Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 98.7%

Time: 14.0s

Alternatives: 23

Speedup: N/A×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Alternative 1: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{r}{s}}\\ 0 - \frac{\frac{\frac{\frac{0.125}{\pi}}{r}}{-t\_0} + 0.125 \cdot \left(0 - \frac{{t\_0}^{-0.3333333333333333}}{\pi \cdot r}\right)}{s} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (exp (/ r s))))
   (-
    0.0
    (/
     (+
      (/ (/ (/ 0.125 PI) r) (- t_0))
      (* 0.125 (- 0.0 (/ (pow t_0 -0.3333333333333333) (* PI r)))))
     s))))

C

float code(float s, float r) {
	float t_0 = expf((r / s));
	return 0.0f - (((((0.125f / ((float) M_PI)) / r) / -t_0) + (0.125f * (0.0f - (powf(t_0, -0.3333333333333333f) / (((float) M_PI) * r))))) / s);
}

Julia

function code(s, r)
	t_0 = exp(Float32(r / s))
	return Float32(Float32(0.0) - Float32(Float32(Float32(Float32(Float32(Float32(0.125) / Float32(pi)) / r) / Float32(-t_0)) + Float32(Float32(0.125) * Float32(Float32(0.0) - Float32((t_0 ^ Float32(-0.3333333333333333)) / Float32(Float32(pi) * r))))) / s))
end

MATLAB

function tmp = code(s, r)
	t_0 = exp((r / s));
	tmp = single(0.0) - (((((single(0.125) / single(pi)) / r) / -t_0) + (single(0.125) * (single(0.0) - ((t_0 ^ single(-0.3333333333333333)) / (single(pi) * r))))) / s);
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \pi}}{s} \]

   2. exp-prod 99.6%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]
8. Step-by-step derivation

   1. clear-num 99.6%

      \[\leadsto \frac{0.125 \cdot \color{blue}{\frac{1}{\frac{r \cdot \pi}{e^{-1 \cdot \frac{r}{s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   2. un-div-inv 99.6%

      \[\leadsto \frac{\color{blue}{\frac{0.125}{\frac{r \cdot \pi}{e^{-1 \cdot \frac{r}{s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   3. div-inv 99.6%

      \[\leadsto \frac{\frac{0.125}{\color{blue}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{-1 \cdot \frac{r}{s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   4. mul-1-neg 99.6%

      \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\color{blue}{-\frac{r}{s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   5. distribute-frac-neg2 99.6%

      \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\color{blue}{\frac{r}{-s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   6. add-sqr-sqrt -0.0%

      \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{r}{-s}} \cdot \sqrt{\frac{r}{-s}}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   7. sqrt-unprod 7.6%

      \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{r}{-s} \cdot \frac{r}{-s}}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   8. distribute-frac-neg2 7.6%

      \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\sqrt{\color{blue}{\left(-\frac{r}{s}\right)} \cdot \frac{r}{-s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   9. distribute-frac-neg2 7.6%

      \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\sqrt{\left(-\frac{r}{s}\right) \cdot \color{blue}{\left(-\frac{r}{s}\right)}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   10. sqr-neg 7.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\sqrt{\color{blue}{\frac{r}{s} \cdot \frac{r}{s}}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   11. sqrt-unprod 7.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{r}{s}} \cdot \sqrt{\frac{r}{s}}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   12. add-sqr-sqrt 7.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\color{blue}{\frac{r}{s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   13. exp-neg 7.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \color{blue}{e^{-\frac{r}{s}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   14. distribute-frac-neg2 7.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\color{blue}{\frac{r}{-s}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   15. add-sqr-sqrt -0.0%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\color{blue}{\sqrt{\frac{r}{-s}} \cdot \sqrt{\frac{r}{-s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   16. sqrt-unprod 99.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\color{blue}{\sqrt{\frac{r}{-s} \cdot \frac{r}{-s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   17. distribute-frac-neg2 99.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\sqrt{\color{blue}{\left(-\frac{r}{s}\right)} \cdot \frac{r}{-s}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   18. distribute-frac-neg2 99.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\sqrt{\left(-\frac{r}{s}\right) \cdot \color{blue}{\left(-\frac{r}{s}\right)}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

9. Applied egg-rr 99.6%

   \[\leadsto \frac{\color{blue}{\frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\frac{r}{s}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]
10. Step-by-step derivation

    1. associate-/r* 99.6%

       \[\leadsto \frac{\color{blue}{\frac{\frac{0.125}{r \cdot \pi}}{e^{\frac{r}{s}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

    2. *-commutative 99.6%

       \[\leadsto \frac{\frac{\frac{0.125}{\color{blue}{\pi \cdot r}}}{e^{\frac{r}{s}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

    3. associate-/r* 99.6%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{0.125}{\pi}}{r}}}{e^{\frac{r}{s}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

11. Simplified 99.6%

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{0.125}{\pi}}{r}}{e^{\frac{r}{s}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

12. Final simplification 99.6%

    \[\leadsto 0 - \frac{\frac{\frac{\frac{0.125}{\pi}}{r}}{-e^{\frac{r}{s}}} + 0.125 \cdot \left(0 - \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{\pi \cdot r}\right)}{s} \]
13. Add Preprocessing

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.125}{\pi} \cdot \sqrt[3]{e^{\frac{r}{-s}}} + \frac{0.125}{\pi \cdot e^{\frac{r}{s}}}}{r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (/
   (+ (* (/ 0.125 PI) (cbrt (exp (/ r (- s))))) (/ 0.125 (* PI (exp (/ r s)))))
   r)
  s))

C

float code(float s, float r) {
	return ((((0.125f / ((float) M_PI)) * cbrtf(expf((r / -s)))) + (0.125f / (((float) M_PI) * expf((r / s))))) / r) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.125) / Float32(pi)) * cbrt(exp(Float32(r / Float32(-s))))) + Float32(Float32(0.125) / Float32(Float32(pi) * exp(Float32(r / s))))) / r) / s)
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \pi}}{s} \]

   2. exp-prod 99.6%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]
8. Step-by-step derivation

   1. clear-num 99.6%

      \[\leadsto \frac{0.125 \cdot \color{blue}{\frac{1}{\frac{r \cdot \pi}{e^{-1 \cdot \frac{r}{s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   2. un-div-inv 99.6%

      \[\leadsto \frac{\color{blue}{\frac{0.125}{\frac{r \cdot \pi}{e^{-1 \cdot \frac{r}{s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   3. div-inv 99.6%

      \[\leadsto \frac{\frac{0.125}{\color{blue}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{-1 \cdot \frac{r}{s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   4. mul-1-neg 99.6%

