Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 4.3s

Alternatives: 12

Speedup: N/A×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.75, \frac{0.16666666666666666}{\pi \cdot s} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.75
  (*
   (/ 0.16666666666666666 (* PI s))
   (/ (exp (* (/ r s) -0.3333333333333333)) r))
  (* 0.25 (/ (exp (/ (- r) s)) (* (* (+ PI PI) s) r)))))

C

float code(float s, float r) {
	return fmaf(0.75f, ((0.16666666666666666f / (((float) M_PI) * s)) * (expf(((r / s) * -0.3333333333333333f)) / r)), (0.25f * (expf((-r / s)) / (((((float) M_PI) + ((float) M_PI)) * s) * r))));
}

Julia

function code(s, r)
	return fma(Float32(0.75), Float32(Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * s)) * Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / r)), Float32(Float32(0.25) * Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(Float32(pi) + 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 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.75, \frac{e^{\frac{-r}{3 \cdot s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right)} \]

4. Taylor expanded in s around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lift-PI.f32 99.5

       \[\leadsto \mathsf{fma}\left(0.75, \frac{0.16666666666666666 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\left(\pi \cdot s\right) \cdot r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{\frac{0.16666666666666666 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\left(\pi \cdot s\right) \cdot r}}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. times-frac N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lift-PI.f32 N/A

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

8. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(0.75, \frac{0.16666666666666666}{\pi \cdot s} \cdot \color{blue}{\frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r}}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]
9. Add Preprocessing

Alternative 2: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot s\right) \cdot r\\ \mathsf{fma}\left(0.75, \frac{0.16666666666666666 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{t\_0}, \frac{e^{\frac{-r}{s}}}{t\_0} \cdot 0.125\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* PI s) r)))
   (fma
    0.75
    (/ (* 0.16666666666666666 (exp (* (/ r s) -0.3333333333333333))) t_0)
    (* (/ (exp (/ (- r) s)) t_0) 0.125))))

C

float code(float s, float r) {
	float t_0 = (((float) M_PI) * s) * r;
	return fmaf(0.75f, ((0.16666666666666666f * expf(((r / s) * -0.3333333333333333f))) / t_0), ((expf((-r / s)) / t_0) * 0.125f));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(Float32(pi) * s) * r)
	return fma(Float32(0.75), Float32(Float32(Float32(0.16666666666666666) * exp(Float32(Float32(r / s) * Float32(-0.3333333333333333)))) / t_0), Float32(Float32(exp(Float32(Float32(-r) / s)) / t_0) * Float32(0.125)))
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.75, \frac{e^{\frac{-r}{3 \cdot s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right)} \]

4. Taylor expanded in s around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lift-PI.f32 99.5

       \[\leadsto \mathsf{fma}\left(0.75, \frac{0.16666666666666666 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\left(\pi \cdot s\right) \cdot r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{\frac{0.16666666666666666 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\left(\pi \cdot s\right) \cdot r}}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

7. Taylor expanded in s around 0

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

9. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(0.75, \frac{0.16666666666666666 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\left(\pi \cdot s\right) \cdot r}, \color{blue}{\frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125}\right) \]
10. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * ((exp((-r / s)) / (single(pi) * r)) + ((exp(((r / s) * single(-0.3333333333333333))) / 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 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. lift-PI.f32 99.5

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

6. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (((exp((-r / s)) + exp(((r / s) * single(-0.3333333333333333)))) / (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. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * ((exp((-r / s)) + exp(((r / s) * single(-0.3333333333333333)))) / 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 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

Alternative 6: 12.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(fma(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(pi) * s)), Float32(0.125), Float32(Float32(0.125) / fma(Float32(Float32(0.3333333333333333) * r), Float32(pi), 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. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. exp-neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 99.5

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \frac{1}{e^{\frac{r}{\color{blue}{3 \cdot s}}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\frac{1}{e^{\frac{r}{3 \cdot s}}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Taylor expanded in r around inf

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

   1. rec-exp N/A

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

   2. distribute-frac-neg N/A

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

   3. lower-/.f32 N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in r around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 12.9

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

9. Applied rewrites 12.9%

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

Alternative 7: 10.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(0.06944444444444445 \cdot \frac{r}{\left(s \cdot s\right) \cdot \pi} - \frac{0.16666666666666666}{\pi \cdot s}, r, \frac{0.25}{\pi}\right)}{r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (/
   (fma
    (-
     (* 0.06944444444444445 (/ r (* (* s s) PI)))
     (/ 0.16666666666666666 (* PI s)))
    r
    (/ 0.25 PI))
   r)
  s))

C

float code(float s, float r) {
	return (fmaf(((0.06944444444444445f * (r / ((s * s) * ((float) M_PI)))) - (0.16666666666666666f / (((float) M_PI) * s))), r, (0.25f / ((float) M_PI))) / r) / s;
}

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(Float32(Float32(0.06944444444444445) * Float32(r / Float32(Float32(s * s) * Float32(pi)))) - Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * s))), r, Float32(Float32(0.25) / Float32(pi))) / r) / s)
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{s} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{s} \]

7. Applied rewrites 10.2%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(0.06944444444444445 \cdot \frac{r}{\left(s \cdot s\right) \cdot \pi} - \frac{0.16666666666666666}{\pi \cdot s}, r, \frac{0.25}{\pi}\right)}{r}}{s} \]
8. Add Preprocessing

Alternative 8: 10.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (-
  (/
   (-
    (-
     (/
      (-
       (- (/ (* (/ r PI) -0.06944444444444445) s))
       (/ 0.16666666666666666 PI))
      s))
    (/ (/ 0.25 r) PI))
   s)))

