Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 18.3s

Alternatives: 15

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)} \cdot \sqrt{{\left(e^{-0.2222222222222222}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ (/ 0.125 PI) s)
  (+
   (/ (exp (/ r (- s))) r)
   (/
    (*
     (pow (cbrt (exp -1.3333333333333333)) (/ (/ r s) 2.0))
     (sqrt (pow (exp -0.2222222222222222) (/ r s))))
    r))))

C

float code(float s, float r) {
	return ((0.125f / ((float) M_PI)) / s) * ((expf((r / -s)) / r) + ((powf(cbrtf(expf(-1.3333333333333333f)), ((r / s) / 2.0f)) * sqrtf(powf(expf(-0.2222222222222222f), (r / s)))) / 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((cbrt(exp(Float32(-1.3333333333333333))) ^ Float32(Float32(r / s) / Float32(2.0))) * sqrt((exp(Float32(-0.2222222222222222)) ^ Float32(r / s)))) / r)))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.0%

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

   2. sqrt-unprod 99.3%

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

   3. prod-exp 99.3%

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

   4. metadata-eval 99.3%

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

5. Applied egg-rr 99.3%

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

   1. sqrt-pow2 99.4%

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

   2. add-cube-cbrt 99.2%

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

   3. unpow-prod-down 99.3%

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

   4. cbrt-unprod 99.6%

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

   5. prod-exp 99.6%

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

   6. metadata-eval 99.6%

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

7. Applied egg-rr 99.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)} \cdot {\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r}\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 99.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}\right)\right)}}{r}\right) \]

   2. expm1-udef 98.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\sqrt[3]{e^{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}\right)} - 1\right)}}{r}\right) \]

9. Applied egg-rr 98.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{{\left(e^{-0.2222222222222222}\right)}^{\left(\frac{r}{s}\right)}}\right)} - 1\right)}}{r}\right) \]
10. Step-by-step derivation

    1. expm1-def 99.6%

       \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(e^{-0.2222222222222222}\right)}^{\left(\frac{r}{s}\right)}}\right)\right)}}{r}\right) \]

    2. expm1-log1p 99.6%

       \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)} \cdot \color{blue}{\sqrt{{\left(e^{-0.2222222222222222}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

11. Simplified 99.6%

    \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)} \cdot \color{blue}{\sqrt{{\left(e^{-0.2222222222222222}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

12. Final simplification 99.6%

    \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(\sqrt[3]{e^{-1.3333333333333333}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)} \cdot \sqrt{{\left(e^{-0.2222222222222222}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-/l* 99.6%

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

   2. *-commutative 99.6%

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

   3. *-commutative 99.6%

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

   4. times-frac 99.6%

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

   5. associate-*l* 99.6%

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

   6. *-commutative 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0 99.6%

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

3. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.3× speedup

Math

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

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(0.125) / Float32(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 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in s around 0 99.3%

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

   1. pow-exp 99.5%

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

   2. *-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in r around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*l/ 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in r around 0 99.5%

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

11. Final simplification 99.5%

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

Alternative 5: 11.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{0.25 \cdot \left(\pi \cdot {r}^{2}\right) + \left(r \cdot \left(\pi \cdot s\right)\right) \cdot 0.5} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ 0.125 (+ (* 0.25 (* PI (pow r 2.0))) (* (* r (* PI s)) 0.5))))

C

float code(float s, float r) {
	return 0.125f / ((0.25f * (((float) M_PI) * powf(r, 2.0f))) + ((r * (((float) M_PI) * s)) * 0.5f));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) / Float32(Float32(Float32(0.25) * Float32(Float32(pi) * (r ^ Float32(2.0)))) + Float32(Float32(r * Float32(Float32(pi) * s)) * Float32(0.5))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) / ((single(0.25) * (single(pi) * (r ^ single(2.0)))) + ((r * (single(pi) * s)) * single(0.5)));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. associate-/l* 10.0%

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

   3. associate-*r/ 10.0%

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

   4. neg-mul-1 10.0%

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

7. Simplified 10.0%

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

8. Taylor expanded in s around inf 11.7%

   \[\leadsto \frac{0.125}{\color{blue}{0.25 \cdot \left({r}^{2} \cdot \pi\right) + 0.5 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)}} \]

9. Final simplification 11.7%

   \[\leadsto \frac{0.125}{0.25 \cdot \left(\pi \cdot {r}^{2}\right) + \left(r \cdot \left(\pi \cdot s\right)\right) \cdot 0.5} \]

Alternative 6: 12.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(\left(\pi \cdot s\right) \cdot 4\right)\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ 1.0 (log1p (expm1 (* r (* (* PI s) 4.0))))))

