Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.6%

Time: 13.7s

Alternatives: 14

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ \mathsf{fma}\left(t\_0, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r}, t\_0 \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. times-frac 99.3%

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

   3. fma-define 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. times-frac 99.3%

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

   10. metadata-eval 99.3%

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

   11. times-frac 99.3%

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

3. Simplified 99.3%

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

   1. pow-exp 99.0%

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

   2. sqr-pow 99.0%

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

   3. pow-prod-down 99.0%

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

   4. prod-exp 99.4%

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

   5. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. times-frac 99.3%

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

   3. fma-define 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. times-frac 99.3%

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

   10. metadata-eval 99.3%

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

   11. times-frac 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around 0 99.2%

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

6. Taylor expanded in r around inf 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. neg-mul-1 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. associate-*r/ 99.3%

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

8. Simplified 99.3%

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

   1. associate-/l* 99.3%

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

   2. *-commutative 99.3%

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

   3. *-un-lft-identity 99.3%

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

   4. pow-exp 99.4%

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

   5. exp-1-e 99.4%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around inf 99.3%

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

6. Final simplification 99.3%

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

Alternative 4: 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 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.8%

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

   1. *-commutative 10.8%

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

   2. associate-*l* 10.8%

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

   3. *-commutative 10.8%

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

7. Simplified 10.8%

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

   1. log1p-expm1-u 47.1%

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

9. Applied egg-rr 47.1%

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

10. Final simplification 47.1%

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

Alternative 5: 11.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.8%

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

   1. log1p-expm1-u 14.3%

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

7. Applied egg-rr 14.3%

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

8. Final simplification 14.3%

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

Alternative 6: 10.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(\frac{\frac{-1}{\pi} - -0.5 \cdot \frac{r}{s \cdot \pi}}{s} + \frac{1}{\pi \cdot r}\right) + 0.125 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (* 0.125 (+ (/ (- (/ -1.0 PI) (* -0.5 (/ r (* s PI)))) s) (/ 1.0 (* PI r))))
   (* 0.125 (/ (exp (* (/ r s) -0.3333333333333333)) (* PI r))))
  s))

C

float code(float s, float r) {
	return ((0.125f * ((((-1.0f / ((float) M_PI)) - (-0.5f * (r / (s * ((float) M_PI))))) / s) + (1.0f / (((float) M_PI) * r)))) + (0.125f * (expf(((r / s) * -0.3333333333333333f)) / (((float) M_PI) * r)))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(Float32(Float32(-1.0) / Float32(pi)) - Float32(Float32(-0.5) * Float32(r / Float32(s * Float32(pi))))) / s) + Float32(Float32(1.0) / Float32(Float32(pi) * r)))) + Float32(Float32(0.125) * Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(Float32(pi) * r)))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) * ((((single(-1.0) / single(pi)) - (single(-0.5) * (r / (s * single(pi))))) / s) + (single(1.0) / (single(pi) * r)))) + (single(0.125) * (exp(((r / s) * single(-0.3333333333333333))) / (single(pi) * r)))) / s;
end

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. times-frac 99.3%

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

   3. fma-define 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. times-frac 99.3%

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

   10. metadata-eval 99.3%

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

   11. times-frac 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around 0 99.2%

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

6. Taylor expanded in s around -inf 12.4%

   \[\leadsto \frac{0.125 \cdot \color{blue}{\left(-1 \cdot \frac{-0.5 \cdot \frac{r}{s \cdot \pi} + \frac{1}{\pi}}{s} + \frac{1}{r \cdot \pi}\right)} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s} \]

7. Final simplification 12.4%

   \[\leadsto \frac{0.125 \cdot \left(\frac{\frac{-1}{\pi} - -0.5 \cdot \frac{r}{s \cdot \pi}}{s} + \frac{1}{\pi \cdot r}\right) + 0.125 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi \cdot r}}{s} \]
8. Add Preprocessing

