Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 8.4s

Alternatives: 17

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{r \cdot -0.3333333333333333}{s} \cdot 0.5}\\ \mathsf{fma}\left(t\_0, \frac{0.75 \cdot t\_0}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (exp (* (/ (* r -0.3333333333333333) s) 0.5))))
   (fma
    t_0
    (/ (* 0.75 t_0) (* (* (* 6.0 PI) s) r))
    (/ (* 0.125 (/ (exp (/ (- r) s)) (* PI s))) r))))

C

float code(float s, float r) {
	float t_0 = expf((((r * -0.3333333333333333f) / s) * 0.5f));
	return fmaf(t_0, ((0.75f * t_0) / (((6.0f * ((float) M_PI)) * s) * r)), ((0.125f * (expf((-r / s)) / (((float) M_PI) * s))) / r));
}

Julia

function code(s, r)
	t_0 = exp(Float32(Float32(Float32(r * Float32(-0.3333333333333333)) / s) * Float32(0.5)))
	return fma(t_0, Float32(Float32(Float32(0.75) * t_0) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)), Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(pi) * s))) / r))
end

Derivation

1. Initial program 99.6%

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

   1. lift-exp.f32 N/A

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

   2. exp-fabs N/A

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

   3. lift-exp.f32 N/A

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

   4. rem-sqrt-square-rev N/A

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

   5. sqrt-prod N/A

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

   6. lower-special-*.f32 N/A

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

3. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\left(\sqrt{e^{\frac{r \cdot -0.3333333333333333}{s}}} \cdot \sqrt{e^{\frac{r \cdot -0.3333333333333333}{s}}}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 99.5

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 99.5

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.5

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. exp-sqrt-rev N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{e^{\frac{\frac{\frac{-1}{3} \cdot r}{s}}{2}}}, \frac{\frac{3}{4} \cdot \sqrt{e^{\frac{\frac{-1}{3} \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

   4. lower-exp.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{e^{\frac{\frac{\frac{-1}{3} \cdot r}{s}}{2}}}, \frac{\frac{3}{4} \cdot \sqrt{e^{\frac{\frac{-1}{3} \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

   5. mult-flip N/A

      \[\leadsto \mathsf{fma}\left(e^{\color{blue}{\frac{\frac{-1}{3} \cdot r}{s} \cdot \frac{1}{2}}}, \frac{\frac{3}{4} \cdot \sqrt{e^{\frac{\frac{-1}{3} \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(e^{\frac{\frac{-1}{3} \cdot r}{s} \cdot \color{blue}{\frac{1}{2}}}, \frac{\frac{3}{4} \cdot \sqrt{e^{\frac{\frac{-1}{3} \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

   7. lower-*.f32 99.5

      \[\leadsto \mathsf{fma}\left(e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s} \cdot 0.5}}, \frac{0.75 \cdot \sqrt{e^{\frac{-0.3333333333333333 \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(e^{\frac{\color{blue}{\frac{-1}{3} \cdot r}}{s} \cdot \frac{1}{2}}, \frac{\frac{3}{4} \cdot \sqrt{e^{\frac{\frac{-1}{3} \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(e^{\frac{\color{blue}{r \cdot \frac{-1}{3}}}{s} \cdot \frac{1}{2}}, \frac{\frac{3}{4} \cdot \sqrt{e^{\frac{\frac{-1}{3} \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

   10. lift-*.f32 99.5

       \[\leadsto \mathsf{fma}\left(e^{\frac{\color{blue}{r \cdot -0.3333333333333333}}{s} \cdot 0.5}, \frac{0.75 \cdot \sqrt{e^{\frac{-0.3333333333333333 \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

9. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\color{blue}{e^{\frac{r \cdot -0.3333333333333333}{s} \cdot 0.5}}, \frac{0.75 \cdot \sqrt{e^{\frac{-0.3333333333333333 \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]
10. Step-by-step derivation

    1. lift-sqrt.f32 N/A

       \[\leadsto \mathsf{fma}\left(e^{\frac{r \cdot \frac{-1}{3}}{s} \cdot \frac{1}{2}}, \frac{\frac{3}{4} \cdot \color{blue}{\sqrt{e^{\frac{\frac{-1}{3} \cdot r}{s}}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

