HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.4%

Time: 20.6s

Alternatives: 8

Speedup: N/A×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (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(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (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(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Alternative 1: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\\ t_1 := 0.6931 - \frac{1}{v}\\ \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{0.2222222222222222 \cdot \left(0.6931 + \left(\frac{-1}{v} + t_0\right)\right)}} \cdot {\left({\left(\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta_O, cosTheta_i, sinTheta_O \cdot \left(-sinTheta_i\right)\right)}{v} + t_1}}\right)}^{4}\right)}^{0.1111111111111111}\right) \cdot \sqrt[3]{\sqrt[3]{e^{t_1 + t_0}}}\right)}^{3} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (- (* cosTheta_O cosTheta_i) (* sinTheta_O sinTheta_i)) v))
        (t_1 (- 0.6931 (/ 1.0 v))))
   (*
    (/ 0.5 v)
    (pow
     (*
      (*
       (cbrt (exp (* 0.2222222222222222 (+ 0.6931 (+ (/ -1.0 v) t_0)))))
       (pow
        (pow
         (cbrt
          (exp
           (+
            (/ (fma cosTheta_O cosTheta_i (* sinTheta_O (- sinTheta_i))) v)
            t_1)))
         4.0)
        0.1111111111111111))
      (cbrt (cbrt (exp (+ t_1 t_0)))))
     3.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = ((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) / v;
	float t_1 = 0.6931f - (1.0f / v);
	return (0.5f / v) * powf(((cbrtf(expf((0.2222222222222222f * (0.6931f + ((-1.0f / v) + t_0))))) * powf(powf(cbrtf(expf(((fmaf(cosTheta_O, cosTheta_i, (sinTheta_O * -sinTheta_i)) / v) + t_1))), 4.0f), 0.1111111111111111f)) * cbrtf(cbrtf(expf((t_1 + t_0))))), 3.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(Float32(cosTheta_O * cosTheta_i) - Float32(sinTheta_O * sinTheta_i)) / v)
	t_1 = Float32(Float32(0.6931) - Float32(Float32(1.0) / v))
	return Float32(Float32(Float32(0.5) / v) * (Float32(Float32(cbrt(exp(Float32(Float32(0.2222222222222222) * Float32(Float32(0.6931) + Float32(Float32(Float32(-1.0) / v) + t_0))))) * ((cbrt(exp(Float32(Float32(fma(cosTheta_O, cosTheta_i, Float32(sinTheta_O * Float32(-sinTheta_i))) / v) + t_1))) ^ Float32(4.0)) ^ Float32(0.1111111111111111))) * cbrt(cbrt(exp(Float32(t_1 + t_0))))) ^ Float32(3.0)))
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.6%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.6%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{\log 1}} \]

   8. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   10. sub-neg 99.6%

       \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + 0.6931} \]

   11. associate-+l+ 99.6%

       \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(\left(-\frac{1}{v}\right) + 0.6931\right)}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)} \]

   2. pow3 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3}} \]

   3. associate-*l/ 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v}} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   4. *-commutative 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   5. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   6. sub-div 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v}} + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   7. +-commutative 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \color{blue}{\left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{3}} \]
7. Step-by-step derivation

   1. pow1/3 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]

   2. add-cube-cbrt 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\color{blue}{\left(\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right) \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}}^{0.3333333333333333}\right)}^{3} \]

   3. unpow-prod-down 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{0.3333333333333333}\right)}}^{3} \]

   4. pow2 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\color{blue}{\left({\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{2}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{0.3333333333333333}\right)}^{3} \]

   5. pow1/3 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\left({\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}}\right)}^{3} \]

8. Applied egg-rr 99.6%

   \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left({\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}}^{3} \]

10. Applied egg-rr 99.6%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\color{blue}{\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{\left(0.6931 + \frac{-1}{v}\right) + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, \left(-sinTheta_i\right) \cdot sinTheta_O\right)}{v}}}\right)}^{4}}} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \left(0.3333333333333333 \cdot \left(\left(0.6931 + \frac{-1}{v}\right) + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, \left(-sinTheta_i\right) \cdot sinTheta_O\right)}{v}\right)\right)}}\right)} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

12. Simplified 99.6%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\color{blue}{\left(\sqrt[3]{e^{0.2222222222222222 \cdot \left(0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}}}\right)} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]
13. Step-by-step derivation

