HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 13.6s

Alternatives: 10

Speedup: 0.4×

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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \frac{-1}{v}}}}\\ \frac{0.5}{v} \cdot {\left(t\_0 \cdot t\_0\right)}^{3} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. +-commutative 99.8%

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

   2. exp-sum 99.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.9%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 99.9%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \cdot \sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}}\right) \cdot \sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}}\right)} \]

   2. pow3 99.9%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}}\right)}^{3}} \]

   3. fma-define 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)}\right) + 0.6931}}\right)}^{3} \]

   4. associate--r+ 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \frac{1}{v}\right)} + 0.6931}}\right)}^{3} \]

   5. associate-+l- 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \left(\frac{1}{v} - 0.6931\right)}}}\right)}^{3} \]

   6. associate-*r/ 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \left(\frac{1}{v} - 0.6931\right)}}\right)}^{3} \]

   7. associate-*r/ 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) - \left(\frac{1}{v} - 0.6931\right)}}\right)}^{3} \]

   8. sub-div 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v}} - \left(\frac{1}{v} - 0.6931\right)}}\right)}^{3} \]

6. Applied egg-rr 99.9%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}}\right)}^{3}} \]
7. Step-by-step derivation

   1. pow1/3 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]

   2. add-sqr-sqrt 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\color{blue}{\left(\sqrt{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}} \cdot \sqrt{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}}\right)}}^{0.3333333333333333}\right)}^{3} \]

   3. unpow-prod-down 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(\sqrt{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}}\right)}^{0.3333333333333333}\right)}}^{3} \]

   4. associate--r- 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\left(\sqrt{e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}}\right)}^{0.3333333333333333}\right)}^{3} \]

   5. sub-div 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\left(\sqrt{e^{\color{blue}{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}} + 0.6931}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}}\right)}^{0.3333333333333333}\right)}^{3} \]

8. Applied egg-rr 99.8%

   \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(\sqrt{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{0.3333333333333333}\right)}}^{3} \]
9. Step-by-step derivation

   1. unpow1/3 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\color{blue}{\sqrt[3]{\sqrt{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}}} \cdot {\left(\sqrt{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{0.3333333333333333}\right)}^{3} \]

   2. +-commutative 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt{e^{\color{blue}{0.6931 + \frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}}}}} \cdot {\left(\sqrt{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{0.3333333333333333}\right)}^{3} \]

   3. associate--l- 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt{e^{0.6931 + \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}}{v}}}} \cdot {\left(\sqrt{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{0.3333333333333333}\right)}^{3} \]

   4. *-commutative 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt{e^{0.6931 + \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i} - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v}}}} \cdot {\left(\sqrt{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{0.3333333333333333}\right)}^{3} \]

   5. *-commutative 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt{e^{0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i - \left(\color{blue}{sinTheta\_O \cdot sinTheta\_i} + 1\right)}{v}}}} \cdot {\left(\sqrt{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{0.3333333333333333}\right)}^{3} \]

   6. +-commutative 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt{e^{0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i - \color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)}}{v}}}} \cdot {\left(\sqrt{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{0.3333333333333333}\right)}^{3} \]

   7. associate--r+ 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt{e^{0.6931 + \frac{\color{blue}{\left(cosTheta\_O \cdot cosTheta\_i - 1\right) - sinTheta\_O \cdot sinTheta\_i}}{v}}}} \cdot {\left(\sqrt{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{0.3333333333333333}\right)}^{3} \]

   8. fma-neg 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt{e^{0.6931 + \frac{\color{blue}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)} - sinTheta\_O \cdot sinTheta\_i}{v}}}} \cdot {\left(\sqrt{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{0.3333333333333333}\right)}^{3} \]

   9. metadata-eval 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, \color{blue}{-1}\right) - sinTheta\_O \cdot sinTheta\_i}{v}}}} \cdot {\left(\sqrt{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{0.3333333333333333}\right)}^{3} \]

10. Simplified 99.9%

    \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right) - sinTheta\_O \cdot sinTheta\_i}{v}}}} \cdot \sqrt[3]{\sqrt{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right) - sinTheta\_O \cdot sinTheta\_i}{v}}}}\right)}}^{3} \]

