Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 12.6s

Alternatives: 18

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(s * Float32(pi)))
	return fma(t_0, Float32((Float32(exp(1)) ^ Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / r), Float32(t_0 * Float32(exp(Float32(r / Float32(-s))) / r)))
end

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.7%

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

   3. fma-define 99.7%

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

   4. associate-*l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

   7. *-commutative 99.6%

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

   8. neg-mul-1 99.6%

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

   9. times-frac 99.6%

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

   10. metadata-eval 99.6%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. exp-prod 99.6%

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

   3. rem-log-exp 99.5%

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

   4. associate-*r/ 99.5%

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

   5. *-commutative 99.5%

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

   6. rem-log-exp 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32((Float32(exp(1)) ^ Float32(r * Float32(Float32(-0.3333333333333333) / s))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + ((single(2.71828182845904523536) ^ (r * (single(-0.3333333333333333) / 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 99.4%

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

   1. pow-to-exp 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-commutative 99.4%

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

   4. rem-log-exp 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/r* 99.6%

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

   2. div-inv 99.6%

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

8. Applied egg-rr 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. exp-prod 99.6%

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

   3. associate-/l* 99.6%

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

10. Applied egg-rr 99.6%

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

    1. exp-1-e 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in s around 0 99.7%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.125f / ((float) M_PI)) / s) * ((expf((r / -s)) / r) + (expf(((r * -0.3333333333333333f) / 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(exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / single(pi)) / s) * ((exp((r / -s)) / r) + (exp(((r * single(-0.3333333333333333)) / 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 99.4%

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

   1. pow-to-exp 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-commutative 99.4%

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

   4. rem-log-exp 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/r* 99.6%

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

   2. div-inv 99.6%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in s around 0 99.6%

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

    1. associate-/l/ 99.6%

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

11. Simplified 99.6%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (exp((single(-0.3333333333333333) * (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 99.4%

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

5. Taylor expanded in r around inf 99.6%

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

Alternative 5: 44.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.8%

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

6. Taylor expanded in s around inf 10.2%

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

   1. pow1 10.2%

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

   2. associate-*r* 10.2%

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

8. Applied egg-rr 10.2%

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

   1. unpow1 10.2%

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

   2. associate-*r* 10.2%

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

   3. *-commutative 10.2%

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

   4. associate-*l* 10.2%

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

10. Simplified 10.2%

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

    1. log1p-expm1-u 44.4%

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

    2. *-commutative 44.4%

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

12. Applied egg-rr 44.4%

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

13. Final simplification 44.4%

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

Alternative 6: 12.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(Float32(pi) * Float32(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 99.4%

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

5. Taylor expanded in r around 0 10.8%

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

6. Taylor expanded in s around inf 10.2%

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

   1. log1p-expm1-u 13.7%

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

   2. associate-*r* 13.7%

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

8. Applied egg-rr 13.7%

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

9. Final simplification 13.7%

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

Alternative 7: 10.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / s) * (single(1.0) / single(pi))) * ((exp((r / -s)) / r) + ((single(1.0) + ((r * single(-0.3333333333333333)) / 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 99.4%

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

   1. pow-to-exp 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-commutative 99.4%

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

   4. rem-log-exp 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/r* 99.6%

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

   2. div-inv 99.6%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in r around 0 11.6%

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

    1. associate-*r/ 11.6%

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

11. Simplified 11.6%

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

12. Final simplification 11.6%

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

Alternative 8: 10.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 11.6%

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

   1. associate-*r/ 11.6%

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

7. Simplified 11.6%

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

8. Final simplification 11.6%

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

Alternative 9: 9.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp(((r * single(-0.3333333333333333)) / s)) / r) + ((single(1.0) - (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 99.4%

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

   1. pow-to-exp 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-commutative 99.4%

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

   4. rem-log-exp 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in r around 0 10.9%

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

   1. mul-1-neg 10.9%

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

   2. unsub-neg 10.9%

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

9. Simplified 10.9%

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

10. Final simplification 10.9%

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

Alternative 10: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / single(pi)) * (((exp((r / -s)) / r) + (single(1.0) / 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 99.4%