Alternative 7: 10.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{\left(\frac{\frac{-1}{\pi} - -0.5 \cdot \frac{r}{s \cdot \pi}}{s} + \frac{1}{\pi \cdot r}\right) + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{\pi \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  0.125
  (/
   (+
    (+ (/ (- (/ -1.0 PI) (* -0.5 (/ r (* s PI)))) s) (/ 1.0 (* PI r)))
    (/ (exp (/ (* r -0.3333333333333333) s)) (* PI r)))
   s)))

C

float code(float s, float r) {
	return 0.125f * ((((((-1.0f / ((float) M_PI)) - (-0.5f * (r / (s * ((float) M_PI))))) / s) + (1.0f / (((float) M_PI) * r))) + (expf(((r * -0.3333333333333333f) / s)) / (((float) M_PI) * r))) / s);
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(-1.0) / Float32(pi)) - Float32(Float32(-0.5) * Float32(r / Float32(s * Float32(pi))))) / s) + Float32(Float32(1.0) / Float32(Float32(pi) * r))) + Float32(exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / Float32(Float32(pi) * r))) / s))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((((((single(-1.0) / single(pi)) - (single(-0.5) * (r / (s * single(pi))))) / s) + (single(1.0) / (single(pi) * r))) + (exp(((r * single(-0.3333333333333333)) / s)) / (single(pi) * r))) / s);
end

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. times-frac 99.3%

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

   3. fma-define 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. times-frac 99.3%

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

   10. metadata-eval 99.3%

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

   11. times-frac 99.3%

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

3. Simplified 99.3%

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

   1. pow-exp 99.0%

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

   2. sqr-pow 99.0%

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

   3. pow-prod-down 99.0%

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

   4. prod-exp 99.4%

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

   5. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in s around 0 99.2%

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

   1. associate-*r/ 99.2%

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

   2. neg-mul-1 99.2%

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

   3. distribute-frac-neg2 99.2%

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

   4. associate-/l* 99.2%

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

   5. exp-prod 98.9%

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

   6. distribute-lft-in 98.9%

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

   7. associate-/l* 98.9%

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

9. Simplified 99.2%

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

10. Taylor expanded in s around -inf 12.4%

    \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(-1 \cdot \frac{-0.5 \cdot \frac{r}{s \cdot \pi} + \frac{1}{\pi}}{s} + \frac{1}{r \cdot \pi}\right)} + \frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r \cdot \pi}}{s} \]

11. Final simplification 12.4%

    \[\leadsto 0.125 \cdot \frac{\left(\frac{\frac{-1}{\pi} - -0.5 \cdot \frac{r}{s \cdot \pi}}{s} + \frac{1}{\pi \cdot r}\right) + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{\pi \cdot r}}{s} \]
12. Add Preprocessing

Alternative 8: 10.1% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125 \cdot \frac{\frac{r}{\pi} \cdot 0.5555555555555556}{s} - \frac{0.16666666666666666}{\pi}}{s} + \frac{\frac{0.25}{r}}{\pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/
    (-
     (* 0.125 (/ (* (/ r PI) 0.5555555555555556) s))
     (/ 0.16666666666666666 PI))
    s)
   (/ (/ 0.25 r) PI))
  s))

C

float code(float s, float r) {
	return ((((0.125f * (((r / ((float) M_PI)) * 0.5555555555555556f) / s)) - (0.16666666666666666f / ((float) M_PI))) / s) + ((0.25f / r) / ((float) M_PI))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(r / Float32(pi)) * Float32(0.5555555555555556)) / s)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s) + Float32(Float32(Float32(0.25) / r) / Float32(pi))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((((single(0.125) * (((r / single(pi)) * single(0.5555555555555556)) / s)) - (single(0.16666666666666666) / single(pi))) / s) + ((single(0.25) / r) / single(pi))) / s;
end

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. times-frac 99.3%

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

   3. fma-define 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. times-frac 99.3%

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

   10. metadata-eval 99.3%

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

   11. times-frac 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around 0 99.2%

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

6. Taylor expanded in r around inf 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. neg-mul-1 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. associate-*r/ 99.3%

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

8. Simplified 99.3%

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

9. Taylor expanded in s around -inf 12.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
10. Step-by-step derivation

    1. mul-1-neg 12.4%

       \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

11. Simplified 12.4%

    \[\leadsto \color{blue}{-\frac{\left(-\frac{0.125 \cdot \frac{\frac{r}{\pi} \cdot 0.5555555555555556}{s} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s}} \]

