Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 33.3s

Alternatives: 22

Speedup: 1.0×

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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + 0.125 \cdot \left(\frac{\frac{e^{\frac{\frac{r \cdot -0.16666666666666666}{{\left(\sqrt[3]{s}\right)}^{2}}}{\sqrt[3]{s}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot {\left(\sqrt[3]{\pi}\right)}^{3}}\right)}^{2}} \cdot \frac{\frac{e^{-0.16666666666666666 \cdot \frac{r}{s}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (*
   0.125
   (*
    (/
     (/
      (exp (/ (/ (* r -0.16666666666666666) (pow (cbrt s) 2.0)) (cbrt s)))
      (sqrt r))
     (pow (cbrt (* s (pow (cbrt PI) 3.0))) 2.0))
    (/ (/ (exp (* -0.16666666666666666 (/ r s))) (sqrt r)) (cbrt (* s PI)))))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (r * (s * (2.0f * ((float) M_PI))))) + (0.125f * (((expf((((r * -0.16666666666666666f) / powf(cbrtf(s), 2.0f)) / cbrtf(s))) / sqrtf(r)) / powf(cbrtf((s * powf(cbrtf(((float) M_PI)), 3.0f))), 2.0f)) * ((expf((-0.16666666666666666f * (r / s))) / sqrtf(r)) / cbrtf((s * ((float) M_PI))))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(0.125) * Float32(Float32(Float32(exp(Float32(Float32(Float32(r * Float32(-0.16666666666666666)) / (cbrt(s) ^ Float32(2.0))) / cbrt(s))) / sqrt(r)) / (cbrt(Float32(s * (cbrt(Float32(pi)) ^ Float32(3.0)))) ^ Float32(2.0))) * Float32(Float32(exp(Float32(Float32(-0.16666666666666666) * Float32(r / s))) / sqrt(r)) / cbrt(Float32(s * Float32(pi)))))))
end

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in r around inf 99.2%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]
3. Step-by-step derivation

   1. associate-/r* 99.2%

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

   2. *-commutative 99.2%

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

   3. pow-exp 97.9%

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

   4. add-sqr-sqrt 98.0%

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

   5. add-cube-cbrt 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}} \cdot \sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\color{blue}{\left(\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}\right) \cdot \sqrt[3]{s \cdot \pi}}} \]

   6. times-frac 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \color{blue}{\left(\frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right)} \]

   7. sqrt-div 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\color{blue}{\frac{\sqrt{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{\sqrt{r}}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   8. sqrt-pow1 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{\left(\frac{-0.3333333333333333}{2}\right)}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   9. pow-exp 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{\color{blue}{e^{\frac{r}{s} \cdot \frac{-0.3333333333333333}{2}}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   10. metadata-eval 97.9%

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{r}{s} \cdot \color{blue}{-0.16666666666666666}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   11. pow2 97.9%

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\color{blue}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

4. Applied egg-rr 99.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \color{blue}{\left(\frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right)} \]
5. Step-by-step derivation

   1. associate-*l/ 99.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\color{blue}{\frac{r \cdot -0.16666666666666666}{s}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   2. add-cube-cbrt 99.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{r \cdot -0.16666666666666666}{\color{blue}{\left(\sqrt[3]{s} \cdot \sqrt[3]{s}\right) \cdot \sqrt[3]{s}}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   3. associate-/r* 99.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\color{blue}{\frac{\frac{r \cdot -0.16666666666666666}{\sqrt[3]{s} \cdot \sqrt[3]{s}}}{\sqrt[3]{s}}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   4. pow2 99.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{\frac{r \cdot -0.16666666666666666}{\color{blue}{{\left(\sqrt[3]{s}\right)}^{2}}}}{\sqrt[3]{s}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

6. Applied egg-rr 99.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\color{blue}{\frac{\frac{r \cdot -0.16666666666666666}{{\left(\sqrt[3]{s}\right)}^{2}}}{\sqrt[3]{s}}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]
7. Step-by-step derivation

   1. add-cube-cbrt 99.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{\frac{r \cdot -0.16666666666666666}{{\left(\sqrt[3]{s}\right)}^{2}}}{\sqrt[3]{s}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)}}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   2. pow3 99.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{\frac{r \cdot -0.16666666666666666}{{\left(\sqrt[3]{s}\right)}^{2}}}{\sqrt[3]{s}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}}}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

8. Applied egg-rr 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{\frac{r \cdot -0.16666666666666666}{{\left(\sqrt[3]{s}\right)}^{2}}}{\sqrt[3]{s}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}}}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