      \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\color{blue}{-\frac{r}{s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   5. distribute-frac-neg2 99.6%

      \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\color{blue}{\frac{r}{-s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   6. add-sqr-sqrt -0.0%

      \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{r}{-s}} \cdot \sqrt{\frac{r}{-s}}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   7. sqrt-unprod 7.6%

      \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{r}{-s} \cdot \frac{r}{-s}}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   8. distribute-frac-neg2 7.6%

      \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\sqrt{\color{blue}{\left(-\frac{r}{s}\right)} \cdot \frac{r}{-s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   9. distribute-frac-neg2 7.6%

      \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\sqrt{\left(-\frac{r}{s}\right) \cdot \color{blue}{\left(-\frac{r}{s}\right)}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   10. sqr-neg 7.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\sqrt{\color{blue}{\frac{r}{s} \cdot \frac{r}{s}}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   11. sqrt-unprod 7.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{r}{s}} \cdot \sqrt{\frac{r}{s}}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   12. add-sqr-sqrt 7.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \frac{1}{e^{\color{blue}{\frac{r}{s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   13. exp-neg 7.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot \color{blue}{e^{-\frac{r}{s}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   14. distribute-frac-neg2 7.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\color{blue}{\frac{r}{-s}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   15. add-sqr-sqrt -0.0%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\color{blue}{\sqrt{\frac{r}{-s}} \cdot \sqrt{\frac{r}{-s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   16. sqrt-unprod 99.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\color{blue}{\sqrt{\frac{r}{-s} \cdot \frac{r}{-s}}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   17. distribute-frac-neg2 99.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\sqrt{\color{blue}{\left(-\frac{r}{s}\right)} \cdot \frac{r}{-s}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

   18. distribute-frac-neg2 99.6%

       \[\leadsto \frac{\frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\sqrt{\left(-\frac{r}{s}\right) \cdot \color{blue}{\left(-\frac{r}{s}\right)}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

9. Applied egg-rr 99.6%

   \[\leadsto \frac{\color{blue}{\frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\frac{r}{s}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]
10. Step-by-step derivation

    1. associate-/r* 99.6%

       \[\leadsto \frac{\color{blue}{\frac{\frac{0.125}{r \cdot \pi}}{e^{\frac{r}{s}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

    2. *-commutative 99.6%

       \[\leadsto \frac{\frac{\frac{0.125}{\color{blue}{\pi \cdot r}}}{e^{\frac{r}{s}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

    3. associate-/r* 99.6%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{0.125}{\pi}}{r}}}{e^{\frac{r}{s}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

11. Simplified 99.6%

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{0.125}{\pi}}{r}}{e^{\frac{r}{s}}}} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}}{s} \]

12. Taylor expanded in r around inf 99.6%

    \[\leadsto \frac{\color{blue}{\frac{0.125 \cdot \left(\frac{1}{\pi} \cdot \sqrt[3]{\frac{1}{e^{\frac{r}{s}}}}\right) + 0.125 \cdot \frac{1}{\pi \cdot e^{\frac{r}{s}}}}{r}}}{s} \]
13. Step-by-step derivation

    1. associate-*r* 99.6%

       \[\leadsto \frac{\frac{\color{blue}{\left(0.125 \cdot \frac{1}{\pi}\right) \cdot \sqrt[3]{\frac{1}{e^{\frac{r}{s}}}}} + 0.125 \cdot \frac{1}{\pi \cdot e^{\frac{r}{s}}}}{r}}{s} \]

    2. associate-*r/ 99.6%

       \[\leadsto \frac{\frac{\color{blue}{\frac{0.125 \cdot 1}{\pi}} \cdot \sqrt[3]{\frac{1}{e^{\frac{r}{s}}}} + 0.125 \cdot \frac{1}{\pi \cdot e^{\frac{r}{s}}}}{r}}{s} \]

    3. metadata-eval 99.6%

       \[\leadsto \frac{\frac{\frac{\color{blue}{0.125}}{\pi} \cdot \sqrt[3]{\frac{1}{e^{\frac{r}{s}}}} + 0.125 \cdot \frac{1}{\pi \cdot e^{\frac{r}{s}}}}{r}}{s} \]

    4. rec-exp 99.6%

       \[\leadsto \frac{\frac{\frac{0.125}{\pi} \cdot \sqrt[3]{\color{blue}{e^{-\frac{r}{s}}}} + 0.125 \cdot \frac{1}{\pi \cdot e^{\frac{r}{s}}}}{r}}{s} \]

    5. distribute-neg-frac2 99.6%

       \[\leadsto \frac{\frac{\frac{0.125}{\pi} \cdot \sqrt[3]{e^{\color{blue}{\frac{r}{-s}}}} + 0.125 \cdot \frac{1}{\pi \cdot e^{\frac{r}{s}}}}{r}}{s} \]

    6. associate-*r/ 99.6%

       \[\leadsto \frac{\frac{\frac{0.125}{\pi} \cdot \sqrt[3]{e^{\frac{r}{-s}}} + \color{blue}{\frac{0.125 \cdot 1}{\pi \cdot e^{\frac{r}{s}}}}}{r}}{s} \]

    7. metadata-eval 99.6%

       \[\leadsto \frac{\frac{\frac{0.125}{\pi} \cdot \sqrt[3]{e^{\frac{r}{-s}}} + \frac{\color{blue}{0.125}}{\pi \cdot e^{\frac{r}{s}}}}{r}}{s} \]

14. Simplified 99.6%

    \[\leadsto \frac{\color{blue}{\frac{\frac{0.125}{\pi} \cdot \sqrt[3]{e^{\frac{r}{-s}}} + \frac{0.125}{\pi \cdot e^{\frac{r}{s}}}}{r}}}{s} \]
15. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{e^{\frac{r}{s}}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (/ 0.25 (exp (/ r s))) (* r (* s (* PI 2.0))))
  (/ (* 0.75 (exp (/ r (* s (- 3.0))))) (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	return ((0.25f / expf((r / s))) / (r * (s * (((float) M_PI) * 2.0f)))) + ((0.75f * expf((r / (s * -3.0f)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / exp(Float32(r / s))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-Float32(3.0)))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / exp((r / s))) / (r * (s * (single(pi) * single(2.0))))) + ((single(0.75) * exp((r / (s * -single(3.0))))) / (r * (s * (single(pi) * single(6.0)))));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Add Preprocessing

3. Taylor expanded in r around inf 99.5%

   \[\leadsto \frac{\color{blue}{0.25 \cdot e^{-1 \cdot \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. neg-mul-1 99.5%

      \[\leadsto \frac{0.25 \cdot e^{\color{blue}{-\frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. rec-exp 99.5%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. associate-*r/ 99.5%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. metadata-eval 99.5%

      \[\leadsto \frac{\frac{\color{blue}{0.25}}{e^{\frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Simplified 99.5%

   \[\leadsto \frac{\color{blue}{\frac{0.25}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

