Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 15.6s

Alternatives: 12

Speedup: 1.0×

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-r}{s}}}{r} \cdot \frac{0.125}{s \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \cdot \frac{0.75}{6 \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (exp (/ (- r) s)) r) (/ 0.125 (* s PI)))
  (* (/ (exp (* (/ r s) -0.3333333333333333)) r) (/ 0.75 (* 6.0 (* s PI))))))

C

float code(float s, float r) {
	return ((expf((-r / s)) / r) * (0.125f / (s * ((float) M_PI)))) + ((expf(((r / s) * -0.3333333333333333f)) / r) * (0.75f / (6.0f * (s * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) / r) * Float32(Float32(0.125) / Float32(s * Float32(pi)))) + Float32(Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / r) * Float32(Float32(0.75) / Float32(Float32(6.0) * Float32(s * Float32(pi))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) / r) * (single(0.125) / (s * single(pi)))) + ((exp(((r / s) * single(-0.3333333333333333))) / r) * (single(0.75) / (single(6.0) * (s * single(pi)))));
end

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. fma-def 99.7%

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

   3. associate-*l* 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. metadata-eval 99.7%

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

   7. associate-/r* 99.7%

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

   8. associate-*l* 99.7%

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

   9. /-rgt-identity 99.7%

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

   10. fma-def 99.7%

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

3. Simplified 99.7%

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

   1. neg-mul-1 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.7%

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

   4. metadata-eval 99.7%

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

   5. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in s around 0 99.7%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]

8. Final simplification 99.7%

   \[\leadsto \frac{e^{\frac{-r}{s}}}{r} \cdot \frac{0.125}{s \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \cdot \frac{0.75}{6 \cdot \left(s \cdot \pi\right)} \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi}}{s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (/ 0.125 PI) s) (/ (exp (/ (- r) s)) r))
  (* (/ 0.75 (* s (* PI 6.0))) (/ (exp (* (/ r s) -0.3333333333333333)) r))))

C

float code(float s, float r) {
	return (((0.125f / ((float) M_PI)) / s) * (expf((-r / s)) / r)) + ((0.75f / (s * (((float) M_PI) * 6.0f))) * (expf(((r / s) * -0.3333333333333333f)) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(0.125) / Float32(pi)) / s) * Float32(exp(Float32(Float32(-r) / s)) / r)) + Float32(Float32(Float32(0.75) / Float32(s * Float32(Float32(pi) * Float32(6.0)))) * Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (((single(0.125) / single(pi)) / s) * (exp((-r / s)) / r)) + ((single(0.75) / (s * (single(pi) * single(6.0)))) * (exp(((r / s) * single(-0.3333333333333333))) / r));
end

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. fma-def 99.7%

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

   3. associate-*l* 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. metadata-eval 99.7%

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

   7. associate-/r* 99.7%

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

   8. associate-*l* 99.7%

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

   9. /-rgt-identity 99.7%

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

   10. fma-def 99.7%

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

3. Simplified 99.7%

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

   1. neg-mul-1 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.7%

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

   4. metadata-eval 99.7%

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

   5. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in s around 0 99.7%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]
8. Step-by-step derivation

   1. *-un-lft-identity 99.7%

      \[\leadsto \color{blue}{\left(1 \cdot \frac{0.125}{s \cdot \pi}\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]

   2. *-commutative 99.7%

      \[\leadsto \color{blue}{\left(\frac{0.125}{s \cdot \pi} \cdot 1\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]

   3. *-commutative 99.7%

      \[\leadsto \left(\frac{0.125}{\color{blue}{\pi \cdot s}} \cdot 1\right) \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]

   4. associate-/r* 99.7%

      \[\leadsto \left(\color{blue}{\frac{\frac{0.125}{\pi}}{s}} \cdot 1\right) \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(\frac{\frac{0.125}{\pi}}{s} \cdot 1\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]

10. Taylor expanded in s around 0 99.7%

    \[\leadsto \left(\frac{\frac{0.125}{\pi}}{s} \cdot 1\right) \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{\color{blue}{6 \cdot \left(s \cdot \pi\right)}} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]
11. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \left(\frac{\frac{0.125}{\pi}}{s} \cdot 1\right) \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{6 \cdot \color{blue}{\left(\pi \cdot s\right)}} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]

    2. associate-*l* 99.7%

       \[\leadsto \left(\frac{\frac{0.125}{\pi}}{s} \cdot 1\right) \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{\color{blue}{\left(6 \cdot \pi\right) \cdot s}} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]

    3. *-commutative 99.7%

       \[\leadsto \left(\frac{\frac{0.125}{\pi}}{s} \cdot 1\right) \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{\color{blue}{s \cdot \left(6 \cdot \pi\right)}} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]

    4. *-commutative 99.7%

       \[\leadsto \left(\frac{\frac{0.125}{\pi}}{s} \cdot 1\right) \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \color{blue}{\left(\pi \cdot 6\right)}} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]

12. Simplified 99.7%

    \[\leadsto \left(\frac{\frac{0.125}{\pi}}{s} \cdot 1\right) \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{\color{blue}{s \cdot \left(\pi \cdot 6\right)}} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]

13. Final simplification 99.7%

    \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r} \]
14. Add Preprocessing

Alternative 4: 11.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in s around inf 9.2%

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

   1. log1p-expm1-u 12.1%

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

8. Applied egg-rr 12.1%

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

9. Final simplification 12.1%

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

Alternative 5: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in r around inf 9.7%

