Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 14.8s

Alternatives: 16

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\frac{0.125}{\pi}}{s} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. 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* 99.5%

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

   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 99.5%

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

3. Simplified 99.6%

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

4. Taylor expanded in s around 0 99.5%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
5. Step-by-step derivation

   1. associate-/r* 99.6%

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

   2. expm1-log1p-u 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.125}{s}}{\pi}\right)\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.125}{s}}{\pi}\right)\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
7. Step-by-step derivation

   1. expm1-log1p-u 99.6%

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

   2. associate-/l/ 99.5%

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

   3. associate-/r* 99.6%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in s around 0 99.6%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\frac{0.125}{\pi}}{s} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
10. Step-by-step derivation

    1. associate-/r* 99.6%

       \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\frac{0.125}{\pi}}{s} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

11. Simplified 99.6%

    \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\frac{0.125}{\pi}}{s} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

12. Final simplification 99.6%

    \[\leadsto \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\frac{0.125}{\pi}}{s} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ \frac{e^{\frac{-r}{s \cdot 3}}}{r} \cdot t_0 + \frac{e^{\frac{-r}{s}}}{r} \cdot t_0 \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* s PI))))
   (+
    (* (/ (exp (/ (- r) (* s 3.0))) r) t_0)
    (* (/ (exp (/ (- r) s)) r) t_0))))

C

float code(float s, float r) {
	float t_0 = 0.125f / (s * ((float) M_PI));
	return ((expf((-r / (s * 3.0f))) / r) * t_0) + ((expf((-r / s)) / r) * t_0);
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(s * Float32(pi)))
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / Float32(s * Float32(3.0)))) / r) * t_0) + Float32(Float32(exp(Float32(Float32(-r) / s)) / r) * t_0))
end

MATLAB

function tmp = code(s, r)
	t_0 = single(0.125) / (s * single(pi));
	tmp = ((exp((-r / (s * single(3.0)))) / r) * t_0) + ((exp((-r / s)) / r) * t_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* 99.5%

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

   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 99.5%

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

3. Simplified 99.6%

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

4. Taylor expanded in s around 0 99.5%

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

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

6. Final simplification 99.5%

   \[\leadsto \frac{e^{\frac{-r}{s \cdot 3}}}{r} \cdot \frac{0.125}{s \cdot \pi} + \frac{e^{\frac{-r}{s}}}{r} \cdot \frac{0.125}{s \cdot \pi} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-r}{s \cdot 3}}}{r} \cdot \frac{0.125}{s \cdot \pi} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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* 99.5%

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

   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 99.5%

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

3. Simplified 99.6%

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

4. Taylor expanded in s around 0 99.5%

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

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
6. Step-by-step derivation

   1. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\frac{0.125}{\pi}}{s} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

8. Final simplification 99.6%

   \[\leadsto \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-r}{s \cdot 3}}}{r} \cdot \frac{0.125}{s \cdot \pi} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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* 99.5%

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

   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 99.5%

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

3. Simplified 99.6%

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

4. Taylor expanded in s around 0 99.5%

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

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
6. Step-by-step derivation

   1. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\frac{0.125}{\pi}}{s} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

8. Taylor expanded in r around 0 99.5%

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

9. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((single(-0.3333333333333333) * (r / s))) / r) + (exp((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 99.2%

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

4. Taylor expanded in r around inf 99.5%

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

5. Final simplification 99.5%

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

Alternative 6: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around inf 99.5%

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

   1. metadata-eval 99.5%

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

   2. times-frac 99.5%

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

   3. neg-mul-1 99.5%

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

   4. frac-2neg 99.5%

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

   5. remove-double-neg 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-neg-in 99.5%

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

   8. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 7: 11.7% accurate, 1.4× speedup

Math

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

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(s * Float32(Float32(pi) * r)))))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 10.6%

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

5. Taylor expanded in s around inf 10.0%

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

   1. log1p-expm1-u 12.7%

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

   2. *-commutative 12.7%

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

   3. associate-*l* 12.7%

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

7. Applied egg-rr 12.7%

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

8. Final simplification 12.7%

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

Alternative 8: 43.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 10.6%

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

5. Taylor expanded in s around inf 10.0%

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

   1. expm1-log1p-u 10.0%

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

   2. expm1-udef 8.3%

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

   3. *-commutative 8.3%

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

   4. associate-*l* 8.3%

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

7. Applied egg-rr 8.3%

   \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)} - 1}} \]
8. Step-by-step derivation