C

float code(float s, float r) {
	return 1.0f / log1pf(expm1f((r * ((((float) M_PI) * s) * 4.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / log1p(expm1(Float32(r * Float32(Float32(Float32(pi) * s) * Float32(4.0))))))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around inf 9.4%

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

   1. clear-num 9.4%

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

   2. inv-pow 9.4%

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

   3. *-commutative 9.4%

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

7. Applied egg-rr 9.4%

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

   1. unpow-1 9.4%

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

   2. associate-/l* 9.4%

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

   3. *-commutative 9.4%

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

9. Simplified 9.4%

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

    1. /-rgt-identity 9.4%

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

    2. log1p-expm1-u 10.8%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{r}{\frac{0.25}{s \cdot \pi}}}{1}\right)\right)}} \]

    3. /-rgt-identity 10.8%

       \[\leadsto \frac{1}{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{r}{\frac{0.25}{s \cdot \pi}}}\right)\right)} \]

    4. div-inv 10.8%

       \[\leadsto \frac{1}{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{r \cdot \frac{1}{\frac{0.25}{s \cdot \pi}}}\right)\right)} \]

    5. clear-num 10.8%

       \[\leadsto \frac{1}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \color{blue}{\frac{s \cdot \pi}{0.25}}\right)\right)} \]

    6. div-inv 10.8%

       \[\leadsto \frac{1}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \color{blue}{\left(\left(s \cdot \pi\right) \cdot \frac{1}{0.25}\right)}\right)\right)} \]

    7. metadata-eval 10.8%

       \[\leadsto \frac{1}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(\left(s \cdot \pi\right) \cdot \color{blue}{4}\right)\right)\right)} \]

11. Applied egg-rr 10.8%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(\left(s \cdot \pi\right) \cdot 4\right)\right)\right)}} \]

12. Final simplification 10.8%

    \[\leadsto \frac{1}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(\left(\pi \cdot s\right) \cdot 4\right)\right)\right)} \]

Alternative 7: 10.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in s around 0 99.3%

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

5. Taylor expanded in r around 0 10.7%

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

6. Final simplification 10.7%

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

Alternative 8: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

6. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. *-commutative 10.0%

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

   3. times-frac 10.1%

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

   4. mul-1-neg 10.1%

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

   5. distribute-neg-frac 10.1%

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

8. Simplified 10.1%

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

9. Final simplification 10.1%

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

Alternative 9: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in r around inf 10.0%

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

   1. associate-*r/ 10.0%

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

   2. neg-mul-1 10.0%

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

7. Simplified 10.0%

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

8. Final simplification 10.0%

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

Alternative 10: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. associate-/l* 10.0%

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

   3. associate-*r/ 10.0%

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

   4. neg-mul-1 10.0%

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

7. Simplified 10.0%

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

8. Taylor expanded in r around inf 10.0%

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

   1. associate-/l* 10.0%

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

   2. associate-/r/ 10.0%

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

   3. mul-1-neg 10.0%

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

   4. distribute-neg-frac 10.0%

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

10. Simplified 10.0%

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

11. Final simplification 10.0%

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

Alternative 11: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in r around inf 10.0%

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

   1. associate-*r/ 10.0%

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

   2. associate-/l* 10.0%

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

   3. associate-*r/ 10.0%

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

   4. neg-mul-1 10.0%

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

7. Simplified 10.0%

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

8. Final simplification 10.0%

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

Alternative 12: 9.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in r around 0 9.4%

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

6. Final simplification 9.4%

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

Alternative 13: 9.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around inf 9.4%

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

   1. clear-num 9.4%

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

   2. inv-pow 9.4%

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

   3. *-commutative 9.4%

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

7. Applied egg-rr 9.4%

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

   1. unpow-1 9.4%

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

   2. associate-/l* 9.4%

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

   3. *-commutative 9.4%

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

9. Simplified 9.4%

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

    1. associate-/r/ 9.4%

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

11. Applied egg-rr 9.4%

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

12. Final simplification 9.4%

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

Alternative 14: 9.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around inf 9.4%

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

6. Final simplification 9.4%

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

Alternative 15: 9.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around inf 9.4%

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

   1. clear-num 9.4%

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

   2. inv-pow 9.4%

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

   3. *-commutative 9.4%

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

7. Applied egg-rr 9.4%

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

   1. unpow-1 9.4%

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

   2. associate-/l* 9.4%

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

   3. *-commutative 9.4%

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

9. Simplified 9.4%

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

10. Taylor expanded in r around 0 9.4%

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

    1. associate-/l/ 9.4%

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

12. Simplified 9.4%

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

13. Final simplification 9.4%

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

Reproduce

herbie shell --seed 2023310 
(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))))