Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 4.5s

Alternatives: 14

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(exp(Float32(Float32(-r) / s)) / Float32(pi)), Float32(0.125), Float32(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(pi)) * Float32(0.125))) / 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 99.6%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-rgt-out N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

5. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 99.6

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

7. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (((exp((r / (single(-3.0) * s))) + exp((-r / s))) / single(pi)) * (single(0.125) / r)) / s;
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 99.6%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f32 N/A

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

5. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. mult-flip-rev N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f32 N/A

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

   10. lower-/.f32 99.6

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

7. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) * (exp((r / (single(-3.0) * s))) + exp((-r / s)))) / (single(pi) * r)) / s;
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 99.6%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f32 N/A

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

5. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. mult-flip-rev N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l/ N/A

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

   7. associate-/l/ N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lower-/.f32 N/A

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

7. Applied rewrites 99.6%

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

Alternative 5: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (((exp((-r / s)) + exp((r / (single(-3.0) * s)))) / single(pi)) * single(0.125)) / (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 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 99.6

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

5. Applied rewrites 99.6%

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

Alternative 6: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f32 N/A

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

5. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. mult-flip-rev N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. frac-times N/A

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

   10. lift-/.f32 N/A

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

7. Applied rewrites 99.6%

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

Alternative 7: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp((r / (single(-3.0) * s))) + exp((-r / s)))) / ((single(pi) * r) * s);
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 99.6%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f32 N/A

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

5. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. mult-flip-rev N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. frac-times N/A

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

7. Applied rewrites 99.6%

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

Alternative 8: 43.6% 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 99.6%

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

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 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. lower-*.f32 N/A

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

   6. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. add-log-exp N/A

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

   4. log-pow-rev N/A

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

   5. lower-log.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. pow-exp N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lower-exp.f32 43.6

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 43.6

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

8. Applied rewrites 43.6%

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

Alternative 9: 9.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s} \cdot \frac{\frac{2 + -1.3333333333333333 \cdot \frac{r}{s}}{\pi}}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (/ 0.125 s) (/ (/ (+ 2.0 (* -1.3333333333333333 (/ r s))) PI) r)))

C

float code(float s, float r) {
	return (0.125f / s) * (((2.0f + (-1.3333333333333333f * (r / s))) / ((float) M_PI)) / r);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / s) * Float32(Float32(Float32(Float32(2.0) + Float32(Float32(-1.3333333333333333) * Float32(r / s))) / Float32(pi)) / r))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / s) * (((single(2.0) + (single(-1.3333333333333333) * (r / s))) / single(pi)) / r);
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-rgt-out N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{8}}{s} \cdot \frac{\frac{\color{blue}{2 + \frac{-4}{3} \cdot \frac{r}{s}}}{\pi}}{r} \]
7. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s} \cdot \frac{\frac{2 + \color{blue}{\frac{-4}{3} \cdot \frac{r}{s}}}{\pi}}{r} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s} \cdot \frac{\frac{2 + \frac{-4}{3} \cdot \color{blue}{\frac{r}{s}}}{\pi}}{r} \]

   3. lower-/.f32 9.0

      \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{2 + -1.3333333333333333 \cdot \frac{r}{\color{blue}{s}}}{\pi}}{r} \]

8. Applied rewrites 9.0%

   \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{\color{blue}{2 + -1.3333333333333333 \cdot \frac{r}{s}}}{\pi}}{r} \]
9. Add Preprocessing

Alternative 10: 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 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in r around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-PI.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. 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 11: 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 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in r around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-PI.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f32 N/A

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

   5. lower-/.f32 9.0

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

8. Applied rewrites 9.0%

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

Alternative 12: 9.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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. lower-*.f32 N/A

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

   6. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. div-flip N/A

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

   5. lower-unsound-/.f32 N/A

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

   6. lower-unsound-/.f32 N/A

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

   7. lower-/.f32 9.0

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 9.0

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

8. Applied rewrites 9.0%

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

Alternative 13: 9.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / ((r * single(pi)) * s);
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 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. lower-*.f32 N/A

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

   6. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

Alternative 14: 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 99.6%

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

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 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 2025159 
(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))))