C

float code(float s, float r) {
	return -((-((-(((r / ((float) M_PI)) * -0.06944444444444445f) / s) - (0.16666666666666666f / ((float) M_PI))) / s) - ((0.25f / r) / ((float) M_PI))) / s);
}

Julia

function code(s, r)
	return Float32(-Float32(Float32(Float32(-Float32(Float32(Float32(-Float32(Float32(Float32(r / Float32(pi)) * Float32(-0.06944444444444445)) / s)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s)) - Float32(Float32(Float32(0.25) / r) / Float32(pi))) / s))
end

MATLAB

function tmp = code(s, r)
	tmp = -((-((-(((r / single(pi)) * single(-0.06944444444444445)) / s) - (single(0.16666666666666666) / single(pi))) / s) - ((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

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}\right) \]

   2. lower-neg.f32 N/A

      \[\leadsto -\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} \]

   3. lower-/.f32 N/A

      \[\leadsto -\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} \]

4. Applied rewrites 10.2%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot \frac{-5}{72}}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   2. lift-PI.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot \frac{-5}{72}}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   3. lift-*.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot \frac{-5}{72}}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   4. *-commutative N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot \frac{-5}{72}}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{r \cdot \mathsf{PI}\left(\right)}}{s} \]

   5. associate-/r* N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot \frac{-5}{72}}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\mathsf{PI}\left(\right)}}{s} \]

   6. lower-/.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot \frac{-5}{72}}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\mathsf{PI}\left(\right)}}{s} \]

   7. lower-/.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot \frac{-5}{72}}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\mathsf{PI}\left(\right)}}{s} \]

   8. lift-PI.f32 10.2

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s} \]

6. Applied rewrites 10.2%

   \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s} \]
7. Add Preprocessing

Alternative 9: 10.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (-
  (/
   (-
    (-
     (/
      (-
       (- (/ (* (/ r PI) -0.06944444444444445) s))
       (/ 0.16666666666666666 PI))
      s))
    (/ 0.25 (* PI r)))
   s)))

C

float code(float s, float r) {
	return -((-((-(((r / ((float) M_PI)) * -0.06944444444444445f) / s) - (0.16666666666666666f / ((float) M_PI))) / s) - (0.25f / (((float) M_PI) * r))) / s);
}

Julia

function code(s, r)
	return Float32(-Float32(Float32(Float32(-Float32(Float32(Float32(-Float32(Float32(Float32(r / Float32(pi)) * Float32(-0.06944444444444445)) / s)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s)) - Float32(Float32(0.25) / Float32(Float32(pi) * r))) / s))
end

MATLAB

function tmp = code(s, r)
	tmp = -((-((-(((r / single(pi)) * single(-0.06944444444444445)) / s) - (single(0.16666666666666666) / single(pi))) / s) - (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

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}\right) \]

   2. lower-neg.f32 N/A

      \[\leadsto -\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} \]

   3. lower-/.f32 N/A

      \[\leadsto -\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} \]

4. Applied rewrites 10.2%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]
5. Add Preprocessing

Alternative 10: 10.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, -0.06944444444444445, \frac{0.16666666666666666}{\pi}\right)}{s} - \frac{0.25}{\pi \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (-
  (/
   (-
    (/ (fma (/ r (* PI s)) -0.06944444444444445 (/ 0.16666666666666666 PI)) s)
    (/ 0.25 (* PI r)))
   s)))

C

float code(float s, float r) {
	return -(((fmaf((r / (((float) M_PI) * s)), -0.06944444444444445f, (0.16666666666666666f / ((float) M_PI))) / s) - (0.25f / (((float) M_PI) * r))) / s);
}

Julia

function code(s, r)
	return Float32(-Float32(Float32(Float32(fma(Float32(r / Float32(Float32(pi) * s)), Float32(-0.06944444444444445), Float32(Float32(0.16666666666666666) / Float32(pi))) / s) - Float32(Float32(0.25) / Float32(Float32(pi) * r))) / s))
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}\right) \]

   2. lower-neg.f32 N/A

      \[\leadsto -\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} \]

   3. lower-/.f32 N/A

      \[\leadsto -\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} \]

4. Applied rewrites 10.2%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]

5. Taylor expanded in s around inf

   \[\leadsto -\frac{\frac{\frac{-5}{72} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto -\frac{\frac{\frac{-5}{72} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   2. *-commutative N/A

      \[\leadsto -\frac{\frac{\frac{r}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{-5}{72} + \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   3. lower-fma.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   4. lower-/.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   5. *-commutative N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{\mathsf{PI}\left(\right) \cdot s}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   6. lift-*.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{\mathsf{PI}\left(\right) \cdot s}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   7. lift-PI.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   8. associate-*r/ N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, \frac{-5}{72}, \frac{\frac{1}{6} \cdot 1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   9. metadata-eval N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   10. lift-/.f32 N/A

       \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   11. lift-PI.f32 10.2

       \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, -0.06944444444444445, \frac{0.16666666666666666}{\pi}\right)}{s} - \frac{0.25}{\pi \cdot r}}{s} \]

7. Applied rewrites 10.2%

   \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, -0.06944444444444445, \frac{0.16666666666666666}{\pi}\right)}{s} - \frac{0.25}{\pi \cdot r}}{s} \]
8. Add Preprocessing

Alternative 11: 9.2% 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. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.2

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

4. Applied rewrites 9.2%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 9.2

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

6. Applied rewrites 9.2%

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

Alternative 12: 9.2% accurate, 6.4× speedup

Math

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

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(0.25) / Float32(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. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.2

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

4. Applied rewrites 9.2%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 9.2

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

6. Applied rewrites 9.2%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 9.2

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

8. Applied rewrites 9.2%

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

Reproduce

herbie shell --seed 2025115 
(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))))