6. Final simplification 99.5%

   \[\leadsto \frac{\frac{0.25}{e^{\frac{r}{s}}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
7. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \frac{e^{\frac{r}{-s}}}{\pi \cdot s} + 0.125 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{\pi \cdot s}}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (* 0.125 (/ (exp (/ r (- s))) (* PI s)))
   (* 0.125 (/ (exp (/ (* r -0.3333333333333333) s)) (* PI s))))
  r))

C

float code(float s, float r) {
	return ((0.125f * (expf((r / -s)) / (((float) M_PI) * s))) + (0.125f * (expf(((r * -0.3333333333333333f) / s)) / (((float) M_PI) * s)))) / r;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) * Float32(exp(Float32(r / Float32(-s))) / Float32(Float32(pi) * s))) + Float32(Float32(0.125) * Float32(exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / Float32(Float32(pi) * s)))) / r)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) * (exp((r / -s)) / (single(pi) * s))) + (single(0.125) * (exp(((r * single(-0.3333333333333333)) / s)) / (single(pi) * s)))) / r;
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around inf 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \pi}}{r}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \pi} + 0.125 \cdot \frac{e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{s \cdot \pi}}{r} \]

7. Applied egg-rr 99.5%

   \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \pi} + 0.125 \cdot \frac{e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{s \cdot \pi}}{r} \]

8. Final simplification 99.5%

   \[\leadsto \frac{0.125 \cdot \frac{e^{\frac{r}{-s}}}{\pi \cdot s} + 0.125 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{\pi \cdot s}}{r} \]
9. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{\frac{e^{\frac{r}{-s}}}{\pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi}}{r \cdot s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  0.125
  (/
   (+ (/ (exp (/ r (- s))) PI) (/ (exp (* (/ r s) -0.3333333333333333)) PI))
   (* r s))))

C

float code(float s, float r) {
	return 0.125f * (((expf((r / -s)) / ((float) M_PI)) + (expf(((r / s) * -0.3333333333333333f)) / ((float) M_PI))) / (r * s));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(Float32(exp(Float32(r / Float32(-s))) / Float32(pi)) + Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(pi))) / Float32(r * s)))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (((exp((r / -s)) / single(pi)) + (exp(((r / s) * single(-0.3333333333333333))) / single(pi))) / (r * s));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \pi}}{s} \]

   2. exp-prod 99.6%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]
8. Step-by-step derivation

   1. div-inv 99.2%

      \[\leadsto \color{blue}{\left(0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}\right) \cdot \frac{1}{s}} \]

   2. distribute-lft-out 99.2%

      \[\leadsto \color{blue}{\left(0.125 \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}\right)\right)} \cdot \frac{1}{s} \]

   3. associate-/r* 99.2%

      \[\leadsto \left(0.125 \cdot \left(\color{blue}{\frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r}}{\pi}} + \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}\right)\right) \cdot \frac{1}{s} \]

   4. associate-*r/ 99.2%

      \[\leadsto \left(0.125 \cdot \left(\frac{\frac{e^{\color{blue}{\frac{-1 \cdot r}{s}}}}{r}}{\pi} + \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}\right)\right) \cdot \frac{1}{s} \]

   5. neg-mul-1 99.2%

      \[\leadsto \left(0.125 \cdot \left(\frac{\frac{e^{\frac{\color{blue}{-r}}{s}}}{r}}{\pi} + \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}\right)\right) \cdot \frac{1}{s} \]

   6. pow-exp 99.1%

      \[\leadsto \left(0.125 \cdot \left(\frac{\frac{e^{\frac{-r}{s}}}{r}}{\pi} + \frac{\color{blue}{e^{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \pi}\right)\right) \cdot \frac{1}{s} \]

9. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\left(0.125 \cdot \left(\frac{\frac{e^{\frac{-r}{s}}}{r}}{\pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \pi}\right)\right) \cdot \frac{1}{s}} \]

10. Taylor expanded in r around inf 99.5%

    \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{\pi} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi}}{r \cdot s}} \]

11. Final simplification 99.5%

    \[\leadsto 0.125 \cdot \frac{\frac{e^{\frac{r}{-s}}}{\pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi}}{r \cdot s} \]
12. Add Preprocessing

Alternative 6: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{e^{\frac{r}{-s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(\pi \cdot s\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  0.125
  (/
   (+ (exp (/ r (- s))) (exp (* (/ r s) -0.3333333333333333)))
   (* r (* PI s)))))

C

float code(float s, float r) {
	return 0.125f * ((expf((r / -s)) + expf(((r / s) * -0.3333333333333333f))) / (r * (((float) M_PI) * s)));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(exp(Float32(r / Float32(-s))) + exp(Float32(Float32(r / s) * Float32(-0.3333333333333333)))) / Float32(r * Float32(Float32(pi) * s))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((r / -s)) + exp(((r / s) * single(-0.3333333333333333)))) / (r * (single(pi) * s)));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around inf 99.5%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]

6. Final simplification 99.5%

   \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(\pi \cdot s\right)} \]
7. Add Preprocessing

Alternative 7: 43.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)}}{s} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ (/ 0.25 (log1p (expm1 (* PI r)))) s))

C

float code(float s, float r) {
	return (0.25f / log1pf(expm1f((((float) M_PI) * r)))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / log1p(expm1(Float32(Float32(pi) * r)))) / s)
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \pi}}{s} \]

   2. exp-prod 99.6%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]

8. Taylor expanded in r around 0 9.2%

   \[\leadsto \frac{\color{blue}{\frac{0.25}{r \cdot \pi}}}{s} \]
9. Step-by-step derivation

   1. log1p-expm1-u 49.5%

      \[\leadsto \frac{\frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}}}{s} \]

10. Applied egg-rr 49.5%

    \[\leadsto \frac{\frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}}}{s} \]

11. Final simplification 49.5%

    \[\leadsto \frac{\frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)}}{s} \]
12. Add Preprocessing

Alternative 8: 10.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi \cdot r} - \frac{\frac{0.125}{\pi} - 0.0625 \cdot \frac{r}{\pi \cdot s}}{s}}{s} + 0.75 \cdot \frac{e^{\frac{r}{s \cdot \left(-3\right)}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (- (/ 0.125 (* PI r)) (/ (- (/ 0.125 PI) (* 0.0625 (/ r (* PI s)))) s)) s)
  (* 0.75 (/ (exp (/ r (* s (- 3.0)))) (* r (* 6.0 (* PI s)))))))