    1. pow1/3 99.6%

       \[\leadsto \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{0.2222222222222222 \cdot \left(0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)\right)}} \cdot \color{blue}{{\left(\sqrt[3]{{\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

    2. pow1/3 99.6%

       \[\leadsto \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{0.2222222222222222 \cdot \left(0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)\right)}} \cdot {\color{blue}{\left({\left({\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}\right)}^{0.3333333333333333}\right)}}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

    3. pow-pow 99.6%

       \[\leadsto \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{0.2222222222222222 \cdot \left(0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)\right)}} \cdot \color{blue}{{\left({\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}\right)}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

14. Applied egg-rr 99.7%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{0.2222222222222222 \cdot \left(0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)\right)}} \cdot \color{blue}{{\left({\left(\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta_O, cosTheta_i, \left(-sinTheta_i\right) \cdot sinTheta_O\right)}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{4}\right)}^{0.1111111111111111}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

15. Final simplification 99.7%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{0.2222222222222222 \cdot \left(0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)\right)}} \cdot {\left({\left(\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta_O, cosTheta_i, sinTheta_O \cdot \left(-sinTheta_i\right)\right)}{v} + \left(0.6931 - \frac{1}{v}\right)}}\right)}^{4}\right)}^{0.1111111111111111}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\left(0.6931 - \frac{1}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}}}}\right)}^{3} \]
16. Add Preprocessing

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\\ t_1 := 0.6931 - \frac{1}{v}\\ \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt[3]{e^{t_1 + t_0}}} \cdot \left(\sqrt[3]{e^{0.2222222222222222 \cdot \left(t_1 - sinTheta_O \cdot \frac{sinTheta_i}{v}\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + t_0\right)}}\right)}^{4}}}\right)\right)}^{3} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (- (* cosTheta_O cosTheta_i) (* sinTheta_O sinTheta_i)) v))
        (t_1 (- 0.6931 (/ 1.0 v))))
   (*
    (/ 0.5 v)
    (pow
     (*
      (cbrt (cbrt (exp (+ t_1 t_0))))
      (*
       (cbrt
        (exp (* 0.2222222222222222 (- t_1 (* sinTheta_O (/ sinTheta_i v))))))
       (cbrt (cbrt (pow (cbrt (exp (+ 0.6931 (+ (/ -1.0 v) t_0)))) 4.0)))))
     3.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = ((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) / v;
	float t_1 = 0.6931f - (1.0f / v);
	return (0.5f / v) * powf((cbrtf(cbrtf(expf((t_1 + t_0)))) * (cbrtf(expf((0.2222222222222222f * (t_1 - (sinTheta_O * (sinTheta_i / v)))))) * cbrtf(cbrtf(powf(cbrtf(expf((0.6931f + ((-1.0f / v) + t_0)))), 4.0f))))), 3.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(Float32(cosTheta_O * cosTheta_i) - Float32(sinTheta_O * sinTheta_i)) / v)
	t_1 = Float32(Float32(0.6931) - Float32(Float32(1.0) / v))
	return Float32(Float32(Float32(0.5) / v) * (Float32(cbrt(cbrt(exp(Float32(t_1 + t_0)))) * Float32(cbrt(exp(Float32(Float32(0.2222222222222222) * Float32(t_1 - Float32(sinTheta_O * Float32(sinTheta_i / v)))))) * cbrt(cbrt((cbrt(exp(Float32(Float32(0.6931) + Float32(Float32(Float32(-1.0) / v) + t_0)))) ^ Float32(4.0)))))) ^ Float32(3.0)))
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.6%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.6%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{\log 1}} \]

   8. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   10. sub-neg 99.6%

       \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + 0.6931} \]

   11. associate-+l+ 99.6%

       \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(\left(-\frac{1}{v}\right) + 0.6931\right)}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)} \]

   2. pow3 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3}} \]

   3. associate-*l/ 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v}} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   4. *-commutative 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   5. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   6. sub-div 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v}} + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   7. +-commutative 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \color{blue}{\left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{3}} \]
7. Step-by-step derivation

   1. pow1/3 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]

   2. add-cube-cbrt 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\color{blue}{\left(\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right) \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}}^{0.3333333333333333}\right)}^{3} \]

   3. unpow-prod-down 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{0.3333333333333333}\right)}}^{3} \]

