Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 4.2s

Alternatives: 15

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: 99.6% 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

\[\begin{array}{l} \\ \mathsf{fma}\left(e^{\frac{r}{-3 \cdot s}}, 0.75 \cdot \frac{1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (exp (/ r (* -3.0 s)))
  (* 0.75 (/ 1.0 (* (* (* 6.0 PI) s) r)))
  (/ (* 0.125 (/ (exp (/ (- r) s)) (* PI s))) r)))

C

float code(float s, float r) {
	return fmaf(expf((r / (-3.0f * s))), (0.75f * (1.0f / (((6.0f * ((float) M_PI)) * s) * r))), ((0.125f * (expf((-r / s)) / (((float) M_PI) * s))) / r));
}

Julia

function code(s, r)
	return fma(exp(Float32(r / Float32(Float32(-3.0) * s))), Float32(Float32(0.75) * Float32(Float32(1.0) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r))), Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(pi) * s))) / r))
end

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   3. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. mult-flip N/A

      \[\leadsto \color{blue}{\left(\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}\right) \cdot \frac{1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}\right)} \cdot \frac{1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(e^{\frac{-r}{3 \cdot s}} \cdot \frac{3}{4}\right)} \cdot \frac{1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. associate-*l* N/A

      \[\leadsto \color{blue}{e^{\frac{-r}{3 \cdot s}} \cdot \left(\frac{3}{4} \cdot \frac{1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}\right)} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\frac{-r}{3 \cdot s}}, \frac{3}{4} \cdot \frac{1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

3. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\frac{r}{-3 \cdot s}}, 0.75 \cdot \frac{1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right)} \]
4. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   3. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{-r}{3 \cdot s}}}{r} \cdot \frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s}, \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

3. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   3. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   10. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot s}}{r}} \]

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{0.125}{\pi \cdot s}, e^{\frac{r}{-3 \cdot s}}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}\right)}{r}} \]
4. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{8}}{\pi \cdot s}, e^{\color{blue}{\frac{r}{-3 \cdot s}}}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}\right)}{r} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{8}}{\pi \cdot s}, e^{\frac{r}{\color{blue}{-3 \cdot s}}}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}\right)}{r} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{8}}{\pi \cdot s}, e^{\color{blue}{\frac{\frac{r}{-3}}{s}}}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}\right)}{r} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{8}}{\pi \cdot s}, e^{\color{blue}{\frac{\frac{r}{-3}}{s}}}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}\right)}{r} \]

   5. lower-/.f32 99.5

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

5. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   3. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   10. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot s}}{r}} \]

3. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot s}}{r}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot s}}{r} + \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot s}}{r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot s}}{r} + \color{blue}{\frac{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(6 \cdot \pi\right) \cdot s}}{r}} \]

   8. div-add-rev N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot s} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(6 \cdot \pi\right) \cdot s}}{r}} \]

   9. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot s} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(6 \cdot \pi\right) \cdot s}}{r}} \]

3. Applied rewrites 99.6%

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

Alternative 6: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   3. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   10. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot s}}{r}} \]

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}}{s}}}{r} \]
5. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}}{\color{blue}{s}}}{r} \]

6. Applied rewrites 99.5%

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

Alternative 7: 97.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}}}{s \cdot r} \]
5. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{8}, \color{blue}{\frac{e^{-1 \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}}, \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{8}, \frac{e^{-1 \cdot \frac{r}{s}}}{\color{blue}{\mathsf{PI}\left(\right)}}, \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   3. lower-exp.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{8}, \frac{e^{-1 \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}, \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{8}, \frac{e^{-1 \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}, \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{8}, \frac{e^{-1 \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}, \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   6. lower-PI.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{8}, \frac{e^{-1 \cdot \frac{r}{s}}}{\pi}, \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   7. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{8}, \frac{e^{-1 \cdot \frac{r}{s}}}{\pi}, \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{8}, \frac{e^{-1 \cdot \frac{r}{s}}}{\pi}, \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   9. lower-exp.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{8}, \frac{e^{-1 \cdot \frac{r}{s}}}{\pi}, \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{8}, \frac{e^{-1 \cdot \frac{r}{s}}}{\pi}, \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   11. lower-/.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{8}, \frac{e^{-1 \cdot \frac{r}{s}}}{\pi}, \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   12. lower-PI.f32 99.5

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

6. Applied rewrites 99.5%

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

Alternative 8: 9.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{e^{\frac{-r}{s}} \cdot \frac{1}{4}}{\pi + \pi} + \frac{\color{blue}{1} \cdot \frac{3}{4}}{6 \cdot \pi}}{s \cdot r} \]
5. Step-by-step derivation

6. Applied rewrites 9.4%

   \[\leadsto \frac{\frac{e^{\frac{-r}{s}} \cdot 0.25}{\pi + \pi} + \frac{\color{blue}{1} \cdot 0.75}{6 \cdot \pi}}{s \cdot r} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-r}{s}} \cdot \frac{1}{4}}{\pi + \pi} + \frac{1 \cdot \frac{3}{4}}{6 \cdot \pi}}{s \cdot r}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{-r}{s}} \cdot \frac{1}{4}}{\pi + \pi} + \frac{1 \cdot \frac{3}{4}}{6 \cdot \pi}}{\color{blue}{s \cdot r}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{-r}{s}} \cdot \frac{1}{4}}{\pi + \pi} + \frac{1 \cdot \frac{3}{4}}{6 \cdot \pi}}{s}}{r}} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{-r}{s}} \cdot \frac{1}{4}}{\pi + \pi} + \frac{1 \cdot \frac{3}{4}}{6 \cdot \pi}}{s}}{r}} \]