   1. expm1-def 10.0%

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

   2. expm1-log1p 10.0%

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

9. Simplified 10.0%

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

    1. log1p-expm1-u 45.9%

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

11. Applied egg-rr 45.9%

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

12. Final simplification 45.9%

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

Alternative 9: 9.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 11.5%

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

5. Final simplification 11.5%

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

Alternative 10: 9.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around inf 99.5%

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

5. Taylor expanded in r around 0 10.8%

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

   1. mul-1-neg 10.8%

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

   2. unsub-neg 10.8%

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

7. Simplified 10.8%

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

8. Final simplification 10.8%

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

Alternative 11: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(-0.125) / r) * ((single(-1.0) + (single(-1.0) / exp((r / s)))) / (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 99.2%

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

4. Taylor expanded in r around 0 10.6%

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

5. Taylor expanded in s around 0 10.6%

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

   1. mul-1-neg 10.6%

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

7. Simplified 10.6%

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

8. Taylor expanded in r around -inf 10.6%

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

   1. associate-*r/ 10.6%

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

   2. times-frac 10.7%

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

   3. sub-neg 10.7%

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

   4. metadata-eval 10.7%

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

   5. +-commutative 10.7%

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

   6. neg-mul-1 10.7%

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

   7. rec-exp 10.7%

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

   8. associate-*r/ 10.7%

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

   9. metadata-eval 10.7%

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

10. Simplified 10.7%

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

11. Final simplification 10.7%

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

Alternative 12: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((-r / s)) + single(1.0)) / (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 99.2%

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

4. Taylor expanded in r around 0 10.6%

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

5. Taylor expanded in s around 0 10.6%

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

   1. mul-1-neg 10.6%

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

7. Simplified 10.6%

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

8. Taylor expanded in r around inf 10.6%

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

9. Final simplification 10.6%

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

Alternative 13: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 10.6%

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

5. Taylor expanded in r around inf 10.6%

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

   1. associate-*r/ 10.6%

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

   2. *-commutative 10.6%

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

   3. associate-*l* 10.6%

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

   4. times-frac 10.6%

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

   5. associate-*r/ 10.6%

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

   6. mul-1-neg 10.6%

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

7. Simplified 10.6%

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

8. Final simplification 10.6%

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

Alternative 14: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 10.6%

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

5. Taylor expanded in r around inf 10.6%

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

   1. associate-*r/ 10.6%

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

   2. *-commutative 10.6%

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

   3. associate-*l* 10.6%

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

   4. times-frac 10.6%

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

   5. associate-*r/ 10.6%

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

   6. mul-1-neg 10.6%

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

7. Simplified 10.6%

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

8. Taylor expanded in s around 0 10.6%

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

   1. associate-*r/ 10.6%

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

   2. *-commutative 10.6%

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

   3. associate-*r* 10.6%

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

   4. associate-/l* 10.6%

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

   5. associate-*r* 10.6%

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

   6. *-commutative 10.6%

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

   7. associate-*r* 10.6%

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

   8. *-commutative 10.6%

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

   9. mul-1-neg 10.6%

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

   10. distribute-neg-frac 10.6%

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

10. Simplified 10.6%

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

11. Final simplification 10.6%

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

Alternative 15: 9.2% accurate, 4.0× 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 99.2%

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

4. Taylor expanded in r around 0 10.6%

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

5. Taylor expanded in s around inf 10.0%

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

6. Final simplification 10.0%

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

Alternative 16: 9.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 10.6%

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

5. Taylor expanded in s around inf 10.0%

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

   1. expm1-log1p-u 10.0%

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

   2. expm1-udef 8.3%

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

   3. *-commutative 8.3%

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

   4. associate-*l* 8.3%

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

7. Applied egg-rr 8.3%

   \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)} - 1}} \]
8. Step-by-step derivation

   1. expm1-def 10.0%

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

   2. expm1-log1p 10.0%

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

9. Simplified 10.0%

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

10. Final simplification 10.0%

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

Reproduce

herbie shell --seed 2023318 
(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))))