   4. pow2 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\color{blue}{\left({\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{2}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{0.3333333333333333}\right)}^{3} \]

   5. pow1/3 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\left({\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}}\right)}^{3} \]

8. Applied egg-rr 99.6%

   \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left({\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}}^{3} \]

10. Applied egg-rr 99.6%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\color{blue}{\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{\left(0.6931 + \frac{-1}{v}\right) + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, \left(-sinTheta_i\right) \cdot sinTheta_O\right)}{v}}}\right)}^{4}}} \cdot \sqrt[3]{e^{0.6666666666666666 \cdot \left(0.3333333333333333 \cdot \left(\left(0.6931 + \frac{-1}{v}\right) + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, \left(-sinTheta_i\right) \cdot sinTheta_O\right)}{v}\right)\right)}}\right)} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

12. Simplified 99.6%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\color{blue}{\left(\sqrt[3]{e^{0.2222222222222222 \cdot \left(0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}}}\right)} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

13. Taylor expanded in cosTheta_O around 0 99.6%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{\color{blue}{0.2222222222222222 \cdot \left(0.6931 - \left(\frac{1}{v} + \frac{sinTheta_O \cdot sinTheta_i}{v}\right)\right)}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]
14. Step-by-step derivation

    1. *-commutative 99.6%

       \[\leadsto \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{\color{blue}{\left(0.6931 - \left(\frac{1}{v} + \frac{sinTheta_O \cdot sinTheta_i}{v}\right)\right) \cdot 0.2222222222222222}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

    2. associate--r+ 99.6%

       \[\leadsto \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{\color{blue}{\left(\left(0.6931 - \frac{1}{v}\right) - \frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \cdot 0.2222222222222222}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

    3. *-commutative 99.6%

       \[\leadsto \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{\left(\left(0.6931 - \frac{1}{v}\right) - \frac{\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}\right) \cdot 0.2222222222222222}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

    4. associate-*l/ 99.6%

       \[\leadsto \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{\left(\left(0.6931 - \frac{1}{v}\right) - \color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}\right) \cdot 0.2222222222222222}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

    5. *-commutative 99.6%

       \[\leadsto \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{\left(\left(0.6931 - \frac{1}{v}\right) - \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}\right) \cdot 0.2222222222222222}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

15. Simplified 99.6%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\left(\sqrt[3]{e^{\color{blue}{\left(\left(0.6931 - \frac{1}{v}\right) - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) \cdot 0.2222222222222222}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

16. Final simplification 99.6%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt[3]{e^{\left(0.6931 - \frac{1}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}}}} \cdot \left(\sqrt[3]{e^{0.2222222222222222 \cdot \left(\left(0.6931 - \frac{1}{v}\right) - sinTheta_O \cdot \frac{sinTheta_i}{v}\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{0.6931 + \left(\frac{-1}{v} + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)}^{4}}}\right)\right)}^{3} \]
17. Add Preprocessing

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{\left(0.6931 - \frac{1}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}}}\\ \frac{0.5}{v} \cdot {\left(\sqrt[3]{t_0} \cdot {\left({t_0}^{2}\right)}^{0.3333333333333333}\right)}^{3} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (cbrt
          (exp
           (+
            (- 0.6931 (/ 1.0 v))
            (/ (- (* cosTheta_O cosTheta_i) (* sinTheta_O sinTheta_i)) v))))))
   (*
    (/ 0.5 v)
    (pow (* (cbrt t_0) (pow (pow t_0 2.0) 0.3333333333333333)) 3.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = cbrtf(expf(((0.6931f - (1.0f / v)) + (((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) / v))));
	return (0.5f / v) * powf((cbrtf(t_0) * powf(powf(t_0, 2.0f), 0.3333333333333333f)), 3.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cbrt(exp(Float32(Float32(Float32(0.6931) - Float32(Float32(1.0) / v)) + Float32(Float32(Float32(cosTheta_O * cosTheta_i) - Float32(sinTheta_O * sinTheta_i)) / v))))
	return Float32(Float32(Float32(0.5) / v) * (Float32(cbrt(t_0) * ((t_0 ^ Float32(2.0)) ^ Float32(0.3333333333333333))) ^ Float32(3.0)))
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.6%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.6%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{\log 1}} \]

   8. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   10. sub-neg 99.6%

       \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + 0.6931} \]