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

   1. associate-*r/ 9.7%

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

   2. mul-1-neg 9.7%

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

   3. rec-exp 9.7%

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

   4. associate-*r/ 9.7%

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

   5. metadata-eval 9.7%

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

   6. associate-*r/ 9.7%

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

   7. metadata-eval 9.7%

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

8. Simplified 9.7%

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

9. Final simplification 9.7%

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

Alternative 6: 9.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{0.125}{r}}{\pi}\\ \frac{t\_0 + \frac{t\_0}{e^{\frac{r}{s}}}}{s} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	t_0 = (single(0.125) / r) / single(pi);
	tmp = (t_0 + (t_0 / exp((r / s)))) / s;
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in s around 0 9.7%

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

   1. associate-*r/ 9.7%

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

   2. mul-1-neg 9.7%

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

   3. rec-exp 9.7%

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

   4. associate-*r/ 9.7%

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

   5. metadata-eval 9.7%

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

   6. associate-*r/ 9.7%

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

   7. metadata-eval 9.7%

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

8. Simplified 9.7%

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

9. Taylor expanded in s around 0 9.7%

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

    1. +-commutative 9.7%

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

    2. associate-*r/ 9.7%

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

    3. metadata-eval 9.7%

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

    4. associate-*r* 9.7%

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

    5. *-commutative 9.7%

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

    6. associate-/r* 9.7%

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

    7. associate-/l/ 9.7%

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

    8. associate-*r/ 9.7%

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

    9. metadata-eval 9.7%

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

    10. associate-/r* 9.7%

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

11. Simplified 9.7%

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

12. Final simplification 9.7%

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

Alternative 7: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in s around 0 9.7%

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

   1. associate-*r/ 9.7%

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

   2. mul-1-neg 9.7%

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

   3. rec-exp 9.7%

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

   4. associate-*r/ 9.7%

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

   5. metadata-eval 9.7%

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

   6. associate-*r/ 9.7%

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

   7. metadata-eval 9.7%

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

8. Simplified 9.7%

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

9. Taylor expanded in r around inf 9.7%

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

    1. associate-*r/ 9.7%

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

    2. metadata-eval 9.7%

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

    3. +-commutative 9.7%

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

    4. associate-*r/ 9.7%

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

    5. metadata-eval 9.7%

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

    6. *-commutative 9.7%

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

11. Simplified 9.7%

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

12. Final simplification 9.7%

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

Alternative 8: 9.0% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{r} \cdot \frac{1}{s \cdot \pi} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (* (/ 0.25 r) (/ 1.0 (* s PI))))

C

float code(float s, float r) {
	return (0.25f / r) * (1.0f / (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / r) * Float32(Float32(1.0) / Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / r) * (single(1.0) / (s * single(pi)));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in s around inf 9.2%

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

   1. associate-/r* 9.2%

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

   2. div-inv 9.2%

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

8. Applied egg-rr 9.2%

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

9. Final simplification 9.2%

   \[\leadsto \frac{0.25}{r} \cdot \frac{1}{s \cdot \pi} \]
10. Add Preprocessing

Alternative 9: 9.0% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s} \cdot \frac{\frac{0.25}{r}}{\pi} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (* (/ 1.0 s) (/ (/ 0.25 r) PI)))

C

float code(float s, float r) {
	return (1.0f / s) * ((0.25f / r) / ((float) M_PI));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(1.0) / s) * Float32(Float32(Float32(0.25) / r) / Float32(pi)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / s) * ((single(0.25) / r) / single(pi));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in s around inf 9.2%

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

   1. associate-/r* 9.2%

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

8. Simplified 9.2%

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

   1. *-un-lft-identity 9.2%

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

   2. times-frac 9.2%

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

10. Applied egg-rr 9.2%

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

11. Final simplification 9.2%

    \[\leadsto \frac{1}{s} \cdot \frac{\frac{0.25}{r}}{\pi} \]
12. Add Preprocessing

Alternative 10: 9.0% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\pi} \cdot \frac{\frac{0.25}{r}}{s} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (* (/ 1.0 PI) (/ (/ 0.25 r) s)))

C

float code(float s, float r) {
	return (1.0f / ((float) M_PI)) * ((0.25f / r) / s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(1.0) / Float32(pi)) * Float32(Float32(Float32(0.25) / r) / s))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / single(pi)) * ((single(0.25) / r) / s);
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in s around inf 9.2%

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

   1. associate-/r* 9.2%

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

8. Simplified 9.2%

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

   1. *-un-lft-identity 9.2%

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

   2. *-commutative 9.2%

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

   3. times-frac 9.2%

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

10. Applied egg-rr 9.2%

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

11. Final simplification 9.2%

    \[\leadsto \frac{1}{\pi} \cdot \frac{\frac{0.25}{r}}{s} \]
12. Add Preprocessing

Alternative 11: 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 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in s around inf 9.2%

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

7. Final simplification 9.2%

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

Alternative 12: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in s around inf 9.2%

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

   1. associate-/r* 9.2%

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

8. Simplified 9.2%

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

9. Taylor expanded in r around 0 9.2%

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

    1. *-commutative 9.2%

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

    2. *-commutative 9.2%

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

    3. associate-*l* 9.2%

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

11. Simplified 9.2%

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

12. Final simplification 9.2%

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

Reproduce

herbie shell --seed 2024026 
(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))))