Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 17.3s

Alternatives: 11

Speedup: 0.5×

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: 99.5% 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: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ \mathsf{fma}\left(t_0, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot {\left(e^{0.5}\right)}^{\left(\frac{r}{\frac{s}{\log \left(\sqrt[3]{e^{-0.6666666666666666}}\right)}}\right)}}{r}, t_0 \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* s PI))))
   (fma
    t_0
    (/
     (*
      (pow (cbrt (exp -1.3333333333333333)) (/ (* r 0.5) s))
      (pow (exp 0.5) (/ r (/ s (log (cbrt (exp -0.6666666666666666)))))))
     r)
    (* t_0 (/ (exp (/ (- r) s)) r)))))

C

float code(float s, float r) {
	float t_0 = 0.125f / (s * ((float) M_PI));
	return fmaf(t_0, ((powf(cbrtf(expf(-1.3333333333333333f)), ((r * 0.5f) / s)) * powf(expf(0.5f), (r / (s / logf(cbrtf(expf(-0.6666666666666666f))))))) / r), (t_0 * (expf((-r / s)) / r)));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(s * Float32(pi)))
	return fma(t_0, Float32(Float32((cbrt(exp(Float32(-1.3333333333333333))) ^ Float32(Float32(r * Float32(0.5)) / s)) * (exp(Float32(0.5)) ^ Float32(r / Float32(s / log(cbrt(exp(Float32(-0.6666666666666666)))))))) / r), Float32(t_0 * Float32(exp(Float32(Float32(-r) / s)) / r)))
end

Derivation

1. Initial program 99.5%

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

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. Step-by-step derivation

   1. pow-exp 99.3%

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

   2. sqr-pow 99.3%

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

   3. pow-prod-down 99.3%

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

   4. prod-exp 99.6%

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

   5. metadata-eval 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   6. associate-/l/ 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r}{2 \cdot s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   7. *-commutative 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{\color{blue}{s \cdot 2}}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
7. Step-by-step derivation

   1. add-cube-cbrt 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\color{blue}{\left(\left(\sqrt[3]{e^{-0.6666666666666666}} \cdot \sqrt[3]{e^{-0.6666666666666666}}\right) \cdot \sqrt[3]{e^{-0.6666666666666666}}\right)}}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   2. unpow-prod-down 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(\sqrt[3]{e^{-0.6666666666666666}} \cdot \sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   3. cbrt-unprod 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\color{blue}{\left(\sqrt[3]{e^{-0.6666666666666666} \cdot e^{-0.6666666666666666}}\right)}}^{\left(\frac{r}{s \cdot 2}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   4. prod-exp 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{\color{blue}{e^{-0.6666666666666666 + -0.6666666666666666}}}\right)}^{\left(\frac{r}{s \cdot 2}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   5. metadata-eval 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{\color{blue}{-1.3333333333333333}}}\right)}^{\left(\frac{r}{s \cdot 2}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   6. associate-/r* 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\color{blue}{\left(\frac{\frac{r}{s}}{2}\right)}} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   7. div-inv 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\color{blue}{\left(\frac{r}{s} \cdot \frac{1}{2}\right)}} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   8. metadata-eval 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r}{s} \cdot \color{blue}{0.5}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   9. associate-/r* 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\color{blue}{\left(\frac{\frac{r}{s}}{2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   10. div-inv 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\color{blue}{\left(\frac{r}{s} \cdot \frac{1}{2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   11. metadata-eval 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s} \cdot \color{blue}{0.5}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

8. Applied egg-rr 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
9. Step-by-step derivation

   1. associate-*l/ 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\color{blue}{\left(\frac{r \cdot 0.5}{s}\right)}} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   2. associate-*l/ 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\color{blue}{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

10. Simplified 99.6%

    \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

11. Taylor expanded in r around inf 99.6%

    \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot \color{blue}{e^{0.5 \cdot \frac{r \cdot \log \left({\left(e^{-0.6666666666666666}\right)}^{0.3333333333333333}\right)}{s}}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
12. Step-by-step derivation