   11. associate-+l+ 99.6%

       \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(\left(-\frac{1}{v}\right) + 0.6931\right)}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)} \]

   2. pow3 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3}} \]

   3. associate-*l/ 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v}} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   4. *-commutative 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   5. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   6. sub-div 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v}} + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   7. +-commutative 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \color{blue}{\left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{3}} \]
7. Step-by-step derivation

   1. pow1/3 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]

   2. add-cube-cbrt 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\color{blue}{\left(\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right) \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}}^{0.3333333333333333}\right)}^{3} \]

   3. unpow-prod-down 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{0.3333333333333333}\right)}}^{3} \]

   4. pow2 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\color{blue}{\left({\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{2}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{0.3333333333333333}\right)}^{3} \]

   5. pow1/3 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\left({\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}}\right)}^{3} \]

8. Applied egg-rr 99.6%

   \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left({\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}}^{3} \]

9. Final simplification 99.6%

   \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt[3]{e^{\left(0.6931 - \frac{1}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}}}} \cdot {\left({\left(\sqrt[3]{e^{\left(0.6931 - \frac{1}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}}}\right)}^{2}\right)}^{0.3333333333333333}\right)}^{3} \]
10. Add Preprocessing

Alternative 4: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(0.6931 - \frac{1}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}}}\right)}^{3} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ 0.5 v)
  (pow
   (cbrt
    (exp
     (+
      (- 0.6931 (/ 1.0 v))
      (/ (- (* cosTheta_O cosTheta_i) (* sinTheta_O sinTheta_i)) v))))
   3.0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * powf(cbrtf(expf(((0.6931f - (1.0f / v)) + (((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) / v)))), 3.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * (cbrt(exp(Float32(Float32(Float32(0.6931) - Float32(Float32(1.0) / v)) + Float32(Float32(Float32(cosTheta_O * cosTheta_i) - Float32(sinTheta_O * sinTheta_i)) / v)))) ^ Float32(3.0)))
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.6%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.6%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{\log 1}} \]

   8. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   10. sub-neg 99.6%

       \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + 0.6931} \]

   11. associate-+l+ 99.6%

       \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(\left(-\frac{1}{v}\right) + 0.6931\right)}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)} \]

   2. pow3 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3}} \]

   3. associate-*l/ 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v}} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   4. *-commutative 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   5. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   6. sub-div 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v}} + \left(\frac{-1}{v} + 0.6931\right)}}\right)}^{3} \]

   7. +-commutative 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \color{blue}{\left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3} \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}}\right)}^{3}} \]

7. Final simplification 99.6%

   \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(0.6931 - \frac{1}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}}}\right)}^{3} \]
8. Add Preprocessing

Alternative 5: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot {e}^{\left(\left(0.6931 - \frac{1}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ 0.5 v)
  (pow
   E
   (+
    (- 0.6931 (/ 1.0 v))
    (/ (- (* cosTheta_O cosTheta_i) (* sinTheta_O sinTheta_i)) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * powf(((float) M_E), ((0.6931f - (1.0f / v)) + (((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * (Float32(exp(1)) ^ Float32(Float32(Float32(0.6931) - Float32(Float32(1.0) / v)) + Float32(Float32(Float32(cosTheta_O * cosTheta_i) - Float32(sinTheta_O * sinTheta_i)) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (single(2.71828182845904523536) ^ ((single(0.6931) - (single(1.0) / v)) + (((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) / v)));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.6%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.6%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{\log 1}} \]

   8. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   10. sub-neg 99.6%

       \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + 0.6931} \]

   11. associate-+l+ 99.6%

       \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(\left(-\frac{1}{v}\right) + 0.6931\right)}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{1 \cdot \left(\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)\right)}} \]

   2. exp-prod 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_O}{v} \cdot cosTheta_i - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)\right)}} \]

   3. associate-*l/ 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v}} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)\right)} \]

   4. *-commutative 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(e^{1}\right)}^{\left(\left(\frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)\right)} \]

   5. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right) + \left(\frac{-1}{v} + 0.6931\right)\right)} \]

   6. sub-div 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(e^{1}\right)}^{\left(\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v}} + \left(\frac{-1}{v} + 0.6931\right)\right)} \]

   7. +-commutative 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(e^{1}\right)}^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \color{blue}{\left(0.6931 + \frac{-1}{v}\right)}\right)} \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right)}} \]