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

5. Taylor expanded in r around 0 10.8%

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

6. Taylor expanded in s around 0 10.8%

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

   1. associate-*r/ 10.8%

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

   2. *-commutative 10.8%

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

   3. times-frac 10.8%

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

   4. mul-1-neg 10.8%

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

   5. distribute-neg-frac2 10.8%

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

8. Simplified 10.8%

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

9. Final simplification 10.8%

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

Alternative 11: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / single(pi)) * (((single(1.0) + 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 99.4%

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

5. Taylor expanded in r around 0 10.8%

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

6. Taylor expanded in s around 0 10.8%

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

   1. associate-*r/ 10.8%

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

   2. *-commutative 10.8%

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

   3. times-frac 10.8%

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

   4. mul-1-neg 10.8%

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

   5. distribute-neg-frac2 10.8%

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

8. Simplified 10.8%

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

9. Taylor expanded in r around inf 10.8%

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

    1. associate-*r/ 10.8%

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

    2. neg-mul-1 10.8%

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

11. Simplified 10.8%

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

12. Final simplification 10.8%

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

Alternative 12: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / single(pi)) * ((single(1.0) + exp((r / -s))) / (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 99.4%

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

5. Taylor expanded in r around 0 10.8%

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

6. Taylor expanded in s around 0 10.8%

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

   1. associate-*r/ 10.8%

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

   2. *-commutative 10.8%

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

   3. times-frac 10.8%

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

   4. mul-1-neg 10.8%

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

   5. distribute-neg-frac2 10.8%

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

8. Simplified 10.8%

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

9. Taylor expanded in r around inf 10.8%

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

    1. associate-*r/ 10.8%

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

    2. neg-mul-1 10.8%

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

    3. *-commutative 10.8%

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

11. Simplified 10.8%

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

12. Final simplification 10.8%

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

Alternative 13: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((single(1.0) + exp((r / -s))) / ((s * single(pi)) * 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 99.4%

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

5. Taylor expanded in r around 0 10.8%

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

6. Taylor expanded in r around inf 10.8%

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

   1. mul-1-neg 10.8%

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

8. Simplified 10.8%

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

9. Final simplification 10.8%

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

Alternative 14: 9.3% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{\pi} \cdot \frac{\frac{2}{r}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / single(pi)) * ((single(2.0) / 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 99.4%

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

5. Taylor expanded in r around 0 10.8%

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

6. Taylor expanded in s around 0 10.8%

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

   1. associate-*r/ 10.8%

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

   2. *-commutative 10.8%

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

   3. times-frac 10.8%

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

   4. mul-1-neg 10.8%

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

   5. distribute-neg-frac2 10.8%

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

8. Simplified 10.8%

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

9. Taylor expanded in r around 0 10.2%

   \[\leadsto \frac{0.125}{\pi} \cdot \frac{\color{blue}{\frac{2}{r}}}{s} \]
10. Add Preprocessing

Alternative 15: 9.3% accurate, 33.0× speedup

Math

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

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(Float32(0.25) / 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 99.4%

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

5. Taylor expanded in r around 0 10.8%

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

6. Taylor expanded in s around 0 10.8%

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

   1. associate-*r/ 10.8%

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

   2. *-commutative 10.8%

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

   3. times-frac 10.8%

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

   4. mul-1-neg 10.8%

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

   5. distribute-neg-frac2 10.8%

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

8. Simplified 10.8%

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

9. Taylor expanded in r around 0 10.2%

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

    1. associate-/l/ 10.2%

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

    2. *-commutative 10.2%

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

    3. associate-/r* 10.2%

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

11. Simplified 10.2%

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

Alternative 16: 9.3% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.8%

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

6. Taylor expanded in s around inf 10.2%

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

   1. associate-/r* 10.2%

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

8. Simplified 10.2%

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

Alternative 17: 9.3% accurate, 33.0× 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 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.8%

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

6. Taylor expanded in s around inf 10.2%

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

   1. pow1 10.2%

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

   2. associate-*r* 10.2%

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

8. Applied egg-rr 10.2%

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

   1. unpow1 10.2%

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

   2. associate-*r* 10.2%

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

   3. *-commutative 10.2%

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

   4. associate-*l* 10.2%

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

10. Simplified 10.2%

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

Alternative 18: 9.3% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.8%

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

6. Taylor expanded in s around inf 10.2%

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

7. Final simplification 10.2%

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

Reproduce

herbie shell --seed 2024108 
(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))))