9. Final simplification 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + 0.125 \cdot \left(\frac{\frac{e^{\frac{\frac{r \cdot -0.16666666666666666}{{\left(\sqrt[3]{s}\right)}^{2}}}{\sqrt[3]{s}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot {\left(\sqrt[3]{\pi}\right)}^{3}}\right)}^{2}} \cdot \frac{\frac{e^{-0.16666666666666666 \cdot \frac{r}{s}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{s \cdot \pi}\\ \frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}} + 0.125 \cdot \left(\frac{\frac{e^{-0.16666666666666666 \cdot \frac{r}{s}}}{\sqrt{r}}}{t_0} \cdot \frac{\frac{e^{\frac{\frac{r \cdot -0.16666666666666666}{{\left(\sqrt[3]{s}\right)}^{2}}}{\sqrt[3]{s}}}}{\sqrt{r}}}{{t_0}^{2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (cbrt (* s PI))))
   (+
    (/ 0.125 (* (* r (* s PI)) (exp (/ r s))))
    (*
     0.125
     (*
      (/ (/ (exp (* -0.16666666666666666 (/ r s))) (sqrt r)) t_0)
      (/
       (/
        (exp (/ (/ (* r -0.16666666666666666) (pow (cbrt s) 2.0)) (cbrt s)))
        (sqrt r))
       (pow t_0 2.0)))))))

C

float code(float s, float r) {
	float t_0 = cbrtf((s * ((float) M_PI)));
	return (0.125f / ((r * (s * ((float) M_PI))) * expf((r / s)))) + (0.125f * (((expf((-0.16666666666666666f * (r / s))) / sqrtf(r)) / t_0) * ((expf((((r * -0.16666666666666666f) / powf(cbrtf(s), 2.0f)) / cbrtf(s))) / sqrtf(r)) / powf(t_0, 2.0f))));
}

Julia

function code(s, r)
	t_0 = cbrt(Float32(s * Float32(pi)))
	return Float32(Float32(Float32(0.125) / Float32(Float32(r * Float32(s * Float32(pi))) * exp(Float32(r / s)))) + Float32(Float32(0.125) * Float32(Float32(Float32(exp(Float32(Float32(-0.16666666666666666) * Float32(r / s))) / sqrt(r)) / t_0) * Float32(Float32(exp(Float32(Float32(Float32(r * Float32(-0.16666666666666666)) / (cbrt(s) ^ Float32(2.0))) / cbrt(s))) / sqrt(r)) / (t_0 ^ Float32(2.0))))))
end

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in r around inf 99.2%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]
3. Step-by-step derivation

   1. associate-/r* 99.2%

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

   2. *-commutative 99.2%

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

   3. pow-exp 97.9%

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

   4. add-sqr-sqrt 98.0%

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

   5. add-cube-cbrt 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}} \cdot \sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\color{blue}{\left(\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}\right) \cdot \sqrt[3]{s \cdot \pi}}} \]

   6. times-frac 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \color{blue}{\left(\frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right)} \]

   7. sqrt-div 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\color{blue}{\frac{\sqrt{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{\sqrt{r}}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   8. sqrt-pow1 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{\left(\frac{-0.3333333333333333}{2}\right)}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   9. pow-exp 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{\color{blue}{e^{\frac{r}{s} \cdot \frac{-0.3333333333333333}{2}}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   10. metadata-eval 97.9%

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{r}{s} \cdot \color{blue}{-0.16666666666666666}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   11. pow2 97.9%

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\color{blue}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

4. Applied egg-rr 99.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \color{blue}{\left(\frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right)} \]
5. Step-by-step derivation

   1. associate-*l/ 99.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\color{blue}{\frac{r \cdot -0.16666666666666666}{s}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   2. add-cube-cbrt 99.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{r \cdot -0.16666666666666666}{\color{blue}{\left(\sqrt[3]{s} \cdot \sqrt[3]{s}\right) \cdot \sqrt[3]{s}}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   3. associate-/r* 99.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\color{blue}{\frac{\frac{r \cdot -0.16666666666666666}{\sqrt[3]{s} \cdot \sqrt[3]{s}}}{\sqrt[3]{s}}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   4. pow2 99.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{\frac{r \cdot -0.16666666666666666}{\color{blue}{{\left(\sqrt[3]{s}\right)}^{2}}}}{\sqrt[3]{s}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

6. Applied egg-rr 99.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\color{blue}{\frac{\frac{r \cdot -0.16666666666666666}{{\left(\sqrt[3]{s}\right)}^{2}}}{\sqrt[3]{s}}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

7. Taylor expanded in r around inf 99.4%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} + 0.125 \cdot \left(\frac{\frac{e^{\frac{\frac{r \cdot -0.16666666666666666}{{\left(\sqrt[3]{s}\right)}^{2}}}{\sqrt[3]{s}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]
8. Step-by-step derivation

