HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 14.6s

Alternatives: 9

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: 99.6% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (cbrt (sqrt (exp (/ (fma cosTheta_O cosTheta_i -1.0) v))))))
   (* (/ 0.5 v) (* (exp 0.6931) (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((fmaf(cosTheta_O, cosTheta_i, -1.0f) / v))));
	return (0.5f / v) * (expf(0.6931f) * 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(fma(cosTheta_O, cosTheta_i, Float32(-1.0)) / v))))
	return Float32(Float32(Float32(0.5) / v) * Float32(exp(Float32(0.6931)) * (Float32(t_0 * 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 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. +-rgt-identity 99.7%

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

   7. metadata-eval 99.7%

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

   8. metadata-eval 99.7%

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

   9. +-rgt-identity 99.7%

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

   10. sub-neg 99.7%

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

   11. associate-+l+ 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

   1. associate--l+ 99.7%

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

   2. associate-*r/ 99.7%

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

   3. exp-sum 99.7%

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

   4. associate-*r/ 99.7%

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

   5. sub-div 99.7%

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

7. Applied egg-rr 99.7%

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

   1. add-cube-cbrt 99.7%

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

   2. pow3 99.7%

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

   3. div-inv 99.7%

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

   4. div-inv 99.7%

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

   5. fma-neg 99.7%

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

   6. metadata-eval 99.7%

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

9. Applied egg-rr 99.7%

   \[\leadsto \frac{0.5}{v} \cdot \left(e^{0.6931} \cdot \color{blue}{{\left(\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta_O, cosTheta_i, -1\right)}{v}}}\right)}^{3}}\right) \]
10. Step-by-step derivation

    1. pow1/3 99.5%

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

    2. rem-cube-cbrt 99.5%

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

    3. add-sqr-sqrt 99.5%

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

    4. unpow-prod-down 99.5%

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

    5. rem-cube-cbrt 99.5%

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

    6. rem-cube-cbrt 99.5%

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

11. Applied egg-rr 99.5%

    \[\leadsto \frac{0.5}{v} \cdot \left(e^{0.6931} \cdot {\color{blue}{\left({\left(\sqrt{e^{\frac{\mathsf{fma}\left(cosTheta_O, cosTheta_i, -1\right)}{v}}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{e^{\frac{\mathsf{fma}\left(cosTheta_O, cosTheta_i, -1\right)}{v}}}\right)}^{0.3333333333333333}\right)}}^{3}\right) \]
12. Step-by-step derivation

    1. unpow1/3 99.6%

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

    2. unpow1/3 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

    \[\leadsto \frac{0.5}{v} \cdot \left(e^{0.6931} \cdot {\left(\sqrt[3]{\sqrt{e^{\frac{\mathsf{fma}\left(cosTheta_O, cosTheta_i, -1\right)}{v}}}} \cdot \sqrt[3]{\sqrt{e^{\frac{\mathsf{fma}\left(cosTheta_O, cosTheta_i, -1\right)}{v}}}}\right)}^{3}\right) \]
15. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (* (exp 0.6931) (pow (cbrt (exp (/ -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) * (expf(0.6931f) * powf(cbrtf(expf((-1.0f / v))), 3.0f));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * Float32(exp(Float32(0.6931)) * (cbrt(exp(Float32(Float32(-1.0) / 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 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. +-rgt-identity 99.7%

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

   7. metadata-eval 99.7%

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

   8. metadata-eval 99.7%

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

   9. +-rgt-identity 99.7%

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

   10. sub-neg 99.7%

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

   11. associate-+l+ 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

   1. associate--l+ 99.7%

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

   2. associate-*r/ 99.7%

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

   3. exp-sum 99.7%

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

   4. associate-*r/ 99.7%

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

   5. sub-div 99.7%

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

7. Applied egg-rr 99.7%

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

   1. add-cube-cbrt 99.7%

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

   2. pow3 99.7%

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

   3. div-inv 99.7%

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

   4. div-inv 99.7%

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

   5. fma-neg 99.7%

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

   6. metadata-eval 99.7%

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

9. Applied egg-rr 99.7%

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

10. Taylor expanded in cosTheta_O around 0 99.5%

    \[\leadsto \frac{0.5}{v} \cdot \left(e^{0.6931} \cdot {\color{blue}{\left({\left(e^{\frac{-1}{v}}\right)}^{0.3333333333333333}\right)}}^{3}\right) \]
11. Step-by-step derivation

    1. unpow1/3 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (* (exp 0.6931) (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(0.6931f) * 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(0.6931e0) * 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) * Float32(exp(Float32(0.6931)) * 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(0.6931)) * 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. 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 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. +-rgt-identity 99.7%

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

   7. metadata-eval 99.7%

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

   8. metadata-eval 99.7%

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

   9. +-rgt-identity 99.7%

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

   10. sub-neg 99.7%

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

   11. associate-+l+ 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

   1. associate--l+ 99.7%

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

   2. associate-*r/ 99.7%

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

   3. exp-sum 99.7%

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

   4. associate-*r/ 99.7%

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

   5. sub-div 99.7%

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in cosTheta_O around 0 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 36.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{-sinTheta_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_i 4.999999999099794e-24)
   (/ (* sinTheta_i (- sinTheta_O)) v)
   (exp (* sinTheta_O (/ (- sinTheta_i) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_i <= 4.999999999099794e-24f) {
		tmp = (sinTheta_i * -sinTheta_O) / v;
	} else {
		tmp = expf((sinTheta_O * (-sinTheta_i / v)));
	}
	return tmp;
}

