Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 4.4s

Alternatives: 10

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: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) + exp((r / (single(-3.0) * s)))) * (single(0.125) / 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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

10. Applied rewrites 99.5%

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

Alternative 6: 16.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. mult-flip N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f32 N/A

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

3. Applied rewrites 97.3%

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

5. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. sinh-+-cosh-rev N/A

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

   3. +-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. mult-flip N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. mult-flip N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. mult-flip N/A

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

   14. lift-/.f32 N/A

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

   15. cosh-neg-rev N/A

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

   16. lift-/.f32 N/A

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

   17. mult-flip N/A

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

   18. lift-*.f32 N/A

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

   19. *-commutative N/A

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

   20. associate-/r* N/A

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

   21. metadata-eval N/A

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

   22. mult-flip N/A

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

   23. associate-*l* N/A

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

   24. lift-*.f32 N/A

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

   25. mult-flip N/A

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

   26. lift-/.f32 N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 16.2

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

10. Applied rewrites 16.2%

    \[\leadsto \frac{\frac{e^{\frac{-r}{s}} + \frac{1}{\color{blue}{1 + 0.3333333333333333 \cdot \frac{r}{s}}}}{\pi \cdot s} \cdot 0.125}{r} \]
11. Add Preprocessing

Alternative 7: 9.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. lift-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. metadata-eval N/A

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

   9. associate-/r* N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-/l* N/A

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

5. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f32 N/A

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

   10. mult-flip N/A

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

   11. lift-/.f32 99.5

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 9.4%

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

Alternative 8: 9.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 9.4%

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

Alternative 9: 8.9% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{0.75}{\pi \cdot s} \cdot e^{\frac{r}{-3 \cdot s}}, 0.16666666666666666, \frac{e^{\frac{-r}{s}}}{\pi \cdot s} \cdot 0.125\right)}}{r} \]

8. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 8.9

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

10. Applied rewrites 8.9%

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

Alternative 10: 8.9% accurate, 6.4× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 8.9%

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

Reproduce

herbie shell --seed 2025140 
(FPCore (s r)
  :name "Disney BSSRDF, PDF of scattering profile"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (< 1e-6 r) (< r 1000000.0)))
  (+ (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r)) (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))