C

float code(float s, float r) {
	return (((0.125f / (((float) M_PI) * r)) - (((0.125f / ((float) M_PI)) - (0.0625f * (r / (((float) M_PI) * s)))) / s)) / s) + (0.75f * (expf((r / (s * -3.0f))) / (r * (6.0f * (((float) M_PI) * s)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(0.125) / Float32(Float32(pi) * r)) - Float32(Float32(Float32(Float32(0.125) / Float32(pi)) - Float32(Float32(0.0625) * Float32(r / Float32(Float32(pi) * s)))) / s)) / s) + Float32(Float32(0.75) * Float32(exp(Float32(r / Float32(s * Float32(-Float32(3.0))))) / Float32(r * Float32(Float32(6.0) * Float32(Float32(pi) * s))))))
end

MATLAB

function tmp = code(s, r)
	tmp = (((single(0.125) / (single(pi) * r)) - (((single(0.125) / single(pi)) - (single(0.0625) * (r / (single(pi) * s)))) / s)) / s) + (single(0.75) * (exp((r / (s * -single(3.0)))) / (r * (single(6.0) * (single(pi) * s)))));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. times-frac 99.6%

      \[\leadsto \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative 99.6%

      \[\leadsto \frac{0.25}{\color{blue}{s \cdot \left(2 \cdot \pi\right)}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. distribute-frac-neg 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\color{blue}{-\frac{r}{s}}}}{r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. associate-/l* 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + \color{blue}{0.75 \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   5. *-commutative 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{\color{blue}{s \cdot 3}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. *-commutative 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{\color{blue}{r \cdot \left(\left(6 \cdot \pi\right) \cdot s\right)}} \]

   7. associate-*l* 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \color{blue}{\left(6 \cdot \left(\pi \cdot s\right)\right)}} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf 10.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.125 \cdot \frac{1}{\pi}}{s} - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 10.9%

      \[\leadsto \color{blue}{\left(-\frac{-1 \cdot \frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.125 \cdot \frac{1}{\pi}}{s} - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   2. mul-1-neg 10.9%

      \[\leadsto \left(-\frac{\color{blue}{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.125 \cdot \frac{1}{\pi}}{s}\right)} - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   3. associate-*r/ 10.9%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \color{blue}{\frac{0.125 \cdot 1}{\pi}}}{s}\right) - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   4. metadata-eval 10.9%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{\color{blue}{0.125}}{\pi}}{s}\right) - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   5. associate-*r/ 10.9%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \color{blue}{\frac{0.125 \cdot 1}{r \cdot \pi}}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   6. metadata-eval 10.9%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \frac{\color{blue}{0.125}}{r \cdot \pi}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

7. Simplified 10.9%

   \[\leadsto \color{blue}{\left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \frac{0.125}{r \cdot \pi}}{s}\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

8. Final simplification 10.9%

   \[\leadsto \frac{\frac{0.125}{\pi \cdot r} - \frac{\frac{0.125}{\pi} - 0.0625 \cdot \frac{r}{\pi \cdot s}}{s}}{s} + 0.75 \cdot \frac{e^{\frac{r}{s \cdot \left(-3\right)}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]
9. Add Preprocessing

Alternative 9: 10.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(\frac{\frac{-1}{\pi} - \frac{r}{\pi \cdot s} \cdot -0.5}{s} + \frac{1}{\pi \cdot r}\right) + 0.125 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (* 0.125 (+ (/ (- (/ -1.0 PI) (* (/ r (* PI s)) -0.5)) s) (/ 1.0 (* PI r))))
   (* 0.125 (/ (exp (* (/ r s) -0.3333333333333333)) (* PI r))))
  s))

C

float code(float s, float r) {
	return ((0.125f * ((((-1.0f / ((float) M_PI)) - ((r / (((float) M_PI) * s)) * -0.5f)) / s) + (1.0f / (((float) M_PI) * r)))) + (0.125f * (expf(((r / s) * -0.3333333333333333f)) / (((float) M_PI) * r)))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(Float32(Float32(-1.0) / Float32(pi)) - Float32(Float32(r / Float32(Float32(pi) * s)) * Float32(-0.5))) / s) + Float32(Float32(1.0) / Float32(Float32(pi) * r)))) + Float32(Float32(0.125) * Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(Float32(pi) * r)))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) * ((((single(-1.0) / single(pi)) - ((r / (single(pi) * s)) * single(-0.5))) / s) + (single(1.0) / (single(pi) * r)))) + (single(0.125) * (exp(((r / s) * single(-0.3333333333333333))) / (single(pi) * r)))) / s;
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]

6. Taylor expanded in s around -inf 10.9%

   \[\leadsto \frac{0.125 \cdot \color{blue}{\left(-1 \cdot \frac{-0.5 \cdot \frac{r}{s \cdot \pi} + \frac{1}{\pi}}{s} + \frac{1}{r \cdot \pi}\right)} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s} \]

7. Final simplification 10.9%

   \[\leadsto \frac{0.125 \cdot \left(\frac{\frac{-1}{\pi} - \frac{r}{\pi \cdot s} \cdot -0.5}{s} + \frac{1}{\pi \cdot r}\right) + 0.125 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi \cdot r}}{s} \]
8. Add Preprocessing

Alternative 10: 10.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{r}{\pi \cdot {s}^{2}} \cdot 0.06944444444444445 + \left(\frac{0.25}{\pi \cdot r} + \frac{-0.16666666666666666}{\pi \cdot s}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (* (/ r (* PI (pow s 2.0))) 0.06944444444444445)
   (+ (/ 0.25 (* PI r)) (/ -0.16666666666666666 (* PI s))))
  s))

C

float code(float s, float r) {
	return (((r / (((float) M_PI) * powf(s, 2.0f))) * 0.06944444444444445f) + ((0.25f / (((float) M_PI) * r)) + (-0.16666666666666666f / (((float) M_PI) * s)))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(r / Float32(Float32(pi) * (s ^ Float32(2.0)))) * Float32(0.06944444444444445)) + Float32(Float32(Float32(0.25) / Float32(Float32(pi) * r)) + Float32(Float32(-0.16666666666666666) / Float32(Float32(pi) * s)))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (((r / (single(pi) * (s ^ single(2.0)))) * single(0.06944444444444445)) + ((single(0.25) / (single(pi) * r)) + (single(-0.16666666666666666) / (single(pi) * s)))) / s;
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 10.9%

   \[\leadsto \color{blue}{\frac{\left(0.006944444444444444 \cdot \frac{r}{{s}^{2} \cdot \pi} + \left(0.0625 \cdot \frac{r}{{s}^{2} \cdot \pi} + 0.25 \cdot \frac{1}{r \cdot \pi}\right)\right) - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]
6. Step-by-step derivation