    1. exp-prod 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot \color{blue}{{\left(e^{0.5}\right)}^{\left(\frac{r \cdot \log \left({\left(e^{-0.6666666666666666}\right)}^{0.3333333333333333}\right)}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

    2. associate-/l* 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot {\left(e^{0.5}\right)}^{\color{blue}{\left(\frac{r}{\frac{s}{\log \left({\left(e^{-0.6666666666666666}\right)}^{0.3333333333333333}\right)}}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

    3. unpow1/3 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot {\left(e^{0.5}\right)}^{\left(\frac{r}{\frac{s}{\log \color{blue}{\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}}}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

13. Simplified 99.6%

    \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot \color{blue}{{\left(e^{0.5}\right)}^{\left(\frac{r}{\frac{s}{\log \left(\sqrt[3]{e^{-0.6666666666666666}}\right)}}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

14. Final simplification 99.6%

    \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot {\left(e^{0.5}\right)}^{\left(\frac{r}{\frac{s}{\log \left(\sqrt[3]{e^{-0.6666666666666666}}\right)}}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
15. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ \mathsf{fma}\left(t_0, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot e^{\frac{r}{s} \cdot -0.1111111111111111}}{r}, t_0 \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* s PI))))
   (fma
    t_0
    (/
     (*
      (pow (cbrt (exp -1.3333333333333333)) (/ (* r 0.5) s))
      (exp (* (/ r s) -0.1111111111111111)))
     r)
    (* t_0 (/ (exp (/ (- r) s)) r)))))

C

float code(float s, float r) {
	float t_0 = 0.125f / (s * ((float) M_PI));
	return fmaf(t_0, ((powf(cbrtf(expf(-1.3333333333333333f)), ((r * 0.5f) / s)) * expf(((r / s) * -0.1111111111111111f))) / r), (t_0 * (expf((-r / s)) / r)));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(s * Float32(pi)))
	return fma(t_0, Float32(Float32((cbrt(exp(Float32(-1.3333333333333333))) ^ Float32(Float32(r * Float32(0.5)) / s)) * exp(Float32(Float32(r / s) * Float32(-0.1111111111111111)))) / r), Float32(t_0 * Float32(exp(Float32(Float32(-r) / s)) / r)))
end

Derivation

1. Initial program 99.5%

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

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. Step-by-step derivation

   1. pow-exp 99.3%

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

   2. sqr-pow 99.3%

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

   3. pow-prod-down 99.3%

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

   4. prod-exp 99.6%

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

   5. metadata-eval 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   6. associate-/l/ 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r}{2 \cdot s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   7. *-commutative 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{\color{blue}{s \cdot 2}}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
7. Step-by-step derivation

   1. add-cube-cbrt 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\color{blue}{\left(\left(\sqrt[3]{e^{-0.6666666666666666}} \cdot \sqrt[3]{e^{-0.6666666666666666}}\right) \cdot \sqrt[3]{e^{-0.6666666666666666}}\right)}}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   2. unpow-prod-down 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(\sqrt[3]{e^{-0.6666666666666666}} \cdot \sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   3. cbrt-unprod 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\color{blue}{\left(\sqrt[3]{e^{-0.6666666666666666} \cdot e^{-0.6666666666666666}}\right)}}^{\left(\frac{r}{s \cdot 2}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   4. prod-exp 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{\color{blue}{e^{-0.6666666666666666 + -0.6666666666666666}}}\right)}^{\left(\frac{r}{s \cdot 2}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   5. metadata-eval 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{\color{blue}{-1.3333333333333333}}}\right)}^{\left(\frac{r}{s \cdot 2}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   6. associate-/r* 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\color{blue}{\left(\frac{\frac{r}{s}}{2}\right)}} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   7. div-inv 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\color{blue}{\left(\frac{r}{s} \cdot \frac{1}{2}\right)}} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   8. metadata-eval 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r}{s} \cdot \color{blue}{0.5}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   9. associate-/r* 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\color{blue}{\left(\frac{\frac{r}{s}}{2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   10. div-inv 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\color{blue}{\left(\frac{r}{s} \cdot \frac{1}{2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   11. metadata-eval 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s} \cdot \color{blue}{0.5}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

