}} + \left(\mathsf{neg}\left(-1\right)\right)}{v} \cdot 2\right) \cdot v} \]

   8. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\frac{\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + \color{blue}{1}}{v} \cdot 2\right) \cdot v} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \left(\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\frac{\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + 1}{v}} \cdot 2\right) \cdot v} \]

7. Simplified 71.2%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{-cosTheta\_i \cdot cosTheta\_O}{-v \cdot \left(v \cdot v\right)} \cdot \mathsf{fma}\left(v, v, 0\right)\right)}{\left(\color{blue}{\frac{1 - \frac{-0.16666666666666666 + \frac{-0.008333333333333333}{v \cdot v}}{v \cdot v}}{v}} \cdot 2\right) \cdot v} \]

8. Final simplification 71.2%

   \[\leadsto \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \left(\mathsf{fma}\left(v, v, 0\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(v \cdot v\right)}\right)}{v \cdot \left(2 \cdot \frac{1 - \frac{-0.16666666666666666 + \frac{-0.008333333333333333}{v \cdot v}}{v \cdot v}}{v}\right)} \]
9. Add Preprocessing

Alternative 12: 70.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(((sintheta_o * -sintheta_i) / v)) * ((costheta_i * costheta_o) / v)) / (v / (0.5e0 / ((1.0e0 + ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v))) / v)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((sinTheta_O * -sinTheta_i) / v)) * ((cosTheta_i * cosTheta_O) / v)) / (v / (single(0.5) / ((single(1.0) + ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v))) / v)));
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-lft-identity N/A

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

   6. metadata-eval N/A

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

   7. associate-/r/ N/A

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

   8. div-inv N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. un-div-inv N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f32 N/A

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

   17. metadata-eval 98.5

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-/.f32 N/A

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

7. Simplified 71.2%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{v}{\frac{0.5}{\color{blue}{\frac{-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{-v}}}}} \]

8. Final simplification 71.2%

   \[\leadsto \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{v}{\frac{0.5}{\frac{1 + \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{v}}}} \]
9. Add Preprocessing

Alternative 13: 70.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft1-in N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-rgt-identity N/A

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

   10. lower-/.f32 N/A

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

5. Simplified 71.2%

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

6. Final simplification 71.2%

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

Alternative 14: 70.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. un-div-inv N/A

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

   8. clear-num N/A

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

   9. associate-*r/ N/A

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

   10. remove-double-div N/A

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

   11. lift-/.f32 N/A

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

   12. frac-times N/A

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

   13. div-inv N/A

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

   14. *-rgt-identity N/A

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

   15. lower-*.f32 N/A

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

4. Applied egg-rr 98.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 4\right) \cdot \frac{1}{\frac{2}{v}}}} \]

5. Taylor expanded in v around inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \left(\frac{\frac{1}{60}}{{v}^{4}} + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \left(\frac{\frac{1}{60}}{{v}^{4}} + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{v}^{2}} + \frac{\frac{1}{60}}{{v}^{4}}\right)}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{v}^{2}} + \frac{\frac{1}{60}}{{v}^{4}}\right)}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + \frac{\frac{1}{60}}{{v}^{4}}\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \left(\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + \frac{\frac{1}{60}}{{v}^{4}}\right)} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \left(\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + \frac{\frac{1}{60}}{{v}^{4}}\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \left(\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + \frac{\frac{1}{60}}{{v}^{4}}\right)} \]

   8. lower-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \left(\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + \frac{\frac{1}{60}}{{v}^{4}}\right)} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \left(\frac{\frac{1}{3}}{v \cdot v} + \color{blue}{\frac{\frac{1}{60}}{{v}^{4}}}\right)} \]

   10. metadata-eval N/A

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

   11. pow-sqr N/A

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

   12. lower-*.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f32 71.2

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

7. Simplified 71.2%

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

8. Final simplification 71.2%

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

Alternative 15: 64.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \frac{v \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) + v \cdot 0}{v \cdot v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (/ (+ (* v (* cosTheta_i cosTheta_O)) (* v 0.0)) (* v 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 (expf(((sinTheta_O * -sinTheta_i) / v)) * (((v * (cosTheta_i * cosTheta_O)) + (v * 0.0f)) / (v * v))) / (2.0f + (0.3333333333333333f / (v * v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(((sintheta_o * -sintheta_i) / v)) * (((v * (costheta_i * costheta_o)) + (v * 0.0e0)) / (v * v))) / (2.0e0 + (0.3333333333333333e0 / (v * v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(Float32(Float32(v * Float32(cosTheta_i * cosTheta_O)) + Float32(v * Float32(0.0))) / Float32(v * v))) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((sinTheta_O * -sinTheta_i) / v)) * (((v * (cosTheta_i * cosTheta_O)) + (v * single(0.0))) / (v * v))) / (single(2.0) + (single(0.3333333333333333) / (v * v)));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. lower-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 65.1