7. Simplified 10.9%

   \[\leadsto \color{blue}{\frac{\frac{r}{{s}^{2} \cdot \pi} \cdot 0.06944444444444445 + \left(\frac{0.25}{r \cdot \pi} + \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s}} \]

8. Final simplification 10.9%

   \[\leadsto \frac{\frac{r}{\pi \cdot {s}^{2}} \cdot 0.06944444444444445 + \left(\frac{0.25}{\pi \cdot r} + \frac{-0.16666666666666666}{\pi \cdot s}\right)}{s} \]
9. Add Preprocessing

Alternative 11: 10.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \frac{1}{\pi \cdot r} - \frac{\frac{1}{\pi} \cdot 0.16666666666666666 - 0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + \frac{r}{\pi} \cdot 0.5}{s}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (* 0.25 (/ 1.0 (* PI r)))
   (/
    (-
     (* (/ 1.0 PI) 0.16666666666666666)
     (* 0.125 (/ (+ (* 0.05555555555555555 (/ r PI)) (* (/ r PI) 0.5)) s)))
    s))
  s))

C

float code(float s, float r) {
	return ((0.25f * (1.0f / (((float) M_PI) * r))) - ((((1.0f / ((float) M_PI)) * 0.16666666666666666f) - (0.125f * (((0.05555555555555555f * (r / ((float) M_PI))) + ((r / ((float) M_PI)) * 0.5f)) / s))) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(Float32(pi) * r))) - Float32(Float32(Float32(Float32(Float32(1.0) / Float32(pi)) * Float32(0.16666666666666666)) - Float32(Float32(0.125) * Float32(Float32(Float32(Float32(0.05555555555555555) * Float32(r / Float32(pi))) + Float32(Float32(r / Float32(pi)) * Float32(0.5))) / s))) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * (single(1.0) / (single(pi) * r))) - ((((single(1.0) / single(pi)) * single(0.16666666666666666)) - (single(0.125) * (((single(0.05555555555555555) * (r / single(pi))) + ((r / single(pi)) * single(0.5))) / s))) / s)) / s;
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf 10.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Final simplification 10.9%

   \[\leadsto \frac{0.25 \cdot \frac{1}{\pi \cdot r} - \frac{\frac{1}{\pi} \cdot 0.16666666666666666 - 0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + \frac{r}{\pi} \cdot 0.5}{s}}{s}}{s} \]
7. Add Preprocessing

Alternative 12: 10.0% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{\pi \cdot s} \cdot \left(\frac{2}{r} + \frac{\frac{0.5 \cdot \left(r + r \cdot 0.1111111111111111\right)}{s} - 1.3333333333333333}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* PI s))
  (+
   (/ 2.0 r)
   (/ (- (/ (* 0.5 (+ r (* r 0.1111111111111111))) s) 1.3333333333333333) s))))

C

float code(float s, float r) {
	return (0.125f / (((float) M_PI) * s)) * ((2.0f / r) + ((((0.5f * (r + (r * 0.1111111111111111f))) / s) - 1.3333333333333333f) / s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(Float32(pi) * s)) * Float32(Float32(Float32(2.0) / r) + Float32(Float32(Float32(Float32(Float32(0.5) * Float32(r + Float32(r * Float32(0.1111111111111111)))) / s) - Float32(1.3333333333333333)) / s)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (single(pi) * s)) * ((single(2.0) / r) + ((((single(0.5) * (r + (r * single(0.1111111111111111)))) / s) - single(1.3333333333333333)) / s));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   2. sqrt-unprod 99.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)} \cdot {\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   3. pow-prod-down 99.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{\color{blue}{{\left(e^{-0.3333333333333333} \cdot e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   4. prod-exp 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\color{blue}{\left(e^{-0.3333333333333333 + -0.3333333333333333}\right)}}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   5. metadata-eval 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

6. Applied egg-rr 99.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

7. Taylor expanded in s around -inf 10.8%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(-1 \cdot \frac{1.3333333333333333 + -1 \cdot \frac{0.5 \cdot r + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r}}{s}}{s} + 2 \cdot \frac{1}{r}\right)} \]
8. Step-by-step derivation

   1. +-commutative 10.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} + -1 \cdot \frac{1.3333333333333333 + -1 \cdot \frac{0.5 \cdot r + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r}}{s}}{s}\right)} \]

   2. mul-1-neg 10.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(2 \cdot \frac{1}{r} + \color{blue}{\left(-\frac{1.3333333333333333 + -1 \cdot \frac{0.5 \cdot r + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r}}{s}}{s}\right)}\right) \]

   3. unsub-neg 10.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} - \frac{1.3333333333333333 + -1 \cdot \frac{0.5 \cdot r + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r}}{s}}{s}\right)} \]

   4. associate-*r/ 10.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\frac{2 \cdot 1}{r}} - \frac{1.3333333333333333 + -1 \cdot \frac{0.5 \cdot r + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r}}{s}}{s}\right) \]

   5. metadata-eval 10.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{2}}{r} - \frac{1.3333333333333333 + -1 \cdot \frac{0.5 \cdot r + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r}}{s}}{s}\right) \]

9. Simplified 10.8%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{0.5 \cdot \left(r + {r}^{2} \cdot \frac{0.1111111111111111}{r}\right)}{s}}{s}\right)} \]

10. Taylor expanded in r around 0 10.8%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{0.5 \cdot \left(r + \color{blue}{0.1111111111111111 \cdot r}\right)}{s}}{s}\right) \]
11. Step-by-step derivation

    1. *-commutative 10.8%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{0.5 \cdot \left(r + \color{blue}{r \cdot 0.1111111111111111}\right)}{s}}{s}\right) \]

12. Simplified 10.8%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{0.5 \cdot \left(r + \color{blue}{r \cdot 0.1111111111111111}\right)}{s}}{s}\right) \]

13. Final simplification 10.8%

    \[\leadsto \frac{0.125}{\pi \cdot s} \cdot \left(\frac{2}{r} + \frac{\frac{0.5 \cdot \left(r + r \cdot 0.1111111111111111\right)}{s} - 1.3333333333333333}{s}\right) \]
14. Add Preprocessing

Alternative 13: 10.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/ 0.25 (* PI r))
   (/
    (+ (/ (* (/ r PI) -0.06944444444444445) s) (/ 0.16666666666666666 PI))
    s))
  s))