12. Final simplification 12.4%

    \[\leadsto \frac{\frac{0.125 \cdot \frac{\frac{r}{\pi} \cdot 0.5555555555555556}{s} - \frac{0.16666666666666666}{\pi}}{s} + \frac{\frac{0.25}{r}}{\pi}}{s} \]
13. Add Preprocessing

Alternative 9: 10.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/ (- (* (/ r (* s PI)) 0.06944444444444445) (/ 0.16666666666666666 PI)) s)
   (/ 0.25 (* PI r)))
  s))

C

float code(float s, float r) {
	return (((((r / (s * ((float) M_PI))) * 0.06944444444444445f) - (0.16666666666666666f / ((float) M_PI))) / s) + (0.25f / (((float) M_PI) * r))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(r / Float32(s * Float32(pi))) * Float32(0.06944444444444445)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s) + Float32(Float32(0.25) / Float32(Float32(pi) * r))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (((((r / (s * single(pi))) * single(0.06944444444444445)) - (single(0.16666666666666666) / single(pi))) / s) + (single(0.25) / (single(pi) * r))) / s;
end

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. times-frac 99.3%

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

   3. fma-define 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. times-frac 99.3%

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

   10. metadata-eval 99.3%

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

   11. times-frac 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around 0 99.2%

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

6. Taylor expanded in s around -inf 12.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(0.006944444444444444 \cdot \frac{r}{s \cdot \pi} + 0.0625 \cdot \frac{r}{s \cdot \pi}\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
7. Step-by-step derivation

   1. mul-1-neg 12.4%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{\left(0.006944444444444444 \cdot \frac{r}{s \cdot \pi} + 0.0625 \cdot \frac{r}{s \cdot \pi}\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

8. Simplified 12.4%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{r \cdot \pi}}{s}} \]

9. Final simplification 12.4%

   \[\leadsto \frac{\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s} \]
10. Add Preprocessing

Alternative 10: 9.6% accurate, 15.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around 0 11.5%

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

6. Taylor expanded in r around inf 11.5%

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

   1. associate-*r/ 11.5%

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

   2. neg-mul-1 11.5%

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

   3. distribute-frac-neg2 11.5%

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

   4. distribute-lft-in 11.5%

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

   5. metadata-eval 11.5%

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

   6. distribute-frac-neg2 11.5%

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

   7. rec-exp 11.5%

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

   8. associate-*r/ 11.5%

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

   9. metadata-eval 11.5%

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

8. Simplified 11.5%

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

9. Taylor expanded in r around 0 11.5%

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

10. Final simplification 11.5%

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

Alternative 11: 9.1% 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 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.8%

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

   1. *-un-lft-identity 10.8%

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

   2. associate-/r* 10.9%

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

7. Applied egg-rr 10.9%

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

8. Final simplification 10.9%

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

Alternative 12: 9.1% accurate, 33.0× speedup

Math

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

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(Float32(0.25) / r) / s) / Float32(pi))
end

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around 0 11.5%

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

6. Taylor expanded in s around inf 10.8%

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

   1. associate-/r* 10.9%

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

   2. associate-/r* 10.9%

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

8. Simplified 10.9%

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

Alternative 13: 9.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.8%

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

   1. *-commutative 10.8%

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

   2. associate-*l* 10.8%

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

   3. *-commutative 10.8%

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

7. Simplified 10.8%

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

8. Taylor expanded in s around 0 10.8%

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

   1. associate-*r* 10.8%

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

   2. *-commutative 10.8%

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

   3. associate-*l* 10.8%

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

   4. associate-/l/ 10.8%

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

10. Simplified 10.8%

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

11. Taylor expanded in r around 0 10.8%

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

    1. *-commutative 10.8%

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

    2. associate-/r* 10.8%

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

13. Simplified 10.8%

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

Alternative 14: 9.1% 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 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.8%

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

6. Final simplification 10.8%

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

Reproduce

herbie shell --seed 2024123 
(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))))