11. Taylor expanded in sinTheta_O around 0 99.9%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\color{blue}{\sqrt{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}}}} \cdot \sqrt[3]{\sqrt{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right) - sinTheta\_O \cdot sinTheta\_i}{v}}}}\right)}^{3} \]

12. Taylor expanded in sinTheta_O around 0 99.9%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}}} \cdot \sqrt[3]{\color{blue}{\sqrt{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}}}}\right)}^{3} \]

13. Final simplification 99.9%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\sqrt{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \frac{-1}{v}}}} \cdot \sqrt[3]{\sqrt{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \frac{-1}{v}}}}\right)}^{3} \]
14. Add Preprocessing

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot {\left({\left(\sqrt[3]{\sqrt[3]{e^{0.6931 + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_i \cdot sinTheta\_O\right) + -1}{v}}}}\right)}^{3}\right)}^{3} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. +-commutative 99.8%

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

   2. exp-sum 99.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.9%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 99.9%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \cdot \sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}}\right) \cdot \sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}}\right)} \]

   2. pow3 99.9%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}}\right)}^{3}} \]

   3. fma-define 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)}\right) + 0.6931}}\right)}^{3} \]

   4. associate--r+ 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \frac{1}{v}\right)} + 0.6931}}\right)}^{3} \]

   5. associate-+l- 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \left(\frac{1}{v} - 0.6931\right)}}}\right)}^{3} \]

   6. associate-*r/ 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \left(\frac{1}{v} - 0.6931\right)}}\right)}^{3} \]

   7. associate-*r/ 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) - \left(\frac{1}{v} - 0.6931\right)}}\right)}^{3} \]

   8. sub-div 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v}} - \left(\frac{1}{v} - 0.6931\right)}}\right)}^{3} \]

6. Applied egg-rr 99.9%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}}\right)}^{3}} \]
7. Step-by-step derivation

   1. add-cube-cbrt 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}}} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}}}\right)}}^{3} \]

   2. pow3 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)}}}\right)}^{3}\right)}}^{3} \]

   3. associate--r- 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\left(\sqrt[3]{\sqrt[3]{e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}}}}\right)}^{3}\right)}^{3} \]

   4. sub-div 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left({\left(\sqrt[3]{\sqrt[3]{e^{\color{blue}{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}} + 0.6931}}}\right)}^{3}\right)}^{3} \]

8. Applied egg-rr 99.9%

   \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}}\right)}^{3}\right)}}^{3} \]

9. Final simplification 99.9%

   \[\leadsto \frac{0.5}{v} \cdot {\left({\left(\sqrt[3]{\sqrt[3]{e^{0.6931 + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_i \cdot sinTheta\_O\right) + -1}{v}}}}\right)}^{3}\right)}^{3} \]
10. Add Preprocessing

Alternative 3: 99.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (pow (cbrt (exp (+ 0.6931 (/ -1.0 v)))) 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)))), 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(0.6931) + Float32(Float32(-1.0) / v)))) ^ Float32(3.0)))
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. +-commutative 99.8%

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

   2. exp-sum 99.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.9%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 99.9%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}} \]
6. Step-by-step derivation

   1. add-cube-cbrt 99.9%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}} \cdot \sqrt[3]{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}}\right) \cdot \sqrt[3]{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}}\right)} \]

   2. pow3 99.9%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}}\right)}^{3}} \]

   3. associate--l+ 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{0.6931 + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)}}}\right)}^{3} \]

   4. *-commutative 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{0.6931 + \left(\frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} - \frac{1}{v}\right)}}\right)}^{3} \]

   5. sub-div 99.9%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{0.6931 + \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}}}}\right)}^{3} \]

7. Applied egg-rr 99.9%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{0.6931 + \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}}}\right)}^{3}} \]

8. Taylor expanded in cosTheta_i around 0 99.9%

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

9. Final simplification 99.9%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (/ (+ (* v 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((((v * 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((((v * 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(Float32(v * Float32(0.6931)) + Float32(-1.0)) / v)))
end

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. +-commutative 99.8%

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

   2. exp-sum 99.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.9%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 99.9%