   1. associate-*r/ 99.2%

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

   2. associate-/l* 99.2%

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

   3. mul-1-neg 99.2%

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

   4. rec-exp 99.1%

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

   5. associate-/r/ 99.2%

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

   6. associate-*l/ 99.2%

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

   7. /-rgt-identity 99.2%

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

9. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}}} + 0.125 \cdot \left(\frac{\frac{e^{\frac{\frac{r \cdot -0.16666666666666666}{{\left(\sqrt[3]{s}\right)}^{2}}}{\sqrt[3]{s}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

10. Final simplification 99.4%

    \[\leadsto \frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}} + 0.125 \cdot \left(\frac{\frac{e^{-0.16666666666666666 \cdot \frac{r}{s}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}} \cdot \frac{\frac{e^{\frac{\frac{r \cdot -0.16666666666666666}{{\left(\sqrt[3]{s}\right)}^{2}}}{\sqrt[3]{s}}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}}\right) \]

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-0.16666666666666666 \cdot \frac{r}{s}}}{\sqrt{r}}\\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + 0.125 \cdot \left(\frac{t_0}{\sqrt[3]{s \cdot \pi}} \cdot \frac{t_0}{{\left(\sqrt[3]{e^{\log \left(s \cdot \pi\right)}}\right)}^{2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ (exp (* -0.16666666666666666 (/ r s))) (sqrt r))))
   (+
    (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
    (*
     0.125
     (*
      (/ t_0 (cbrt (* s PI)))
      (/ t_0 (pow (cbrt (exp (log (* s PI)))) 2.0)))))))

C

float code(float s, float r) {
	float t_0 = expf((-0.16666666666666666f * (r / s))) / sqrtf(r);
	return ((0.25f * expf((-r / s))) / (r * (s * (2.0f * ((float) M_PI))))) + (0.125f * ((t_0 / cbrtf((s * ((float) M_PI)))) * (t_0 / powf(cbrtf(expf(logf((s * ((float) M_PI))))), 2.0f))));
}

Julia

function code(s, r)
	t_0 = Float32(exp(Float32(Float32(-0.16666666666666666) * Float32(r / s))) / sqrt(r))
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(0.125) * Float32(Float32(t_0 / cbrt(Float32(s * Float32(pi)))) * Float32(t_0 / (cbrt(exp(log(Float32(s * Float32(pi))))) ^ Float32(2.0))))))
end

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in r around inf 99.2%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]
3. Step-by-step derivation

   1. associate-/r* 99.2%

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

   2. *-commutative 99.2%

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

   3. pow-exp 97.9%

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

   4. add-sqr-sqrt 98.0%

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

   5. add-cube-cbrt 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}} \cdot \sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\color{blue}{\left(\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}\right) \cdot \sqrt[3]{s \cdot \pi}}} \]

   6. times-frac 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \color{blue}{\left(\frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right)} \]

   7. sqrt-div 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\color{blue}{\frac{\sqrt{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{\sqrt{r}}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   8. sqrt-pow1 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{\left(\frac{-0.3333333333333333}{2}\right)}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   9. pow-exp 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{\color{blue}{e^{\frac{r}{s} \cdot \frac{-0.3333333333333333}{2}}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   10. metadata-eval 97.9%

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{r}{s} \cdot \color{blue}{-0.16666666666666666}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   11. pow2 97.9%

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\color{blue}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

4. Applied egg-rr 99.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \color{blue}{\left(\frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right)} \]
5. Step-by-step derivation

   1. add-exp-log 99.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{{\left(\sqrt[3]{\color{blue}{e^{\log \left(s \cdot \pi\right)}}}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

6. Applied egg-rr 99.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{{\left(\sqrt[3]{\color{blue}{e^{\log \left(s \cdot \pi\right)}}}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

7. Final simplification 99.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + 0.125 \cdot \left(\frac{\frac{e^{-0.16666666666666666 \cdot \frac{r}{s}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}} \cdot \frac{\frac{e^{-0.16666666666666666 \cdot \frac{r}{s}}}{\sqrt{r}}}{{\left(\sqrt[3]{e^{\log \left(s \cdot \pi\right)}}\right)}^{2}}\right) \]

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-0.16666666666666666 \cdot \frac{r}{s}}}{\sqrt{r}}\\ t_1 := \sqrt[3]{s \cdot \pi}\\ \frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}} + 0.125 \cdot \left(\frac{t_0}{t_1} \cdot \frac{t_0}{{t_1}^{2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ (exp (* -0.16666666666666666 (/ r s))) (sqrt r)))
        (t_1 (cbrt (* s PI))))
   (+
    (/ 0.125 (* (* r (* s PI)) (exp (/ r s))))
    (* 0.125 (* (/ t_0 t_1) (/ t_0 (pow t_1 2.0)))))))

