Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 5.5s

Alternatives: 18

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.125f / ((((float) M_PI) * s) * expf((r / 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(0.125) / Float32(Float32(Float32(pi) * s) * exp(Float32(r / 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(0.125) / ((single(pi) * s) * exp((r / s)))) / r) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.5%

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

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

   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. frac-times N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*r/ N/A

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

   16. lower-/.f32 N/A

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

3. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return fmaf((0.125f / (((float) M_PI) * s)), expf((r / (-3.0f * s))), (0.125f / ((((float) M_PI) * s) * expf((r / 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(Float32(Float32(pi) * s) * exp(Float32(r / s))))) / r)
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 7: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

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

5. Applied rewrites 99.4%

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

Alternative 8: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 99.4%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lower-/.f32 N/A

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

7. Applied rewrites 99.5%

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

Alternative 9: 44.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-r}{s}}\\ \mathbf{if}\;\frac{0.25 \cdot t\_0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0.0002500000118743628:\\ \;\;\;\;\frac{0.25}{\log \left(e^{\pi \cdot r}\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t\_0}{\pi}, 0.125, \frac{1 + -0.3333333333333333 \cdot \frac{r}{s}}{\pi} \cdot 0.125\right)}{s \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (exp (/ (- r) s))))
   (if (<=
        (+
         (/ (* 0.25 t_0) (* (* (* 2.0 PI) s) r))
         (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
        0.0002500000118743628)
     (/ 0.25 (* (log (exp (* PI r))) s))
     (/
      (fma
       (/ t_0 PI)
       0.125
       (* (/ (+ 1.0 (* -0.3333333333333333 (/ r s))) PI) 0.125))
      (* s r)))))

C

float code(float s, float r) {
	float t_0 = expf((-r / s));
	float tmp;
	if ((((0.25f * t_0) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r))) <= 0.0002500000118743628f) {
		tmp = 0.25f / (logf(expf((((float) M_PI) * r))) * s);
	} else {
		tmp = fmaf((t_0 / ((float) M_PI)), 0.125f, (((1.0f + (-0.3333333333333333f * (r / s))) / ((float) M_PI)) * 0.125f)) / (s * r);
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = exp(Float32(Float32(-r) / s))
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(Float32(0.25) * t_0) / 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))) <= Float32(0.0002500000118743628))
		tmp = Float32(Float32(0.25) / Float32(log(exp(Float32(Float32(pi) * r))) * s));
	else
		tmp = Float32(fma(Float32(t_0 / Float32(pi)), Float32(0.125), Float32(Float32(Float32(Float32(1.0) + Float32(Float32(-0.3333333333333333) * Float32(r / s))) / Float32(pi)) * Float32(0.125))) / Float32(s * r));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 2.50000012e-4

   1. Initial program 99.5%

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

   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 9.0

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

   4. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
   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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f32 N/A

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

      6. lower-*.f32 9.0

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 9.0

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

   6. Applied rewrites 9.0%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f32 N/A

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

      4. add-log-exp N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right) \cdot s} \]

      5. log-pow-rev N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right) \cdot s} \]

      6. lower-log.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right) \cdot s} \]

      7. lift-PI.f32 N/A

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

      8. pow-exp N/A

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

      9. lift-*.f32 N/A

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

      10. lower-exp.f32 43.0

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

   8. Applied rewrites 43.0%

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

   if 2.50000012e-4 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 99.5%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in s around inf

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 9.7

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

   6. Applied rewrites 9.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, 0.125, \frac{\color{blue}{1 + -0.3333333333333333 \cdot \frac{r}{s}}}{\pi} \cdot 0.125\right)}{s \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 43.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.0

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

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 9.0

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-log-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right) \cdot s} \]

   5. log-pow-rev N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right) \cdot s} \]

   6. lower-log.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right) \cdot s} \]

   7. lift-PI.f32 N/A

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

   8. pow-exp N/A

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

   9. lift-*.f32 N/A

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

   10. lower-exp.f32 43.0

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

8. Applied rewrites 43.0%

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

Alternative 11: 9.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, -0.16666666666666666, \frac{0.25}{\pi}\right)}{s}}{r} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in r around 0