8. Applied egg-rr 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
9. Step-by-step derivation

   1. associate-*l/ 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\color{blue}{\left(\frac{r \cdot 0.5}{s}\right)}} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   2. associate-*l/ 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\color{blue}{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

10. Simplified 99.6%

    \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

11. Taylor expanded in r around inf 99.6%

    \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot \color{blue}{e^{0.5 \cdot \frac{r \cdot \log \left({\left(e^{-0.6666666666666666}\right)}^{0.3333333333333333}\right)}{s}}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
12. Step-by-step derivation

    1. log-pow 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot e^{0.5 \cdot \frac{r \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{-0.6666666666666666}\right)\right)}}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

    2. rem-log-exp 99.7%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot e^{0.5 \cdot \frac{r \cdot \left(0.3333333333333333 \cdot \color{blue}{-0.6666666666666666}\right)}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

    3. metadata-eval 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot e^{0.5 \cdot \frac{r \cdot \color{blue}{-0.2222222222222222}}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

    4. associate-*l/ 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot e^{0.5 \cdot \color{blue}{\left(\frac{r}{s} \cdot -0.2222222222222222\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

    5. associate-*l* 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot e^{\color{blue}{\left(0.5 \cdot \frac{r}{s}\right) \cdot -0.2222222222222222}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

    6. *-commutative 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot e^{\color{blue}{\left(\frac{r}{s} \cdot 0.5\right)} \cdot -0.2222222222222222}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

    7. associate-*l* 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot e^{\color{blue}{\frac{r}{s} \cdot \left(0.5 \cdot -0.2222222222222222\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

    8. metadata-eval 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot e^{\frac{r}{s} \cdot \color{blue}{-0.1111111111111111}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

13. Simplified 99.6%

    \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot \color{blue}{e^{\frac{r}{s} \cdot -0.1111111111111111}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

14. Final simplification 99.6%

    \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)} \cdot e^{\frac{r}{s} \cdot -0.1111111111111111}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
15. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ \mathsf{fma}\left(t_0, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, t_0 \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* s PI))))
   (fma
    t_0
    (/ (pow (exp -0.6666666666666666) (/ r (* s 2.0))) r)
    (* t_0 (/ (exp (/ (- r) s)) r)))))

C

float code(float s, float r) {
	float t_0 = 0.125f / (s * ((float) M_PI));
	return fmaf(t_0, (powf(expf(-0.6666666666666666f), (r / (s * 2.0f))) / r), (t_0 * (expf((-r / s)) / r)));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(s * Float32(pi)))
	return fma(t_0, Float32((exp(Float32(-0.6666666666666666)) ^ Float32(r / Float32(s * Float32(2.0)))) / r), Float32(t_0 * Float32(exp(Float32(Float32(-r) / s)) / r)))
end

Derivation

1. Initial program 99.5%

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

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. Step-by-step derivation

   1. pow-exp 99.3%

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

   2. sqr-pow 99.3%

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

   3. pow-prod-down 99.3%

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

   4. prod-exp 99.6%

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

   5. metadata-eval 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   6. associate-/l/ 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r}{2 \cdot s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   7. *-commutative 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{\color{blue}{s \cdot 2}}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

7. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
8. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * ((single(1.0) / single(pi)) / s)) * ((exp((r / -s)) / r) + (exp((r / (s / single(-0.3333333333333333)))) / 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 r around inf 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-/r/ 99.5%