    2. lift-exp.f32 N/A

       \[\leadsto \mathsf{fma}\left(e^{\frac{r \cdot \frac{-1}{3}}{s} \cdot \frac{1}{2}}, \frac{\frac{3}{4} \cdot \sqrt{\color{blue}{e^{\frac{\frac{-1}{3} \cdot r}{s}}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

    3. exp-sqrt-rev N/A

       \[\leadsto \mathsf{fma}\left(e^{\frac{r \cdot \frac{-1}{3}}{s} \cdot \frac{1}{2}}, \frac{\frac{3}{4} \cdot \color{blue}{e^{\frac{\frac{\frac{-1}{3} \cdot r}{s}}{2}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

    4. lower-exp.f32 N/A

       \[\leadsto \mathsf{fma}\left(e^{\frac{r \cdot \frac{-1}{3}}{s} \cdot \frac{1}{2}}, \frac{\frac{3}{4} \cdot \color{blue}{e^{\frac{\frac{\frac{-1}{3} \cdot r}{s}}{2}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

    5. mult-flip N/A

       \[\leadsto \mathsf{fma}\left(e^{\frac{r \cdot \frac{-1}{3}}{s} \cdot \frac{1}{2}}, \frac{\frac{3}{4} \cdot e^{\color{blue}{\frac{\frac{-1}{3} \cdot r}{s} \cdot \frac{1}{2}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(e^{\frac{r \cdot \frac{-1}{3}}{s} \cdot \frac{1}{2}}, \frac{\frac{3}{4} \cdot e^{\frac{\frac{-1}{3} \cdot r}{s} \cdot \color{blue}{\frac{1}{2}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

    7. lower-*.f32 99.6

       \[\leadsto \mathsf{fma}\left(e^{\frac{r \cdot -0.3333333333333333}{s} \cdot 0.5}, \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s} \cdot 0.5}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

    8. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(e^{\frac{r \cdot \frac{-1}{3}}{s} \cdot \frac{1}{2}}, \frac{\frac{3}{4} \cdot e^{\frac{\color{blue}{\frac{-1}{3} \cdot r}}{s} \cdot \frac{1}{2}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(e^{\frac{r \cdot \frac{-1}{3}}{s} \cdot \frac{1}{2}}, \frac{\frac{3}{4} \cdot e^{\frac{\color{blue}{r \cdot \frac{-1}{3}}}{s} \cdot \frac{1}{2}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

    10. lift-*.f32 99.6

        \[\leadsto \mathsf{fma}\left(e^{\frac{r \cdot -0.3333333333333333}{s} \cdot 0.5}, \frac{0.75 \cdot e^{\frac{\color{blue}{r \cdot -0.3333333333333333}}{s} \cdot 0.5}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]

11. Applied rewrites 99.6%

    \[\leadsto \mathsf{fma}\left(e^{\frac{r \cdot -0.3333333333333333}{s} \cdot 0.5}, \frac{0.75 \cdot \color{blue}{e^{\frac{r \cdot -0.3333333333333333}{s} \cdot 0.5}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot s}}{r}\right) \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{e^{\frac{-0.3333333333333333 \cdot r}{s}}}\\ \mathsf{fma}\left(t\_0, \frac{0.75 \cdot t\_0}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{\frac{0.125}{\left(e^{\frac{r}{s}} \cdot \pi\right) \cdot s}}{r}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (sqrt (exp (/ (* -0.3333333333333333 r) s)))))
   (fma
    t_0
    (/ (* 0.75 t_0) (* (* (* 6.0 PI) s) r))
    (/ (/ 0.125 (* (* (exp (/ r s)) PI) s)) r))))

C

float code(float s, float r) {
	float t_0 = sqrtf(expf(((-0.3333333333333333f * r) / s)));
	return fmaf(t_0, ((0.75f * t_0) / (((6.0f * ((float) M_PI)) * s) * r)), ((0.125f / ((expf((r / s)) * ((float) M_PI)) * s)) / r));
}