8. Applied rewrites 9.4%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(e^{\frac{-r}{s}}, \frac{0.25}{\pi + \pi}, 1 \cdot \frac{0.75}{6 \cdot \pi}\right)}{s}}{r}} \]
9. Add Preprocessing

Alternative 9: 9.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{\frac{-r}{s}} \cdot 0.25}{\pi + \pi} + \frac{0.75}{6 \cdot \pi}}{s \cdot r} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{e^{\frac{-r}{s}} \cdot \frac{1}{4}}{\pi + \pi} + \frac{\color{blue}{1} \cdot \frac{3}{4}}{6 \cdot \pi}}{s \cdot r} \]
5. Step-by-step derivation

6. Applied rewrites 9.4%

   \[\leadsto \frac{\frac{e^{\frac{-r}{s}} \cdot 0.25}{\pi + \pi} + \frac{\color{blue}{1} \cdot 0.75}{6 \cdot \pi}}{s \cdot r} \]

7. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{e^{\frac{-r}{s}} \cdot \frac{1}{4}}{\pi + \pi} + \frac{\color{blue}{\frac{3}{4}}}{6 \cdot \pi}}{s \cdot r} \]
8. Step-by-step derivation

9. Applied rewrites 9.4%

   \[\leadsto \frac{\frac{e^{\frac{-r}{s}} \cdot 0.25}{\pi + \pi} + \frac{\color{blue}{0.75}}{6 \cdot \pi}}{s \cdot r} \]
10. Add Preprocessing

Alternative 10: 9.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   3. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r} + \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   10. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left(6 \cdot \pi\right) \cdot s} \cdot e^{\frac{-r}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot s}}{r}} \]

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{0.125}{\pi \cdot s}, e^{\frac{r}{-3 \cdot s}}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}\right)}{r}} \]
4. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{8}}{\pi \cdot s}, e^{\color{blue}{\frac{r}{-3 \cdot s}}}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}\right)}{r} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{8}}{\pi \cdot s}, e^{\frac{r}{\color{blue}{-3 \cdot s}}}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}\right)}{r} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{8}}{\pi \cdot s}, e^{\color{blue}{\frac{\frac{r}{-3}}{s}}}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}\right)}{r} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{8}}{\pi \cdot s}, e^{\color{blue}{\frac{\frac{r}{-3}}{s}}}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}\right)}{r} \]

   5. lower-/.f32 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 9.4%

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

Alternative 11: 9.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around inf

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}}{6 \cdot \pi}}{s \cdot r} \]
5. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8}}{\color{blue}{\mathsf{PI}\left(\right)}} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}}{6 \cdot \pi}}{s \cdot r} \]

   2. lower-PI.f32 9.1

      \[\leadsto \frac{\frac{0.125}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{6 \cdot \pi}}{s \cdot r} \]

6. Applied rewrites 9.1%

   \[\leadsto \frac{\color{blue}{\frac{0.125}{\pi}} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{6 \cdot \pi}}{s \cdot r} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}}{6 \cdot \pi}}{s \cdot r}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}}{6 \cdot \pi}}{\color{blue}{s \cdot r}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}}{6 \cdot \pi}}{s}}{r}} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}}{6 \cdot \pi}}{s}}{r}} \]

8. Applied rewrites 9.1%

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

Alternative 12: 9.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around inf

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}}{6 \cdot \pi}}{s \cdot r} \]
5. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8}}{\color{blue}{\mathsf{PI}\left(\right)}} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}}{6 \cdot \pi}}{s \cdot r} \]

   2. lower-PI.f32 9.1

      \[\leadsto \frac{\frac{0.125}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{6 \cdot \pi}}{s \cdot r} \]

6. Applied rewrites 9.1%

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

7. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{\frac{1}{8}}{\pi} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}}}{s \cdot r} \]
8. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8}}{\pi} + \frac{1}{8} \cdot \color{blue}{\frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}}}{s \cdot r} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8}}{\pi} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\color{blue}{\mathsf{PI}\left(\right)}}}{s \cdot r} \]

   3. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8}}{\pi} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}}{s \cdot r} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8}}{\pi} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}}{s \cdot r} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8}}{\pi} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\mathsf{PI}\left(\right)}}{s \cdot r} \]

   6. lower-PI.f32 9.1

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

9. Applied rewrites 9.1%

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

Alternative 13: 8.9% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / s) / (r * 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. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-PI.f32 8.9

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

4. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \pi\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\pi}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot s\right) \cdot \color{blue}{\pi}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   6. lower-*.f32 8.9

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

6. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \color{blue}{\pi}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot \color{blue}{\left(r \cdot \pi\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot \color{blue}{\left(r \cdot \pi\right)}} \]

   5. lower-*.f32 8.9

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

8. Applied rewrites 8.9%

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

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{s \cdot \left(r \cdot \pi\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot \color{blue}{\left(r \cdot \pi\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s}}{\color{blue}{r \cdot \pi}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s}}{\color{blue}{r \cdot \pi}} \]

   5. lower-/.f32 8.9

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

10. Applied rewrites 8.9%

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

Alternative 14: 8.9% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / (s * (r * 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. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-PI.f32 8.9

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

4. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \pi\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\pi}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot s\right) \cdot \color{blue}{\pi}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   6. lower-*.f32 8.9

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

6. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \color{blue}{\pi}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot \color{blue}{\left(r \cdot \pi\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot \color{blue}{\left(r \cdot \pi\right)}} \]

   5. lower-*.f32 8.9

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

8. Applied rewrites 8.9%

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

Alternative 15: 8.9% accurate, 6.4× 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%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-PI.f32 8.9

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

4. Applied rewrites 8.9%

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

Reproduce

herbie shell --seed 2025140 
(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))))