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

   1. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{r}{s \cdot \mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{\color{blue}{s \cdot \mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \color{blue}{\mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   4. lower-PI.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \pi}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \pi}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \pi}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   7. lower-PI.f32 9.0

      \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, 0.25 \cdot \frac{1}{\pi}\right)}{s \cdot r} \]

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

8. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, -0.16666666666666666, \frac{0.25}{\pi}\right)}{s}}{r}} \]
9. Add Preprocessing

Alternative 12: 9.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, -0.16666666666666666, \frac{0.25}{\pi}\right)}{r}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in r around 0

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

   1. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{r}{s \cdot \mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{\color{blue}{s \cdot \mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \color{blue}{\mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   4. lower-PI.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \pi}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \pi}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \pi}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   7. lower-PI.f32 9.0

      \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, 0.25 \cdot \frac{1}{\pi}\right)}{s \cdot r} \]

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

8. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, -0.16666666666666666, \frac{0.25}{\pi}\right)}{r}}{s}} \]
9. Add Preprocessing

Alternative 13: 9.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, -0.16666666666666666, \frac{0.25}{\pi}\right)}{s \cdot r} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in r around 0

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

   1. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{r}{s \cdot \mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{\color{blue}{s \cdot \mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \color{blue}{\mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   4. lower-PI.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \pi}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \pi}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \pi}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   7. lower-PI.f32 9.0

      \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, 0.25 \cdot \frac{1}{\pi}\right)}{s \cdot r} \]

6. Applied rewrites 9.0%

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

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 9.0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot \pi}, \color{blue}{-0.16666666666666666}, 0.25 \cdot \frac{1}{\pi}\right)}{s \cdot r} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot \pi}, \frac{-1}{6}, \frac{1}{4} \cdot \frac{1}{\pi}\right)}{s \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, \frac{-1}{6}, \frac{1}{4} \cdot \frac{1}{\pi}\right)}{s \cdot r} \]

   6. lift-*.f32 9.0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, -0.16666666666666666, 0.25 \cdot \frac{1}{\pi}\right)}{s \cdot r} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, \frac{-1}{6}, \frac{1}{4} \cdot \frac{1}{\pi}\right)}{s \cdot r} \]

   8. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, \frac{-1}{6}, \frac{1}{4} \cdot \frac{1}{\pi}\right)}{s \cdot r} \]

   9. mult-flip-rev N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, \frac{-1}{6}, \frac{\frac{1}{4}}{\pi}\right)}{s \cdot r} \]

   10. lower-/.f32 9.0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, -0.16666666666666666, \frac{0.25}{\pi}\right)}{s \cdot r} \]

8. Applied rewrites 9.0%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, \color{blue}{-0.16666666666666666}, \frac{0.25}{\pi}\right)}{s \cdot r} \]
9. Add Preprocessing

Alternative 14: 9.0% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.0

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

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 9.0

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 9.0

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

8. Applied rewrites 9.0%

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

Alternative 15: 9.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in r around 0

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

   1. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{r}{s \cdot \mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{\color{blue}{s \cdot \mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \color{blue}{\mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   4. lower-PI.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \pi}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \pi}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \pi}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s \cdot r} \]

   7. lower-PI.f32 9.0

      \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, 0.25 \cdot \frac{1}{\pi}\right)}{s \cdot r} \]

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

8. Applied rewrites 9.0%

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

9. Taylor expanded in s around inf

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

    1. lower-/.f32 N/A

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

    2. lower-PI.f32 9.1

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

11. Applied rewrites 9.1%

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

Alternative 16: 9.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-PI.f32 9.0

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

6. Applied rewrites 9.0%

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

Alternative 17: 9.0% accurate, 6.4× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.0

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

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.0

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

6. Applied rewrites 9.0%

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

Alternative 18: 9.0% 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 99.5%

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

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 9.0

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

4. Applied rewrites 9.0%

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

Reproduce

herbie shell --seed 2025156 
(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))))