Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 35.4s

Alternatives: 10

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\frac{0.125}{s}}{\pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (/ 0.125 s) PI)
  (/ (pow (exp -0.6666666666666666) (/ (* r 0.5) s)) r)
  (* (/ 0.125 (* s PI)) (/ (exp (/ (- r) s)) r))))

C

float code(float s, float r) {
	return fmaf(((0.125f / s) / ((float) M_PI)), (powf(expf(-0.6666666666666666f), ((r * 0.5f) / s)) / r), ((0.125f / (s * ((float) M_PI))) * (expf((-r / s)) / r)));
}

Julia

function code(s, r)
	return fma(Float32(Float32(Float32(0.125) / s) / Float32(pi)), Float32((exp(Float32(-0.6666666666666666)) ^ Float32(Float32(r * Float32(0.5)) / s)) / r), Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(exp(Float32(Float32(-r) / s)) / r)))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.5%

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

   1. pow-exp 99.3%

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

   2. sqr-pow 99.3%

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

   3. pow-prod-down 99.3%

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

   4. prod-exp 99.7%

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

   5. metadata-eval 99.7%

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

   6. div-inv 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r}{s} \cdot \frac{1}{2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   7. metadata-eval 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot \color{blue}{0.5}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

8. Simplified 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
9. Step-by-step derivation

   1. expm1-log1p-u 99.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{s \cdot \pi}\right)\right)}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

10. Applied egg-rr 99.8%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{s \cdot \pi}\right)\right)}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
11. Step-by-step derivation

    1. expm1-log1p-u 99.7%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.125}{s \cdot \pi}}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

    2. associate-/r* 99.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.125}{s}}{\pi}}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

12. Applied egg-rr 99.8%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.125}{s}}{\pi}}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

13. Final simplification 99.8%

    \[\leadsto \mathsf{fma}\left(\frac{\frac{0.125}{s}}{\pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
14. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ \mathsf{fma}\left(t_0, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, t_0 \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* s PI))))
   (fma
    t_0
    (/ (pow (exp -0.6666666666666666) (/ (* r 0.5) 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(expf(-0.6666666666666666f), ((r * 0.5f) / 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((exp(Float32(-0.6666666666666666)) ^ Float32(Float32(r * Float32(0.5)) / s)) / r), Float32(t_0 * Float32(exp(Float32(Float32(-r) / s)) / r)))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.5%

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

   1. pow-exp 99.3%

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

   2. sqr-pow 99.3%

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

   3. pow-prod-down 99.3%

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

   4. prod-exp 99.7%

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

   5. metadata-eval 99.7%

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

   6. div-inv 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r}{s} \cdot \frac{1}{2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   7. metadata-eval 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot \color{blue}{0.5}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

8. Simplified 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

9. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
10. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-r}{s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{-1}{s} \cdot \left(r \cdot 0.3333333333333333\right)}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (/ (- r) s)) 0.25) (* r (* s (* PI 2.0))))
  (/
   (* 0.75 (exp (* (/ -1.0 s) (* r 0.3333333333333333))))
   (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	return ((expf((-r / s)) * 0.25f) / (r * (s * (((float) M_PI) * 2.0f)))) + ((0.75f * expf(((-1.0f / s) * (r * 0.3333333333333333f)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) * Float32(0.25)) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(Float32(-1.0) / s) * Float32(r * Float32(0.3333333333333333))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) * single(0.25)) / (r * (s * (single(pi) * single(2.0))))) + ((single(0.75) * exp(((single(-1.0) / s) * (r * single(0.3333333333333333))))) / (r * (s * (single(pi) * single(6.0)))));
end

Derivation

1. Initial program 99.6%

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

   1. neg-mul-1 99.6%

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

   2. *-commutative 99.6%

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

   3. times-frac 99.6%

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

   4. add-sqr-sqrt 99.5%

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

   5. sqrt-unprod 99.6%

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

   6. sqr-neg 99.6%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 7.0%

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

   9. div-inv 7.0%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 99.6%

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

   12. sqr-neg 99.6%

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

   13. sqrt-unprod 99.6%

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

   14. add-sqr-sqrt 99.6%

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

   15. metadata-eval 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-/r/ 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. fma-def 99.6%