C

float code(float s, float r) {
	return ((0.25f / (((float) M_PI) * r)) - (((((r / ((float) M_PI)) * -0.06944444444444445f) / s) + (0.16666666666666666f / ((float) M_PI))) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(Float32(pi) * r)) - Float32(Float32(Float32(Float32(Float32(r / Float32(pi)) * Float32(-0.06944444444444445)) / s) + Float32(Float32(0.16666666666666666) / Float32(pi))) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (single(pi) * r)) - (((((r / single(pi)) * single(-0.06944444444444445)) / s) + (single(0.16666666666666666) / single(pi))) / s)) / s;
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf 10.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. mul-1-neg 10.9%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

7. Simplified 10.9%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{r \cdot \pi}}{s}} \]

8. Final simplification 10.9%

   \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s} \]
9. Add Preprocessing

Alternative 14: 10.0% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{\pi \cdot s} \cdot \left(\frac{2}{r} + \frac{\frac{r \cdot 0.5555555555555556}{s} - 1.3333333333333333}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* PI s))
  (+ (/ 2.0 r) (/ (- (/ (* r 0.5555555555555556) s) 1.3333333333333333) s))))

C

float code(float s, float r) {
	return (0.125f / (((float) M_PI) * s)) * ((2.0f / r) + ((((r * 0.5555555555555556f) / s) - 1.3333333333333333f) / s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(Float32(pi) * s)) * Float32(Float32(Float32(2.0) / r) + Float32(Float32(Float32(Float32(r * Float32(0.5555555555555556)) / s) - Float32(1.3333333333333333)) / s)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (single(pi) * s)) * ((single(2.0) / r) + ((((r * single(0.5555555555555556)) / s) - single(1.3333333333333333)) / s));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   2. sqrt-unprod 99.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)} \cdot {\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   3. pow-prod-down 99.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{\color{blue}{{\left(e^{-0.3333333333333333} \cdot e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   4. prod-exp 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\color{blue}{\left(e^{-0.3333333333333333 + -0.3333333333333333}\right)}}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   5. metadata-eval 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

6. Applied egg-rr 99.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

7. Taylor expanded in s around -inf 10.8%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(-1 \cdot \frac{1.3333333333333333 + -1 \cdot \frac{0.5 \cdot r + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r}}{s}}{s} + 2 \cdot \frac{1}{r}\right)} \]
8. Step-by-step derivation

   1. +-commutative 10.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} + -1 \cdot \frac{1.3333333333333333 + -1 \cdot \frac{0.5 \cdot r + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r}}{s}}{s}\right)} \]

   2. mul-1-neg 10.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(2 \cdot \frac{1}{r} + \color{blue}{\left(-\frac{1.3333333333333333 + -1 \cdot \frac{0.5 \cdot r + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r}}{s}}{s}\right)}\right) \]

   3. unsub-neg 10.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} - \frac{1.3333333333333333 + -1 \cdot \frac{0.5 \cdot r + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r}}{s}}{s}\right)} \]

   4. associate-*r/ 10.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\frac{2 \cdot 1}{r}} - \frac{1.3333333333333333 + -1 \cdot \frac{0.5 \cdot r + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r}}{s}}{s}\right) \]

   5. metadata-eval 10.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{2}}{r} - \frac{1.3333333333333333 + -1 \cdot \frac{0.5 \cdot r + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r}}{s}}{s}\right) \]

9. Simplified 10.8%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{0.5 \cdot \left(r + {r}^{2} \cdot \frac{0.1111111111111111}{r}\right)}{s}}{s}\right)} \]

10. Taylor expanded in r around 0 10.8%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \color{blue}{0.5555555555555556 \cdot \frac{r}{s}}}{s}\right) \]
11. Step-by-step derivation

    1. associate-*r/ 10.8%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \color{blue}{\frac{0.5555555555555556 \cdot r}{s}}}{s}\right) \]

    2. *-commutative 10.8%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{\color{blue}{r \cdot 0.5555555555555556}}{s}}{s}\right) \]

12. Simplified 10.8%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \color{blue}{\frac{r \cdot 0.5555555555555556}{s}}}{s}\right) \]

13. Final simplification 10.8%

    \[\leadsto \frac{0.125}{\pi \cdot s} \cdot \left(\frac{2}{r} + \frac{\frac{r \cdot 0.5555555555555556}{s} - 1.3333333333333333}{s}\right) \]
14. Add Preprocessing

Alternative 15: 9.1% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.25}{r}}{\pi} - \frac{\frac{0.16666666666666666}{s}}{\pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (- (/ (/ 0.25 r) PI) (/ (/ 0.16666666666666666 s) PI)) s))

C

float code(float s, float r) {
	return (((0.25f / r) / ((float) M_PI)) - ((0.16666666666666666f / s) / ((float) M_PI))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(0.25) / r) / Float32(pi)) - Float32(Float32(Float32(0.16666666666666666) / s) / Float32(pi))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (((single(0.25) / r) / single(pi)) - ((single(0.16666666666666666) / s) / single(pi))) / s;
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \pi}}{s} \]

   2. exp-prod 99.6%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]
8. Step-by-step derivation

   1. div-inv 99.2%

      \[\leadsto \color{blue}{\left(0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}\right) \cdot \frac{1}{s}} \]

   2. distribute-lft-out 99.2%

      \[\leadsto \color{blue}{\left(0.125 \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}\right)\right)} \cdot \frac{1}{s} \]

   3. associate-/r* 99.2%

      \[\leadsto \left(0.125 \cdot \left(\color{blue}{\frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r}}{\pi}} + \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}\right)\right) \cdot \frac{1}{s} \]

   4. associate-*r/ 99.2%

      \[\leadsto \left(0.125 \cdot \left(\frac{\frac{e^{\color{blue}{\frac{-1 \cdot r}{s}}}}{r}}{\pi} + \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}\right)\right) \cdot \frac{1}{s} \]

   5. neg-mul-1 99.2%

      \[\leadsto \left(0.125 \cdot \left(\frac{\frac{e^{\frac{\color{blue}{-r}}{s}}}{r}}{\pi} + \frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r \cdot \pi}\right)\right) \cdot \frac{1}{s} \]

   6. pow-exp 99.1%

      \[\leadsto \left(0.125 \cdot \left(\frac{\frac{e^{\frac{-r}{s}}}{r}}{\pi} + \frac{\color{blue}{e^{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \pi}\right)\right) \cdot \frac{1}{s} \]

9. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\left(0.125 \cdot \left(\frac{\frac{e^{\frac{-r}{s}}}{r}}{\pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \pi}\right)\right) \cdot \frac{1}{s}} \]

10. Taylor expanded in s around inf 9.8%

    \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s}} \]
11. Step-by-step derivation

    1. associate-*r/ 9.8%

       \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{s \cdot \pi}}}{s} \]

