Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 19.4s

Alternatives: 13

Speedup: 0.7×

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. times-frac 99.5%

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

   2. fma-def 99.5%

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

   3. associate-*l* 99.5%

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

   4. associate-/r* 99.5%

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

   5. *-commutative 99.5%

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

   6. metadata-eval 99.5%

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

   7. times-frac 99.5%

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

   8. associate-*l* 99.5%

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

   9. associate-/r* 99.5%

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

   10. metadata-eval 99.5%

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

   11. *-commutative 99.5%

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

3. Simplified 99.4%

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

   1. pow-exp 99.1%

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

   2. sqr-pow 99.2%

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

   3. pow-prod-down 99.1%

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

   4. prod-exp 99.5%

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

   5. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

   1. add-exp-log 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. times-frac 99.5%

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

   2. fma-def 99.5%

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

   3. associate-*l* 99.5%

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

   4. associate-/r* 99.5%

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

   5. *-commutative 99.5%

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

   6. metadata-eval 99.5%

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

   7. times-frac 99.5%

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

   8. associate-*l* 99.5%

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

   9. associate-/r* 99.5%

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

   10. metadata-eval 99.5%

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

   11. *-commutative 99.5%

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

3. Simplified 99.4%

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

   1. pow-exp 99.1%

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

   2. sqr-pow 99.2%

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

   3. pow-prod-down 99.1%

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

   4. prod-exp 99.5%

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

   5. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-r}{s}}}{r} \cdot \frac{0.25}{\left(s \cdot \pi\right) \cdot 2} + \left(0.75 \cdot \frac{1}{s \cdot \left(\pi \cdot 6\right)}\right) \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (exp (/ (- r) s)) r) (/ 0.25 (* (* s PI) 2.0)))
  (* (* 0.75 (/ 1.0 (* s (* PI 6.0)))) (/ (exp (/ (- r) (* s 3.0))) r))))

C

float code(float s, float r) {
	return ((expf((-r / s)) / r) * (0.25f / ((s * ((float) M_PI)) * 2.0f))) + ((0.75f * (1.0f / (s * (((float) M_PI) * 6.0f)))) * (expf((-r / (s * 3.0f))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) / r) * Float32(Float32(0.25) / Float32(Float32(s * Float32(pi)) * Float32(2.0)))) + Float32(Float32(Float32(0.75) * Float32(Float32(1.0) / Float32(s * Float32(Float32(pi) * Float32(6.0))))) * Float32(exp(Float32(Float32(-r) / Float32(s * Float32(3.0)))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) / r) * (single(0.25) / ((s * single(pi)) * single(2.0)))) + ((single(0.75) * (single(1.0) / (s * (single(pi) * single(6.0))))) * (exp((-r / (s * single(3.0)))) / r));
end

Derivation

1. Initial program 99.5%

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

   1. times-frac 99.5%

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

   2. fma-def 99.5%

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

   3. /-rgt-identity 99.5%

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

   4. fma-def 99.5%

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

   5. /-rgt-identity 99.5%

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

   6. associate-*l* 99.5%

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

   7. times-frac 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*r* 99.5%

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

6. Simplified 99.5%

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

   1. div-inv 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. times-frac 99.5%

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

   2. fma-def 99.5%

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

   3. /-rgt-identity 99.5%

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

   4. fma-def 99.5%

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

   5. /-rgt-identity 99.5%

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

   6. associate-*l* 99.5%

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

   7. times-frac 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*r* 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in s around 0 99.5%

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

8. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around 0 99.1%

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

5. Taylor expanded in r around inf 99.4%

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

6. Final simplification 99.4%

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

Alternative 6: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

   1. add-cube-cbrt 99.1%

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

   2. pow3 99.1%

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

   3. associate-/l/ 99.1%

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

   4. *-commutative 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in r around inf 99.3%

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

   1. rem-cube-cbrt 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-/r* 99.4%

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 7: 43.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.3%

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

5. Taylor expanded in s around inf 7.9%

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

   1. log1p-expm1-u 45.8%

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

   2. *-commutative 45.8%

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

7. Applied egg-rr 45.8%

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

8. Final simplification 45.8%

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

Alternative 8: 9.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(0.75 \cdot \frac{1}{s \cdot \left(\pi \cdot 6\right)}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (* 0.75 (/ 1.0 (* s (* PI 6.0)))) (+ (/ (exp (/ r (- s))) r) (/ 1.0 r))))

C

float code(float s, float r) {
	return (0.75f * (1.0f / (s * (((float) M_PI) * 6.0f)))) * ((expf((r / -s)) / r) + (1.0f / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.75) * Float32(Float32(1.0) / Float32(s * Float32(Float32(pi) * Float32(6.0))))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(1.0) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.75) * (single(1.0) / (s * (single(pi) * single(6.0))))) * ((exp((r / -s)) / r) + (single(1.0) / r));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.3%

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

   1. associate-/l/ 8.3%

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

   2. metadata-eval 8.3%

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

   3. *-commutative 8.3%

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

   4. associate-/r* 8.3%

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

   5. div-inv 8.3%

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

   6. associate-*r* 8.3%

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

   7. *-commutative 8.3%

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

   8. *-commutative 8.3%

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

6. Applied egg-rr 8.3%

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

7. Final simplification 8.3%

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

Alternative 9: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.3%

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

   1. div-inv 8.3%

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

6. Applied egg-rr 8.3%

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

7. Final simplification 8.3%

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

Alternative 10: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.3%

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

5. Taylor expanded in r around inf 8.3%

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

   1. +-commutative 8.3%

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

   2. associate-*r/ 8.3%

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

   3. neg-mul-1 8.3%

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

   4. *-commutative 8.3%

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

   5. associate-*l* 8.3%

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

   6. *-commutative 8.3%

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

7. Simplified 8.3%

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

8. Final simplification 8.3%

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

Alternative 11: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.3%

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

5. Taylor expanded in s around 0 8.3%

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

6. Taylor expanded in r around inf 8.3%

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

   1. associate-*r/ 8.3%

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

   2. +-commutative 8.3%

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

   3. distribute-lft-in 8.3%

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

   4. metadata-eval 8.3%

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

   5. associate-*r/ 8.3%

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

   6. mul-1-neg 8.3%

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

   7. associate-*r* 8.3%

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

   8. *-commutative 8.3%

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

8. Simplified 8.3%

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

9. Final simplification 8.3%

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

Alternative 12: 9.0% 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 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.3%

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

5. Taylor expanded in r around 0 8.0%

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

6. Final simplification 8.0%

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

Alternative 13: 9.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.3%

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

5. Taylor expanded in s around inf 7.9%

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

6. Final simplification 7.9%

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

Reproduce

herbie shell --seed 2023279 
(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))))