7. Final simplification 99.6%

   \[\leadsto \frac{0.5}{v} \cdot {e}^{\left(\left(0.6931 - \frac{1}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v}\right)} \]
8. Add Preprocessing

Alternative 6: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{0.6931 - \frac{1}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (- 0.6931 (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf((0.6931f - (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 = (0.5e0 / v) * exp((0.6931e0 - (1.0e0 / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) - Float32(Float32(1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((single(0.6931) - (single(1.0) / v)));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in sinTheta_i around 0 99.6%

   \[\leadsto \color{blue}{e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i}{v}\right)\right) - \frac{1}{v}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 99.6%

      \[\leadsto e^{\color{blue}{1 \cdot \left(\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i}{v}\right)\right) - \frac{1}{v}\right)}} \]

   2. exp-prod 99.6%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i}{v}\right)\right) - \frac{1}{v}\right)}} \]

   3. +-commutative 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(0.6931 + \color{blue}{\left(\frac{cosTheta_O \cdot cosTheta_i}{v} + \log \left(\frac{0.5}{v}\right)\right)}\right) - \frac{1}{v}\right)} \]

   4. associate-/l* 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(0.6931 + \left(\color{blue}{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}} + \log \left(\frac{0.5}{v}\right)\right)\right) - \frac{1}{v}\right)} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(0.6931 + \left(\frac{cosTheta_O}{\frac{v}{cosTheta_i}} + \log \left(\frac{0.5}{v}\right)\right)\right) - \frac{1}{v}\right)}} \]

6. Taylor expanded in cosTheta_O around 0 99.6%

   \[\leadsto \color{blue}{e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}}} \]
7. Step-by-step derivation

   1. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \left(-\frac{1}{v}\right)}} \]

   2. exp-sum 99.2%

      \[\leadsto \color{blue}{e^{0.6931 + \log \left(\frac{0.5}{v}\right)} \cdot e^{-\frac{1}{v}}} \]

   3. prod-exp 99.2%

      \[\leadsto \color{blue}{\left(e^{0.6931} \cdot e^{\log \left(\frac{0.5}{v}\right)}\right)} \cdot e^{-\frac{1}{v}} \]

   4. rem-exp-log 99.2%

      \[\leadsto \left(e^{0.6931} \cdot \color{blue}{\frac{0.5}{v}}\right) \cdot e^{-\frac{1}{v}} \]

   5. *-commutative 99.2%

      \[\leadsto \color{blue}{\left(\frac{0.5}{v} \cdot e^{0.6931}\right)} \cdot e^{-\frac{1}{v}} \]

   6. associate-*l* 99.6%

      \[\leadsto \color{blue}{\frac{0.5}{v} \cdot \left(e^{0.6931} \cdot e^{-\frac{1}{v}}\right)} \]

   7. distribute-neg-frac 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \left(e^{0.6931} \cdot e^{\color{blue}{\frac{-1}{v}}}\right) \]

   8. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \left(e^{0.6931} \cdot e^{\frac{\color{blue}{-1}}{v}}\right) \]

   9. exp-sum 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{0.6931 + \frac{-1}{v}}} \]

8. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}}} \]

9. Final simplification 99.6%

   \[\leadsto \frac{0.5}{v} \cdot e^{0.6931 - \frac{1}{v}} \]
10. Add Preprocessing

Alternative 7: 97.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ e^{\frac{-1}{v}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((-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(((-1.0e0) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in sinTheta_i around 0 99.6%

   \[\leadsto \color{blue}{e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_O \cdot cosTheta_i}{v}\right)\right) - \frac{1}{v}}} \]

4. Taylor expanded in cosTheta_O around 0 99.6%

   \[\leadsto \color{blue}{e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}}} \]

5. Taylor expanded in v around 0 97.2%

   \[\leadsto e^{\color{blue}{\frac{-1}{v}}} \]

6. Final simplification 97.2%

   \[\leadsto e^{\frac{-1}{v}} \]
7. Add Preprocessing

Alternative 8: 6.4% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

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 = 1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in v around 0 97.2%

   \[\leadsto e^{\color{blue}{\frac{cosTheta_O \cdot cosTheta_i - \left(1 + sinTheta_O \cdot sinTheta_i\right)}{v}}} \]

4. Taylor expanded in v around inf 6.4%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 6.4%

   \[\leadsto 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (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))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))