C

float code(float s, float r) {
	float t_0 = expf((-0.16666666666666666f * (r / s))) / sqrtf(r);
	float t_1 = cbrtf((s * ((float) M_PI)));
	return (0.125f / ((r * (s * ((float) M_PI))) * expf((r / s)))) + (0.125f * ((t_0 / t_1) * (t_0 / powf(t_1, 2.0f))));
}

Julia

function code(s, r)
	t_0 = Float32(exp(Float32(Float32(-0.16666666666666666) * Float32(r / s))) / sqrt(r))
	t_1 = cbrt(Float32(s * Float32(pi)))
	return Float32(Float32(Float32(0.125) / Float32(Float32(r * Float32(s * Float32(pi))) * exp(Float32(r / s)))) + Float32(Float32(0.125) * Float32(Float32(t_0 / t_1) * Float32(t_0 / (t_1 ^ Float32(2.0))))))
end

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in r around inf 99.2%

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

3. Taylor expanded in r around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. associate-/l* 99.2%

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

   3. mul-1-neg 99.2%

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

   4. rec-exp 99.1%

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

   5. associate-/r/ 99.2%

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

   6. associate-*l/ 99.2%

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

   7. /-rgt-identity 99.2%

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

5. Simplified 99.2%

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

   1. associate-/r* 99.2%

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

   2. *-commutative 99.2%

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

   3. pow-exp 97.9%

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

   4. add-sqr-sqrt 98.0%

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

   5. add-cube-cbrt 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}} \cdot \sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\color{blue}{\left(\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}\right) \cdot \sqrt[3]{s \cdot \pi}}} \]

   6. times-frac 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \color{blue}{\left(\frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right)} \]

   7. sqrt-div 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\color{blue}{\frac{\sqrt{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}}{\sqrt{r}}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   8. sqrt-pow1 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{\color{blue}{{\left(e^{\frac{r}{s}}\right)}^{\left(\frac{-0.3333333333333333}{2}\right)}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   9. pow-exp 97.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{\color{blue}{e^{\frac{r}{s} \cdot \frac{-0.3333333333333333}{2}}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   10. metadata-eval 97.9%

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{r}{s} \cdot \color{blue}{-0.16666666666666666}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi} \cdot \sqrt[3]{s \cdot \pi}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

   11. pow2 97.9%

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \left(\frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\color{blue}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}}} \cdot \frac{\sqrt{\frac{{\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}}{r}}}{\sqrt[3]{s \cdot \pi}}\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto \frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}} + 0.125 \cdot \color{blue}{\left(\frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}} \cdot \frac{\frac{e^{\frac{r}{s} \cdot -0.16666666666666666}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}}\right)} \]

8. Final simplification 99.4%

   \[\leadsto \frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}} + 0.125 \cdot \left(\frac{\frac{e^{-0.16666666666666666 \cdot \frac{r}{s}}}{\sqrt{r}}}{\sqrt[3]{s \cdot \pi}} \cdot \frac{\frac{e^{-0.16666666666666666 \cdot \frac{r}{s}}}{\sqrt{r}}}{{\left(\sqrt[3]{s \cdot \pi}\right)}^{2}}\right) \]

Alternative 5: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{{\left(\sqrt[3]{\left(s \cdot \pi\right) \cdot 6}\right)}^{3}} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ 0.25 (/ (* r (* s (* 2.0 PI))) (exp (/ (- r) s))))
  (*
   (/ 0.75 (pow (cbrt (* (* s PI) 6.0)) 3.0))
   (exp (- (* (/ r s) -0.3333333333333333) (log r))))))

C

float code(float s, float r) {
	return (0.25f / ((r * (s * (2.0f * ((float) M_PI)))) / expf((-r / s)))) + ((0.75f / powf(cbrtf(((s * ((float) M_PI)) * 6.0f)), 3.0f)) * expf((((r / s) * -0.3333333333333333f) - logf(r))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / Float32(Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi)))) / exp(Float32(Float32(-r) / s)))) + Float32(Float32(Float32(0.75) / (cbrt(Float32(Float32(s * Float32(pi)) * Float32(6.0))) ^ Float32(3.0))) * exp(Float32(Float32(Float32(r / s) * Float32(-0.3333333333333333)) - log(r)))))
end

Derivation

1. Initial program 99.3%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. *-commutative 99.2%

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

   4. times-frac 99.2%

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

   5. associate-*l* 99.2%

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

   6. *-commutative 99.2%

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

3. Simplified 99.2%

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

   1. add-exp-log 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. *-commutative 99.1%