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

   3. associate-*l* 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. metadata-eval 99.6%

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

   7. associate-/r* 99.6%

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

   8. associate-*l* 99.6%

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

   9. /-rgt-identity 99.6%

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

   10. fma-def 99.6%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around 0 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. fma-def 99.6%

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

   3. associate-*l* 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. metadata-eval 99.6%

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

   7. associate-/r* 99.6%

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

   8. associate-*l* 99.6%

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

   9. /-rgt-identity 99.6%

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

   10. fma-def 99.6%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*l* 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in s around 0 99.6%

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

9. Final simplification 99.6%

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

Alternative 7: 93.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (exp((single(-0.3333333333333333) * (r / s))) / (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 99.3%

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

   1. pow-to-exp 99.3%

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

   2. rem-log-exp 99.5%

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

   3. metadata-eval 99.5%

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

   4. times-frac 99.6%

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

   5. neg-mul-1 99.6%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 7.0%

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

   8. sqr-neg 7.0%

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

   9. sqrt-unprod 7.0%

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

   10. add-sqr-sqrt 7.0%

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

   11. frac-2neg 7.0%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 99.6%

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

   14. sqr-neg 99.6%

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

   15. sqrt-unprod 99.5%

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

   16. add-sqr-sqrt 99.6%

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

   17. *-commutative 99.6%

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

   18. distribute-rgt-neg-in 99.6%

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

   19. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in r around 0 13.5%

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

8. Taylor expanded in s around 0 94.3%

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

9. Final simplification 94.3%

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

Alternative 8: 9.4% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (((single(1.0) / r) + (single(1.0) / (r * ((r / s) + single(1.0))))) / (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 99.3%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in s around 0 8.7%

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

7. Taylor expanded in r around 0 8.7%

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

8. Final simplification 8.7%

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

Alternative 9: 9.4% accurate, 13.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / r) + ((single(0.125) / r) / ((r / s) + single(1.0)))) / (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 99.3%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in s around 0 8.7%

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

   1. expm1-log1p-u 8.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot \frac{\frac{1}{r} + \frac{1}{r \cdot e^{\frac{r}{s}}}}{s \cdot \pi}\right)\right)} \]

   2. expm1-udef 9.0%

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

   3. associate-*r/ 9.0%

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

   4. *-commutative 9.0%

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

   5. times-frac 9.0%

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

   6. reciprocal-define 9.0%

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

   7. associate-/r* 9.0%

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

   8. reciprocal-define 9.0%

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

8. Applied egg-rr 9.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.125}{\pi} \cdot \frac{\mathsf{reciprocal}\left(r\right) + \frac{\mathsf{reciprocal}\left(r\right)}{e^{\frac{r}{s}}}}{s}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 8.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{\pi} \cdot \frac{\mathsf{reciprocal}\left(r\right) + \frac{\mathsf{reciprocal}\left(r\right)}{e^{\frac{r}{s}}}}{s}\right)\right)} \]

   2. expm1-log1p 8.7%

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

   3. times-frac 8.7%

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

   4. distribute-lft-in 8.7%

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

   5. reciprocal-define 8.7%

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

   6. associate-*r/ 8.7%

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

   7. metadata-eval 8.7%

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

   8. associate-*r/ 8.7%

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

   9. reciprocal-define 8.7%

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

   10. associate-*r/ 8.7%

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

   11. metadata-eval 8.7%

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

   12. *-commutative 8.7%

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

10. Simplified 8.7%

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

11. Taylor expanded in r around 0 8.7%

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

12. Final simplification 8.7%

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

Alternative 10: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in s around inf 8.3%

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

7. Final simplification 8.3%

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

Reproduce

herbie shell --seed 2024024 
(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))))