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

5. Simplified 65.1%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. frac-2neg N/A

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

   3. neg-sub0 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\color{blue}{0 - cosTheta\_i \cdot cosTheta\_O}}{\mathsf{neg}\left(v\right)}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   4. div-sub N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(v\right)} - \frac{cosTheta\_i \cdot cosTheta\_O}{\mathsf{neg}\left(v\right)}\right)}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   5. frac-sub N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(v\right)\right) - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(v\right)\right)}}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   6. sqr-neg N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{0 \cdot \left(\mathsf{neg}\left(v\right)\right) - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{v \cdot v}}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{0 \cdot \left(\mathsf{neg}\left(v\right)\right) - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{v \cdot v}}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(v\right)\right) - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v \cdot v}}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   9. lower--.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\color{blue}{0 \cdot \left(\mathsf{neg}\left(v\right)\right) - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}}{v \cdot v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{\color{blue}{0 \cdot \left(\mathsf{neg}\left(v\right)\right)} - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v \cdot v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   11. lower-neg.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{0 \cdot \color{blue}{\left(\mathsf{neg}\left(v\right)\right)} - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v \cdot v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{0 \cdot \left(\mathsf{neg}\left(v\right)\right) - \color{blue}{\left(\mathsf{neg}\left(v\right)\right) \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}}{v \cdot v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   13. lower-neg.f32 65.2

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 \cdot \left(-v\right) - \color{blue}{\left(-v\right)} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v \cdot v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]

7. Applied egg-rr 65.2%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{0 \cdot \left(-v\right) - \left(-v\right) \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v \cdot v}}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]

8. Final simplification 65.2%

   \[\leadsto \frac{e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}} \cdot \frac{v \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) + v \cdot 0}{v \cdot v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]
9. Add Preprocessing

Alternative 16: 64.0% accurate, 5.8× speedup

Math

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

FPCore

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. lower-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 65.1

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

5. Simplified 65.1%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. associate-/l/ N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. times-frac N/A

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

7. Applied egg-rr 65.2%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot cosTheta\_O}{\frac{0.3333333333333333}{v \cdot v} + 2} \cdot \frac{cosTheta\_i}{v}} \]

8. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 65.2

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

10. Simplified 65.2%

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

11. Final simplification 65.2%

    \[\leadsto \frac{cosTheta\_i}{v} \cdot \frac{cosTheta\_O}{2 + \frac{0.3333333333333333}{v \cdot v}} \]
12. Add Preprocessing

Alternative 17: 64.0% accurate, 6.6× speedup

Math

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

FPCore

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. lower-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 65.1

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

5. Simplified 65.1%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 65.2

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

8. Simplified 65.2%

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

9. Final simplification 65.2%

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

Alternative 18: 58.8% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

5. Simplified 59.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 59.6

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

8. Simplified 59.6%

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

   1. lift-*.f32 N/A

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

   2. associate-*r/ N/A

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

   3. clear-num N/A

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

   4. lower-/.f32 N/A

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

   5. *-rgt-identity N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 60.1

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

10. Applied egg-rr 60.1%

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

11. Final simplification 60.1%

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

Alternative 19: 58.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

5. Simplified 59.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 59.6

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

8. Simplified 59.6%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 60.1

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

10. Applied egg-rr 60.1%

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

Alternative 20: 58.3% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

5. Simplified 59.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 59.6

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

8. Simplified 59.6%

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

   1. lift-*.f32 N/A

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

   2. associate-*r/ N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 59.6

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

10. Applied egg-rr 59.6%

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

Alternative 21: 58.2% 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 98.5%

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

3. Taylor expanded in v around inf

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

5. Simplified 59.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 59.6

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

8. Simplified 59.6%

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

9. Final simplification 59.6%

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

Reproduce

herbie shell --seed 2024219 
(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)))