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

   4. distribute-frac-neg 99.1%

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

   5. log-div 99.0%

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

   6. add-log-exp 99.3%

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

   7. neg-mul-1 99.3%

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

   8. times-frac 99.3%

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

   9. metadata-eval 99.3%

      \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s} - \log r} \]

   10. rem-log-exp 99.1%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\color{blue}{\log \left(e^{-0.3333333333333333}\right)} \cdot \frac{r}{s} - \log r} \]

   11. div-exp 98.8%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \color{blue}{\frac{e^{\log \left(e^{-0.3333333333333333}\right) \cdot \frac{r}{s}}}{e^{\log r}}} \]

   12. pow-to-exp 98.8%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{\color{blue}{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}{e^{\log r}} \]

   13. pow-exp 99.0%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{e^{\log r}} \]

   14. div-exp 99.3%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r}} \]

   15. *-commutative 99.3%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333} - \log r} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \color{blue}{e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r}} \]
6. Step-by-step derivation

   1. add-cube-cbrt 99.3%

      \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{\color{blue}{\left(\sqrt[3]{6 \cdot \left(\pi \cdot s\right)} \cdot \sqrt[3]{6 \cdot \left(\pi \cdot s\right)}\right) \cdot \sqrt[3]{6 \cdot \left(\pi \cdot s\right)}}} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   2. pow3 99.3%

      \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{\color{blue}{{\left(\sqrt[3]{6 \cdot \left(\pi \cdot s\right)}\right)}^{3}}} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   3. *-commutative 99.3%

      \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{{\left(\sqrt[3]{6 \cdot \color{blue}{\left(s \cdot \pi\right)}}\right)}^{3}} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

7. Applied egg-rr 99.3%

   \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{\color{blue}{{\left(\sqrt[3]{6 \cdot \left(s \cdot \pi\right)}\right)}^{3}}} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

8. Final simplification 99.3%

   \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{{\left(\sqrt[3]{\left(s \cdot \pi\right) \cdot 6}\right)}^{3}} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

Alternative 6: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \cdot \frac{0.75}{\left(s \cdot \pi\right) \cdot 6} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ 0.25 (/ (* r (* s (* 2.0 PI))) (exp (/ (- r) s))))
  (*
   (exp (- (* (/ r s) -0.3333333333333333) (log r)))
   (/ 0.75 (* (* s PI) 6.0)))))

C

float code(float s, float r) {
	return (0.25f / ((r * (s * (2.0f * ((float) M_PI)))) / expf((-r / s)))) + (expf((((r / s) * -0.3333333333333333f) - logf(r))) * (0.75f / ((s * ((float) M_PI)) * 6.0f)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / Float32(Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi)))) / exp(Float32(Float32(-r) / s)))) + Float32(exp(Float32(Float32(Float32(r / s) * Float32(-0.3333333333333333)) - log(r))) * Float32(Float32(0.75) / Float32(Float32(s * Float32(pi)) * Float32(6.0)))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / ((r * (s * (single(2.0) * single(pi)))) / exp((-r / s)))) + (exp((((r / s) * single(-0.3333333333333333)) - log(r))) * (single(0.75) / ((s * single(pi)) * single(6.0))));
end

Derivation

1. Initial program 99.3%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. *-commutative 99.2%

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

   4. times-frac 99.2%

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

   5. associate-*l* 99.2%

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

   6. *-commutative 99.2%

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

3. Simplified 99.2%

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

   1. add-exp-log 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. *-commutative 99.1%

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

   4. distribute-frac-neg 99.1%

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

   5. log-div 99.0%

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

   6. add-log-exp 99.3%

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

   7. neg-mul-1 99.3%

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

   8. times-frac 99.3%

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

   9. metadata-eval 99.3%

      \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s} - \log r} \]

   10. rem-log-exp 99.1%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\color{blue}{\log \left(e^{-0.3333333333333333}\right)} \cdot \frac{r}{s} - \log r} \]

   11. div-exp 98.8%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \color{blue}{\frac{e^{\log \left(e^{-0.3333333333333333}\right) \cdot \frac{r}{s}}}{e^{\log r}}} \]

   12. pow-to-exp 98.8%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{\color{blue}{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}{e^{\log r}} \]

   13. pow-exp 99.0%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{e^{\log r}} \]

   14. div-exp 99.3%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r}} \]

   15. *-commutative 99.3%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333} - \log r} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \color{blue}{e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r}} \]

6. Final simplification 99.3%

   \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \cdot \frac{0.75}{\left(s \cdot \pi\right) \cdot 6} \]