    2. metadata-eval 9.8%

       \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi} - \frac{\color{blue}{0.16666666666666666}}{s \cdot \pi}}{s} \]

    3. associate-*r/ 9.8%

       \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{r \cdot \pi}} - \frac{0.16666666666666666}{s \cdot \pi}}{s} \]

    4. metadata-eval 9.8%

       \[\leadsto \frac{\frac{\color{blue}{0.25}}{r \cdot \pi} - \frac{0.16666666666666666}{s \cdot \pi}}{s} \]

    5. associate-/r* 9.9%

       \[\leadsto \frac{\color{blue}{\frac{\frac{0.25}{r}}{\pi}} - \frac{0.16666666666666666}{s \cdot \pi}}{s} \]

    6. associate-/r* 9.9%

       \[\leadsto \frac{\frac{\frac{0.25}{r}}{\pi} - \color{blue}{\frac{\frac{0.16666666666666666}{s}}{\pi}}}{s} \]

12. Simplified 9.9%

    \[\leadsto \color{blue}{\frac{\frac{\frac{0.25}{r}}{\pi} - \frac{\frac{0.16666666666666666}{s}}{\pi}}{s}} \]
13. Add Preprocessing

Alternative 16: 9.1% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\pi \cdot r} - \frac{0.16666666666666666}{\pi \cdot s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (- (/ 0.25 (* PI r)) (/ 0.16666666666666666 (* PI s))) s))

C

float code(float s, float r) {
	return ((0.25f / (((float) M_PI) * r)) - (0.16666666666666666f / (((float) M_PI) * s))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(Float32(pi) * r)) - Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * s))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (single(pi) * r)) - (single(0.16666666666666666) / (single(pi) * s))) / s;
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 9.8%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. associate-*r/ 9.8%

      \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{s \cdot \pi}}}{s} \]

   2. metadata-eval 9.8%

      \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi} - \frac{\color{blue}{0.16666666666666666}}{s \cdot \pi}}{s} \]

   3. associate-*r/ 9.8%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{r \cdot \pi}} - \frac{0.16666666666666666}{s \cdot \pi}}{s} \]

   4. metadata-eval 9.8%

      \[\leadsto \frac{\frac{\color{blue}{0.25}}{r \cdot \pi} - \frac{0.16666666666666666}{s \cdot \pi}}{s} \]

7. Simplified 9.8%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]

8. Final simplification 9.8%

   \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{0.16666666666666666}{\pi \cdot s}}{s} \]
9. Add Preprocessing

Alternative 17: 9.1% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{\pi \cdot s} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (/ 0.125 (* PI s)) (- (/ 2.0 r) (/ 1.3333333333333333 s))))

C

float code(float s, float r) {
	return (0.125f / (((float) M_PI) * s)) * ((2.0f / r) - (1.3333333333333333f / s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(Float32(pi) * s)) * Float32(Float32(Float32(2.0) / r) - Float32(Float32(1.3333333333333333) / s)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (single(pi) * s)) * ((single(2.0) / r) - (single(1.3333333333333333) / s));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   2. sqrt-unprod 99.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)} \cdot {\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   3. pow-prod-down 99.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{\color{blue}{{\left(e^{-0.3333333333333333} \cdot e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   4. prod-exp 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\color{blue}{\left(e^{-0.3333333333333333 + -0.3333333333333333}\right)}}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   5. metadata-eval 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

6. Applied egg-rr 99.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

7. Taylor expanded in s around inf 9.8%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} - 1.3333333333333333 \cdot \frac{1}{s}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 9.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\frac{2 \cdot 1}{r}} - 1.3333333333333333 \cdot \frac{1}{s}\right) \]

   2. metadata-eval 9.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{2}}{r} - 1.3333333333333333 \cdot \frac{1}{s}\right) \]

   3. associate-*r/ 9.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \color{blue}{\frac{1.3333333333333333 \cdot 1}{s}}\right) \]

   4. metadata-eval 9.8%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{\color{blue}{1.3333333333333333}}{s}\right) \]

9. Simplified 9.8%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(\frac{2}{r} - \frac{1.3333333333333333}{s}\right)} \]

10. Final simplification 9.8%

    \[\leadsto \frac{0.125}{\pi \cdot s} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333}{s}\right) \]
11. Add Preprocessing

Alternative 18: 9.0% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{s}{\frac{\frac{0.25}{r}}{\pi}}} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 1.0 (/ s (/ (/ 0.25 r) PI))))

C

float code(float s, float r) {
	return 1.0f / (s / ((0.25f / r) / ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(s / Float32(Float32(Float32(0.25) / r) / Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(1.0) / (s / ((single(0.25) / r) / single(pi)));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \pi}}{s} \]

   2. exp-prod 99.6%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]

8. Taylor expanded in r around 0 9.2%

   \[\leadsto \frac{\color{blue}{\frac{0.25}{r \cdot \pi}}}{s} \]
9. Step-by-step derivation

   1. clear-num 9.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{s}{\frac{0.25}{r \cdot \pi}}}} \]

   2. inv-pow 9.2%

      \[\leadsto \color{blue}{{\left(\frac{s}{\frac{0.25}{r \cdot \pi}}\right)}^{-1}} \]

   3. *-commutative 9.2%

      \[\leadsto {\left(\frac{s}{\frac{0.25}{\color{blue}{\pi \cdot r}}}\right)}^{-1} \]

10. Applied egg-rr 9.2%

    \[\leadsto \color{blue}{{\left(\frac{s}{\frac{0.25}{\pi \cdot r}}\right)}^{-1}} \]
11. Step-by-step derivation

    1. unpow-1 9.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{s}{\frac{0.25}{\pi \cdot r}}}} \]

    2. *-commutative 9.2%

       \[\leadsto \frac{1}{\frac{s}{\frac{0.25}{\color{blue}{r \cdot \pi}}}} \]

    3. associate-/r* 9.2%

       \[\leadsto \frac{1}{\frac{s}{\color{blue}{\frac{\frac{0.25}{r}}{\pi}}}} \]

12. Simplified 9.2%

    \[\leadsto \color{blue}{\frac{1}{\frac{s}{\frac{\frac{0.25}{r}}{\pi}}}} \]
13. Add Preprocessing

Alternative 19: 9.0% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\pi \cdot r\right) \cdot \frac{s}{0.25}} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 1.0 (* (* PI r) (/ s 0.25))))

C

float code(float s, float r) {
	return 1.0f / ((((float) M_PI) * r) * (s / 0.25f));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(Float32(Float32(pi) * r) * Float32(s / Float32(0.25))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(1.0) / ((single(pi) * r) * (s / single(0.25)));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \pi}}{s} \]

   2. exp-prod 99.6%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]