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
    real(4) :: tmp
    if (sintheta_i <= 4.999999999099794e-24) then
        tmp = (sintheta_i * -sintheta_o) / v
    else
        tmp = exp((sintheta_o * (-sintheta_i / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_i <= Float32(4.999999999099794e-24))
		tmp = Float32(Float32(sinTheta_i * Float32(-sinTheta_O)) / v);
	else
		tmp = exp(Float32(sinTheta_O * Float32(Float32(-sinTheta_i) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_i <= single(4.999999999099794e-24))
		tmp = (sinTheta_i * -sinTheta_O) / v;
	else
		tmp = exp((sinTheta_O * (-sinTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_i < 5e-24

   1. Initial program 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in sinTheta_i around inf 11.2%

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

   6. Taylor expanded in sinTheta_O around 0 6.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
   7. Step-by-step derivation

      1. mul-1-neg 6.4%

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

      2. associate-/l* 6.4%

         \[\leadsto 1 + \left(-\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right) \]

      3. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

      4. associate-/l* 6.4%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

      5. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

   8. Simplified 6.4%

      \[\leadsto \color{blue}{1 - sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

   9. Taylor expanded in sinTheta_O around inf 44.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   if 5e-24 < sinTheta_i

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in sinTheta_i around inf 19.1%

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

      1. associate-*r/ 19.1%

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

      2. neg-mul-1 19.1%

         \[\leadsto e^{\color{blue}{-sinTheta_O \cdot \frac{sinTheta_i}{v}}} \]

      3. *-commutative 19.1%

         \[\leadsto e^{-\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}} \]

      4. distribute-rgt-neg-in 19.1%

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

   7. Simplified 19.1%

      \[\leadsto e^{\color{blue}{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{-sinTheta_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 36.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}\\ \mathbf{if}\;sinTheta_i \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* sinTheta_i (- sinTheta_O)) v)))
   (if (<= sinTheta_i 4.999999999099794e-24) t_0 (exp t_0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (sinTheta_i * -sinTheta_O) / v;
	float tmp;
	if (sinTheta_i <= 4.999999999099794e-24f) {
		tmp = t_0;
	} else {
		tmp = expf(t_0);
	}
	return tmp;
}

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
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (sintheta_i * -sintheta_o) / v
    if (sintheta_i <= 4.999999999099794e-24) then
        tmp = t_0
    else
        tmp = exp(t_0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(sinTheta_i * Float32(-sinTheta_O)) / v)
	tmp = Float32(0.0)
	if (sinTheta_i <= Float32(4.999999999099794e-24))
		tmp = t_0;
	else
		tmp = exp(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (sinTheta_i * -sinTheta_O) / v;
	tmp = single(0.0);
	if (sinTheta_i <= single(4.999999999099794e-24))
		tmp = t_0;
	else
		tmp = exp(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_i < 5e-24

   1. Initial program 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in sinTheta_i around inf 11.2%

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

   6. Taylor expanded in sinTheta_O around 0 6.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
   7. Step-by-step derivation

      1. mul-1-neg 6.4%

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

      2. associate-/l* 6.4%

         \[\leadsto 1 + \left(-\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right) \]

      3. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

      4. associate-/l* 6.4%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

      5. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

   8. Simplified 6.4%

      \[\leadsto \color{blue}{1 - sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

   9. Taylor expanded in sinTheta_O around inf 44.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   if 5e-24 < sinTheta_i

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in sinTheta_i around inf 19.1%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}\\ \end{array} \]
5. 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. 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 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. +-rgt-identity 99.7%

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

   7. metadata-eval 99.7%

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

   8. metadata-eval 99.7%

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

   9. +-rgt-identity 99.7%

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

   10. sub-neg 99.7%

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

   11. associate-+l+ 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

6. Taylor expanded in cosTheta_O around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 7: 19.8% 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(Float32(-sinTheta_i) / 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 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around inf 13.6%

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

6. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
7. Step-by-step derivation

   1. mul-1-neg 6.3%

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

   2. associate-/l* 6.3%

      \[\leadsto 1 + \left(-\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right) \]

   3. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

   4. associate-/l* 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   5. associate-*r/ 6.3%

      \[\leadsto 1 - \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

8. Simplified 6.3%

   \[\leadsto \color{blue}{1 - sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

9. Taylor expanded in sinTheta_O around inf 36.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
10. Step-by-step derivation

    1. mul-1-neg 36.6%

       \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

    2. associate-*r/ 19.5%

       \[\leadsto -\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

    3. distribute-rgt-neg-in 19.5%

       \[\leadsto \color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

    4. distribute-neg-frac 19.5%

       \[\leadsto sinTheta_O \cdot \color{blue}{\frac{-sinTheta_i}{v}} \]

11. Simplified 19.5%

    \[\leadsto \color{blue}{sinTheta_O \cdot \frac{-sinTheta_i}{v}} \]

12. Final simplification 19.5%

    \[\leadsto sinTheta_O \cdot \frac{-sinTheta_i}{v} \]
13. Add Preprocessing

Alternative 8: 38.2% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v} \end{array} \]

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 * Float32(-sinTheta_O)) / 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 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around inf 13.6%

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

6. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
7. Step-by-step derivation

   1. mul-1-neg 6.3%

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

   2. associate-/l* 6.3%

      \[\leadsto 1 + \left(-\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right) \]

   3. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

   4. associate-/l* 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   5. associate-*r/ 6.3%

      \[\leadsto 1 - \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

8. Simplified 6.3%

   \[\leadsto \color{blue}{1 - sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

9. Taylor expanded in sinTheta_O around inf 36.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

10. Final simplification 36.6%

    \[\leadsto \frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v} \]
11. Add Preprocessing

Alternative 9: 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%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around inf 13.6%

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

6. Taylor expanded in sinTheta_O around 0 6.5%

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

7. Final simplification 6.5%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(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))))))