Alternative 7: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \cdot \frac{0.75}{\left(s \cdot \pi\right) \cdot 6} + \frac{0.25}{e^{\frac{r}{s}} \cdot \left(r \cdot \left(\pi \cdot \left(s \cdot 2\right)\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (*
   (exp (- (* (/ r s) -0.3333333333333333) (log r)))
   (/ 0.75 (* (* s PI) 6.0)))
  (/ 0.25 (* (exp (/ r s)) (* r (* PI (* s 2.0)))))))

C

float code(float s, float r) {
	return (expf((((r / s) * -0.3333333333333333f) - logf(r))) * (0.75f / ((s * ((float) M_PI)) * 6.0f))) + (0.25f / (expf((r / s)) * (r * (((float) M_PI) * (s * 2.0f)))));
}

Julia

function code(s, r)
	return Float32(Float32(exp(Float32(Float32(Float32(r / s) * Float32(-0.3333333333333333)) - log(r))) * Float32(Float32(0.75) / Float32(Float32(s * Float32(pi)) * Float32(6.0)))) + Float32(Float32(0.25) / Float32(exp(Float32(r / s)) * Float32(r * Float32(Float32(pi) * Float32(s * Float32(2.0)))))))
end

MATLAB

function tmp = code(s, r)
	tmp = (exp((((r / s) * single(-0.3333333333333333)) - log(r))) * (single(0.75) / ((s * single(pi)) * single(6.0)))) + (single(0.25) / (exp((r / s)) * (r * (single(pi) * (s * single(2.0))))));
end

Derivation

1. Initial program 99.3%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. *-commutative 99.2%

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

   4. times-frac 99.2%

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

   5. associate-*l* 99.2%

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

   6. *-commutative 99.2%

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

3. Simplified 99.2%

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

   1. add-exp-log 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. *-commutative 99.1%

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

   4. distribute-frac-neg 99.1%

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

   5. log-div 99.0%

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

   6. add-log-exp 99.3%

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

   7. neg-mul-1 99.3%

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

   8. times-frac 99.3%

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

   9. metadata-eval 99.3%

      \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s} - \log r} \]

   10. rem-log-exp 99.1%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\color{blue}{\log \left(e^{-0.3333333333333333}\right)} \cdot \frac{r}{s} - \log r} \]

   11. div-exp 98.8%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \color{blue}{\frac{e^{\log \left(e^{-0.3333333333333333}\right) \cdot \frac{r}{s}}}{e^{\log r}}} \]

   12. pow-to-exp 98.8%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{\color{blue}{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}{e^{\log r}} \]

   13. pow-exp 99.0%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{e^{\log r}} \]

   14. div-exp 99.3%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r}} \]

   15. *-commutative 99.3%

       \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333} - \log r} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{0.25}{\frac{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \color{blue}{e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r}} \]
6. Step-by-step derivation

   1. div-inv 99.3%

      \[\leadsto \frac{0.25}{\color{blue}{\left(r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)\right) \cdot \frac{1}{e^{\frac{-r}{s}}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   2. *-commutative 99.3%

      \[\leadsto \frac{0.25}{\color{blue}{\left(\left(s \cdot \left(2 \cdot \pi\right)\right) \cdot r\right)} \cdot \frac{1}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   3. *-commutative 99.3%

      \[\leadsto \frac{0.25}{\left(\color{blue}{\left(\left(2 \cdot \pi\right) \cdot s\right)} \cdot r\right) \cdot \frac{1}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   4. *-commutative 99.3%

      \[\leadsto \frac{0.25}{\color{blue}{\left(r \cdot \left(\left(2 \cdot \pi\right) \cdot s\right)\right)} \cdot \frac{1}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   5. *-commutative 99.3%

      \[\leadsto \frac{0.25}{\left(r \cdot \color{blue}{\left(s \cdot \left(2 \cdot \pi\right)\right)}\right) \cdot \frac{1}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   6. associate-*r* 99.3%

      \[\leadsto \frac{0.25}{\left(r \cdot \color{blue}{\left(\left(s \cdot 2\right) \cdot \pi\right)}\right) \cdot \frac{1}{e^{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   7. rec-exp 99.3%

      \[\leadsto \frac{0.25}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot \color{blue}{e^{-\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   8. add-sqr-sqrt -0.0%

      \[\leadsto \frac{0.25}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot e^{-\frac{\color{blue}{\sqrt{-r} \cdot \sqrt{-r}}}{s}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   9. sqrt-unprod 6.7%

      \[\leadsto \frac{0.25}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot e^{-\frac{\color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}}{s}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   10. sqr-neg 6.7%

       \[\leadsto \frac{0.25}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot e^{-\frac{\sqrt{\color{blue}{r \cdot r}}}{s}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   11. sqrt-unprod 6.7%