Julia

function code(s, r)
	t_0 = sqrt(exp(Float32(Float32(Float32(-0.3333333333333333) * r) / s)))
	return fma(t_0, Float32(Float32(Float32(0.75) * t_0) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)), Float32(Float32(Float32(0.125) / Float32(Float32(exp(Float32(r / s)) * Float32(pi)) * s)) / r))
end

Derivation

1. Initial program 99.6%

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

   1. lift-exp.f32 N/A

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

   2. exp-fabs N/A

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

   3. lift-exp.f32 N/A

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

   4. rem-sqrt-square-rev N/A

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

   5. sqrt-prod N/A

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

   6. lower-special-*.f32 N/A

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

3. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\left(\sqrt{e^{\frac{r \cdot -0.3333333333333333}{s}}} \cdot \sqrt{e^{\frac{r \cdot -0.3333333333333333}{s}}}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 99.5

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 99.5

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.5

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. add-to-fraction N/A

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

   7. metadata-eval N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lower-/.f32 N/A

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

5. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. distribute-rgt-out N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

5. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / r) * (((exp(((single(-0.3333333333333333) * r) / s)) + exp((-r / s))) / 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-exp.f32 N/A

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

   2. exp-fabs N/A

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

   3. lift-exp.f32 N/A

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

   4. rem-sqrt-square-rev N/A

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

   5. sqrt-prod N/A

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

   6. lower-special-*.f32 N/A

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-exp.f32 N/A

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

   2. exp-fabs N/A

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

   3. lift-exp.f32 N/A

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

   4. rem-sqrt-square-rev N/A

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

   5. sqrt-prod N/A

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

   6. lower-special-*.f32 N/A

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

Alternative 7: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp(((single(-0.3333333333333333) * r) / s)) + exp((-r / s))) / (single(pi) * r)) * (single(0.125) / 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-exp.f32 N/A

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

   2. exp-fabs N/A

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

   3. lift-exp.f32 N/A

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

   4. rem-sqrt-square-rev N/A

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

   5. sqrt-prod N/A

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

   6. lower-special-*.f32 N/A

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.5

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

   4. lift-/.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. lower-/.f32 N/A

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

   8. lower-*.f32 99.5

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

7. Applied rewrites 99.5%

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

Alternative 8: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-exp.f32 N/A

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

   2. exp-fabs N/A

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

   3. lift-exp.f32 N/A

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

   4. rem-sqrt-square-rev N/A

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

   5. sqrt-prod N/A

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

   6. lower-special-*.f32 N/A

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/l/ N/A

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

   6. lift-/.f32 N/A

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

   7. frac-times N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 99.5

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

7. Applied rewrites 99.5%

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

Alternative 9: 43.2% 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 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 9.1

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

6. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-exp N/A

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

   9. lift-*.f32 N/A

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

   10. lower-exp.f32 43.2

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

8. Applied rewrites 43.2%

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

Alternative 10: 10.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. add-log-exp N/A

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

   8. log-pow-rev N/A

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

   9. lower-log.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. pow-exp N/A

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

   12. lift-*.f32 N/A

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

   13. lower-exp.f32 10.2

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 10.2

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

6. Applied rewrites 10.2%

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

Alternative 11: 9.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-PI.f32 9.1

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

6. Applied rewrites 9.1%

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

Alternative 12: 9.1% accurate, 5.7× speedup

Math

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

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(Float32(Float32(0.25) / 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 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 9.1

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

6. Applied rewrites 9.1%

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

Alternative 13: 9.1% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 9.1

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

6. Applied rewrites 9.1%

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

Alternative 14: 9.1% accurate, 6.0× speedup

Math

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

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) / s) / Float32(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. 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 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 9.1

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

6. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 9.1

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

8. Applied rewrites 9.1%

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

Alternative 15: 9.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / ((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 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 9.1

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

6. Applied rewrites 9.1%

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

Alternative 16: 9.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 9.1

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

6. Applied rewrites 9.1%

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

Alternative 17: 9.1% 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 9.1

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

4. Applied rewrites 9.1%

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

Reproduce

herbie shell --seed 2025151 
(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))))