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

7. Simplified 99.5%

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

   1. clear-num 9.5%

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

   2. associate-/r/ 9.5%

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

   3. *-commutative 9.5%

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

   4. associate-/r* 9.5%

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

9. Applied egg-rr 99.6%

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

10. Final simplification 99.6%

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

Alternative 5: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (exp((r / (s / single(-0.3333333333333333)))) / 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 r around inf 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-/r/ 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 6: 43.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / Float32(s * log1p(expm1(Float32(Float32(pi) * 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 r around 0 9.5%

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

   1. clear-num 9.5%

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

   2. associate-/r/ 9.5%

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

   3. *-commutative 9.5%

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

   4. associate-/r* 9.5%

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

7. Applied egg-rr 9.5%

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

8. Taylor expanded in s around inf 9.0%

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

   1. associate-*r* 9.0%

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

   2. *-commutative 9.0%

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

   3. associate-*l* 9.0%

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

10. Simplified 9.0%

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

12. Applied egg-rr 44.6%

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

13. Final simplification 44.6%

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

Alternative 7: 15.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + ((single(1.0) / (single(1.0) + ((r / s) * single(0.3333333333333333)))) / 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. Step-by-step derivation

   1. pow-to-exp 99.3%

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

   2. rem-log-exp 99.5%

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

   3. metadata-eval 99.5%

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

   4. times-frac 99.5%

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

   5. neg-mul-1 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-frac-neg 99.5%

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

   8. *-commutative 99.5%

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

   9. exp-neg 99.5%

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

   10. add-sqr-sqrt 99.5%

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

   11. sqrt-unprod 99.5%

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

   12. sqr-neg 99.5%

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

   13. sqrt-unprod -0.0%

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

   14. add-sqr-sqrt 7.8%

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

   15. associate-/l/ 7.8%

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

   16. exp-cbrt 7.8%

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

   17. add-sqr-sqrt -0.0%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in r around 0 15.1%

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

   1. *-commutative 15.1%

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

9. Simplified 15.1%

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

10. Final simplification 15.1%

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

Alternative 8: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 0 9.9%

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

   1. associate-*r/ 9.9%

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

7. Simplified 9.9%

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

8. Final simplification 9.9%

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

Alternative 9: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * ((single(1.0) / single(pi)) / s)) * ((exp((-r / s)) + single(1.0)) / 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 r around 0 9.5%

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

   1. clear-num 9.5%

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

   2. associate-/r/ 9.5%

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

   3. *-commutative 9.5%

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

   4. associate-/r* 9.5%

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

7. Applied egg-rr 9.5%

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

8. Taylor expanded in r around inf 9.5%

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

   1. associate-*r/ 9.5%

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

   2. neg-mul-1 9.5%

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

10. Simplified 9.5%

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

11. Final simplification 9.5%

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

Alternative 10: 9.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot r} \cdot \frac{e^{\frac{-r}{s}} + 1}{\pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 0 9.5%

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

   1. clear-num 9.5%

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

   2. associate-/r/ 9.5%

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

   3. *-commutative 9.5%

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

   4. associate-/r* 9.5%

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

7. Applied egg-rr 9.5%

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

8. Taylor expanded in r around inf 9.5%

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

   1. associate-*r/ 9.5%

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

   2. associate-*r* 9.5%

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

   3. times-frac 9.5%

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

   4. associate-*r/ 9.5%

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

   5. neg-mul-1 9.5%

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

10. Simplified 9.5%

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

11. Final simplification 9.5%

    \[\leadsto \frac{0.125}{s \cdot r} \cdot \frac{e^{\frac{-r}{s}} + 1}{\pi} \]
12. Add Preprocessing

Alternative 11: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / ((s * single(pi)) * 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 r around 0 9.5%

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

6. Taylor expanded in s around inf 9.0%

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

7. Final simplification 9.0%

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

Reproduce

herbie shell --seed 2024019 
(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))))