       \[\leadsto \frac{0.25}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot e^{-\frac{\color{blue}{\sqrt{r} \cdot \sqrt{r}}}{s}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   12. add-sqr-sqrt 6.7%

       \[\leadsto \frac{0.25}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot e^{-\frac{\color{blue}{r}}{s}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   13. distribute-frac-neg 6.7%

       \[\leadsto \frac{0.25}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot e^{\color{blue}{\frac{-r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   14. add-sqr-sqrt -0.0%

       \[\leadsto \frac{0.25}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot e^{\frac{\color{blue}{\sqrt{-r} \cdot \sqrt{-r}}}{s}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   15. sqrt-unprod 99.3%

       \[\leadsto \frac{0.25}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot e^{\frac{\color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}}{s}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   16. sqr-neg 99.3%

       \[\leadsto \frac{0.25}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot e^{\frac{\sqrt{\color{blue}{r \cdot r}}}{s}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   17. sqrt-unprod 99.3%

       \[\leadsto \frac{0.25}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot e^{\frac{\color{blue}{\sqrt{r} \cdot \sqrt{r}}}{s}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

   18. add-sqr-sqrt 99.3%

       \[\leadsto \frac{0.25}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot e^{\frac{\color{blue}{r}}{s}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

7. Applied egg-rr 99.3%

   \[\leadsto \frac{0.25}{\color{blue}{\left(r \cdot \left(\left(s \cdot 2\right) \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \]

8. Final simplification 99.3%

   \[\leadsto e^{\frac{r}{s} \cdot -0.3333333333333333 - \log r} \cdot \frac{0.75}{\left(s \cdot \pi\right) \cdot 6} + \frac{0.25}{e^{\frac{r}{s}} \cdot \left(r \cdot \left(\pi \cdot \left(s \cdot 2\right)\right)\right)} \]

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/ (* 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 * (2.0f * ((float) M_PI))))) + ((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(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(s * 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(2.0) * single(pi))))) + ((single(0.75) * exp((-r / (s * single(3.0))))) / (r * (s * (single(pi) * single(6.0)))));
end

Derivation

1. Initial program 99.3%

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

2. Final simplification 99.3%

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

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in r around inf 99.3%

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

   1. associate-*r/ 99.2%

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

   2. associate-/l* 99.2%

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

   3. mul-1-neg 99.2%

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

   4. rec-exp 99.1%

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

   5. associate-/r/ 99.2%

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

   6. associate-*l/ 99.2%

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

   7. /-rgt-identity 99.2%

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

4. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 10: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(s \cdot \pi\right)\\ \frac{0.125}{t_0 \cdot e^{\frac{r}{s}}} + 0.125 \cdot \frac{e^{-\frac{r}{s \cdot \left(--3\right)}}}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* r (* s PI))))
   (+
    (/ 0.125 (* t_0 (exp (/ r s))))
    (* 0.125 (/ (exp (- (/ r (* s (- -3.0))))) t_0)))))

C

float code(float s, float r) {
	float t_0 = r * (s * ((float) M_PI));
	return (0.125f / (t_0 * expf((r / s)))) + (0.125f * (expf(-(r / (s * -(-3.0f)))) / t_0));
}

Julia

function code(s, r)
	t_0 = Float32(r * Float32(s * Float32(pi)))
	return Float32(Float32(Float32(0.125) / Float32(t_0 * exp(Float32(r / s)))) + Float32(Float32(0.125) * Float32(exp(Float32(-Float32(r / Float32(s * Float32(-Float32(-3.0)))))) / t_0)))
end

MATLAB

function tmp = code(s, r)
	t_0 = r * (s * single(pi));
	tmp = (single(0.125) / (t_0 * exp((r / s)))) + (single(0.125) * (exp(-(r / (s * -single(-3.0)))) / t_0));
end

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in r around inf 99.2%

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

3. Taylor expanded in r around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. associate-/l* 99.2%

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

   3. mul-1-neg 99.2%

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

   4. rec-exp 99.1%

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

   5. associate-/r/ 99.2%

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

   6. associate-*l/ 99.2%

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

   7. /-rgt-identity 99.2%

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

5. Simplified 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-/r/ 99.2%

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

   3. frac-2neg 99.2%

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

   4. div-inv 99.2%

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

   5. metadata-eval 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 11: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in r around inf 99.2%

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

3. Taylor expanded in r around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. associate-/l* 99.2%

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

   3. mul-1-neg 99.2%

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

   4. rec-exp 99.1%

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

   5. associate-/r/ 99.2%

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

   6. associate-*l/ 99.2%

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

   7. /-rgt-identity 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 12: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.8%