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

6. Taylor expanded in cosTheta_O around 0 99.9%

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

7. Taylor expanded in v around 0 99.9%

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

8. Final simplification 99.9%

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

Alternative 5: 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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. +-commutative 99.8%

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

   2. exp-sum 99.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.9%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 99.9%

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

6. Taylor expanded in cosTheta_O around 0 99.9%

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

7. Final simplification 99.9%

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

Alternative 6: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. +-commutative 99.8%

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

   2. exp-sum 99.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.9%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 99.9%

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

6. Taylor expanded in cosTheta_O around 0 99.9%

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

7. Taylor expanded in v around 0 98.6%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\frac{-1}{v}}} \]
8. Add Preprocessing

Alternative 7: 98.0% 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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.8%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.8%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in cosTheta_O around 0 99.8%

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

7. Taylor expanded in v around 0 98.5%

   \[\leadsto e^{\color{blue}{\frac{-1}{v}}} \]
8. Add Preprocessing

Alternative 8: 38.9% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta\_i \cdot sinTheta\_O}{-v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (sinTheta_i * sinTheta_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 = (sintheta_i * sintheta_o) / -v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.8%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.8%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 15.0%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
6. Step-by-step derivation

   1. mul-1-neg 15.0%

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

   2. *-commutative 15.0%

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

   3. associate-*r/ 15.0%

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

7. Simplified 15.0%

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

8. Taylor expanded in sinTheta_i around 0 6.2%

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

   1. neg-mul-1 6.2%

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

   2. *-commutative 6.2%

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

   3. associate-*r/ 6.2%

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

   4. unsub-neg 6.2%

      \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

   5. associate-*r/ 6.2%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

   6. *-commutative 6.2%

      \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

   7. associate-/l* 6.2%

      \[\leadsto 1 - \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

10. Simplified 6.2%

    \[\leadsto \color{blue}{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

11. Taylor expanded in sinTheta_O around inf 40.8%

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

    1. associate-*r/ 40.8%

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

    2. neg-mul-1 40.8%

       \[\leadsto \frac{\color{blue}{-sinTheta\_O \cdot sinTheta\_i}}{v} \]

    3. distribute-lft-neg-in 40.8%

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

13. Simplified 40.8%

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

14. Final simplification 40.8%

    \[\leadsto \frac{sinTheta\_i \cdot sinTheta\_O}{-v} \]
15. Add Preprocessing

Alternative 9: 20.1% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ sinTheta\_O \cdot \frac{sinTheta\_i}{-v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.8%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.8%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 15.0%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
6. Step-by-step derivation

   1. mul-1-neg 15.0%

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

   2. *-commutative 15.0%

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

   3. associate-*r/ 15.0%

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

7. Simplified 15.0%

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

8. Taylor expanded in sinTheta_i around 0 6.2%

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

   1. neg-mul-1 6.2%

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

   2. *-commutative 6.2%

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

   3. associate-*r/ 6.2%

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

   4. unsub-neg 6.2%

      \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

   5. associate-*r/ 6.2%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

   6. *-commutative 6.2%

      \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

   7. associate-/l* 6.2%

      \[\leadsto 1 - \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

10. Simplified 6.2%

    \[\leadsto \color{blue}{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

11. Taylor expanded in sinTheta_O around inf 40.8%

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

    1. associate-*r/ 20.0%

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

    2. neg-mul-1 20.0%

       \[\leadsto \color{blue}{-sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

    3. distribute-rgt-neg-in 20.0%

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

    4. distribute-neg-frac2 20.0%

       \[\leadsto sinTheta\_O \cdot \color{blue}{\frac{sinTheta\_i}{-v}} \]

13. Simplified 20.0%

    \[\leadsto \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}} \]
14. Add Preprocessing

Alternative 10: 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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.8%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.8%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 15.0%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
6. Step-by-step derivation

   1. mul-1-neg 15.0%

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

   2. *-commutative 15.0%

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

   3. associate-*r/ 15.0%

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

7. Simplified 15.0%

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

8. Taylor expanded in sinTheta_i around 0 6.4%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024098 
(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))))))