8. Taylor expanded in r around 0 9.2%

   \[\leadsto \frac{\color{blue}{\frac{0.25}{r \cdot \pi}}}{s} \]
9. Step-by-step derivation

   1. clear-num 9.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{s}{\frac{0.25}{r \cdot \pi}}}} \]

   2. inv-pow 9.2%

      \[\leadsto \color{blue}{{\left(\frac{s}{\frac{0.25}{r \cdot \pi}}\right)}^{-1}} \]

   3. *-commutative 9.2%

      \[\leadsto {\left(\frac{s}{\frac{0.25}{\color{blue}{\pi \cdot r}}}\right)}^{-1} \]

10. Applied egg-rr 9.2%

    \[\leadsto \color{blue}{{\left(\frac{s}{\frac{0.25}{\pi \cdot r}}\right)}^{-1}} \]
11. Step-by-step derivation

    1. unpow-1 9.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{s}{\frac{0.25}{\pi \cdot r}}}} \]

    2. associate-/r/ 9.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{s}{0.25} \cdot \left(\pi \cdot r\right)}} \]

    3. *-commutative 9.2%

       \[\leadsto \frac{1}{\frac{s}{0.25} \cdot \color{blue}{\left(r \cdot \pi\right)}} \]

12. Simplified 9.2%

    \[\leadsto \color{blue}{\frac{1}{\frac{s}{0.25} \cdot \left(r \cdot \pi\right)}} \]

13. Final simplification 9.2%

    \[\leadsto \frac{1}{\left(\pi \cdot r\right) \cdot \frac{s}{0.25}} \]
14. Add Preprocessing

Alternative 20: 9.0% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \frac{\pi}{\frac{0.25}{r}}} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 1.0 (* s (/ PI (/ 0.25 r)))))

C

float code(float s, float r) {
	return 1.0f / (s * (((float) M_PI) / (0.25f / r)));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(pi) / Float32(Float32(0.25) / r))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(1.0) / (s * (single(pi) / (single(0.25) / r)));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 9.2%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
6. Step-by-step derivation

   1. div-inv 9.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)}} \]

7. Applied egg-rr 9.2%

   \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)}} \]
8. Step-by-step derivation

   1. associate-*r/ 9.2%

      \[\leadsto \color{blue}{\frac{0.25 \cdot 1}{r \cdot \left(s \cdot \pi\right)}} \]

   2. metadata-eval 9.2%

      \[\leadsto \frac{\color{blue}{0.25}}{r \cdot \left(s \cdot \pi\right)} \]

   3. associate-/r* 9.2%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi}} \]

9. Simplified 9.2%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi}} \]
10. Step-by-step derivation

    1. clear-num 9.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{s \cdot \pi}{\frac{0.25}{r}}}} \]

    2. inv-pow 9.2%

       \[\leadsto \color{blue}{{\left(\frac{s \cdot \pi}{\frac{0.25}{r}}\right)}^{-1}} \]

11. Applied egg-rr 9.2%

    \[\leadsto \color{blue}{{\left(\frac{s \cdot \pi}{\frac{0.25}{r}}\right)}^{-1}} \]
12. Step-by-step derivation

    1. *-un-lft-identity 9.2%

       \[\leadsto \color{blue}{1 \cdot {\left(\frac{s \cdot \pi}{\frac{0.25}{r}}\right)}^{-1}} \]

    2. unpow-1 9.2%

       \[\leadsto 1 \cdot \color{blue}{\frac{1}{\frac{s \cdot \pi}{\frac{0.25}{r}}}} \]

    3. associate-/l* 9.2%

       \[\leadsto 1 \cdot \frac{1}{\color{blue}{s \cdot \frac{\pi}{\frac{0.25}{r}}}} \]

13. Applied egg-rr 9.2%

    \[\leadsto \color{blue}{1 \cdot \frac{1}{s \cdot \frac{\pi}{\frac{0.25}{r}}}} \]
14. Step-by-step derivation

    1. *-lft-identity 9.2%

       \[\leadsto \color{blue}{\frac{1}{s \cdot \frac{\pi}{\frac{0.25}{r}}}} \]

15. Simplified 9.2%

    \[\leadsto \color{blue}{\frac{1}{s \cdot \frac{\pi}{\frac{0.25}{r}}}} \]
16. Add Preprocessing

Alternative 21: 9.0% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\pi \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ (/ 0.25 (* PI r)) s))

C

float code(float s, float r) {
	return (0.25f / (((float) M_PI) * r)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / Float32(Float32(pi) * r)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / (single(pi) * r)) / s;
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \pi}}{s} \]

   2. exp-prod 99.6%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{r \cdot \pi}}{s} \]

8. Taylor expanded in r around 0 9.2%

   \[\leadsto \frac{\color{blue}{\frac{0.25}{r \cdot \pi}}}{s} \]

9. Final simplification 9.2%

   \[\leadsto \frac{\frac{0.25}{\pi \cdot r}}{s} \]
10. Add Preprocessing

Alternative 22: 9.0% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r}}{\pi \cdot s} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ (/ 0.25 r) (* PI s)))

C

float code(float s, float r) {
	return (0.25f / r) / (((float) M_PI) * s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / r) / Float32(Float32(pi) * s))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / r) / (single(pi) * s);
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 9.2%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
6. Step-by-step derivation

   1. div-inv 9.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)}} \]

7. Applied egg-rr 9.2%

   \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)}} \]
8. Step-by-step derivation

   1. associate-*r/ 9.2%

      \[\leadsto \color{blue}{\frac{0.25 \cdot 1}{r \cdot \left(s \cdot \pi\right)}} \]

   2. metadata-eval 9.2%

      \[\leadsto \frac{\color{blue}{0.25}}{r \cdot \left(s \cdot \pi\right)} \]

   3. associate-/r* 9.2%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi}} \]

9. Simplified 9.2%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi}} \]

10. Final simplification 9.2%

    \[\leadsto \frac{\frac{0.25}{r}}{\pi \cdot s} \]
11. Add Preprocessing

Alternative 23: 9.0% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{r \cdot \left(\pi \cdot s\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (* r (* PI s))))

C

float code(float s, float r) {
	return 0.25f / (r * (((float) M_PI) * s));
}

Julia

function code(s, r)
	return Float32(Float32(0.25) / Float32(r * Float32(Float32(pi) * s)))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / (r * (single(pi) * s));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 9.2%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]

6. Final simplification 9.2%

   \[\leadsto \frac{0.25}{r \cdot \left(\pi \cdot s\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(FPCore (s r)
  :name "Disney BSSRDF, PDF of scattering profile"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (< 1e-6 r) (< r 1000000.0)))
  (+ (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r)) (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))