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

   1. pow-exp 99.1%

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

   2. *-commutative 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

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

Alternative 13: 11.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.8%

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

4. Taylor expanded in r around 0 8.7%

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

5. Taylor expanded in s around inf 8.4%

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

   1. log1p-expm1-u 10.5%

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

7. Applied egg-rr 10.5%

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

8. Final simplification 10.5%

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

Alternative 14: 45.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.8%

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

4. Taylor expanded in r around 0 8.7%

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

5. Taylor expanded in s around inf 8.4%

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

   1. expm1-log1p-u 8.4%

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

   2. expm1-udef 9.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{r \cdot \left(s \cdot \pi\right)}\right)} - 1} \]

   3. *-commutative 9.1%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{\left(s \cdot \pi\right) \cdot r}}\right)} - 1 \]

   4. associate-/r* 9.1%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{0.25}{s \cdot \pi}}{r}}\right)} - 1 \]

7. Applied egg-rr 9.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0.25}{s \cdot \pi}}{r}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 8.4%

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

   2. expm1-log1p 8.4%

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

   3. associate-/r* 8.4%

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

   4. associate-*l* 8.4%

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

9. Simplified 8.4%

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

    1. log1p-expm1-u 43.6%

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

11. Applied egg-rr 43.6%

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

12. Final simplification 43.6%

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

Alternative 15: 9.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.8%

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

4. Taylor expanded in r around 0 8.7%

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

5. Taylor expanded in s around 0 8.7%

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

   1. associate-*r/ 8.7%

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

   2. neg-mul-1 8.7%

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

7. Simplified 8.7%

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

8. Final simplification 8.7%

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

Alternative 16: 9.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.8%

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

4. Taylor expanded in r around 0 8.7%

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

5. Taylor expanded in s around 0 8.7%

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

6. Final simplification 8.7%

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

Alternative 17: 9.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.8%

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

4. Taylor expanded in r around 0 8.7%

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

5. Final simplification 8.7%

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

Alternative 18: 9.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.8%

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

4. Taylor expanded in r around 0 8.7%

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

5. Taylor expanded in r around -inf 8.7%

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

   1. associate-*r/ 8.7%

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

   2. sub-neg 8.7%

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

   3. metadata-eval 8.7%

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

   4. distribute-lft-in 8.7%

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

   5. mul-1-neg 8.7%

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

   6. rec-exp 8.7%

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

   7. associate-*r/ 8.7%

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

   8. metadata-eval 8.7%

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

   9. metadata-eval 8.7%

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

7. Simplified 8.7%

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

8. Final simplification 8.7%

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

Alternative 19: 9.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.8%

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

4. Taylor expanded in r around 0 8.7%

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

5. Taylor expanded in r around inf 8.7%

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

   1. associate-*r/ 8.7%

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

   2. neg-mul-1 8.7%

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

7. Simplified 8.7%

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

8. Final simplification 8.7%

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

Alternative 20: 8.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.8%

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

4. Taylor expanded in r around 0 8.7%

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

5. Taylor expanded in s around inf 8.4%

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

6. Final simplification 8.4%

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

Alternative 21: 8.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.8%

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

4. Taylor expanded in r around 0 8.7%

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

5. Taylor expanded in s around inf 8.4%

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

   1. expm1-log1p-u 8.4%

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

   2. expm1-udef 9.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{r \cdot \left(s \cdot \pi\right)}\right)} - 1} \]

   3. *-commutative 9.1%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{\left(s \cdot \pi\right) \cdot r}}\right)} - 1 \]

   4. associate-/r* 9.1%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{0.25}{s \cdot \pi}}{r}}\right)} - 1 \]

7. Applied egg-rr 9.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0.25}{s \cdot \pi}}{r}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 8.4%

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

   2. expm1-log1p 8.4%

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

   3. associate-/r* 8.4%

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

   4. associate-*l* 8.4%

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

9. Simplified 8.4%

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

10. Taylor expanded in s around 0 8.4%

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

    1. associate-/r* 8.4%

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

    2. *-commutative 8.4%

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

12. Simplified 8.4%

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

13. Final simplification 8.4%

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

Alternative 22: 8.7% accurate, 4.0× speedup

Math

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

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(Float32(0.25) / 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 99.3%

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

3. Simplified 98.8%

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

4. Taylor expanded in r around 0 8.7%

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

5. Taylor expanded in s around inf 8.4%

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

   1. *-un-lft-identity 8.4%

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

   2. *-commutative 8.4%

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

   3. *-commutative 8.4%

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

   4. associate-/r* 8.4%

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

7. Applied egg-rr 8.4%

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

8. Final simplification 8.4%

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

Reproduce

herbie shell --seed 2023308 
(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))))