Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.8s

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

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

FPCore

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

C

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

Julia

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

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

4. Taylor expanded in r around 0

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

   1. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-PI.f32 99.6

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

9. Applied rewrites 99.6%

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

Alternative 2: 10.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, r, s \cdot 0.75\right), s, \left(r \cdot r\right) \cdot 0.041666666666666664\right)}{s \cdot s}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/
   (/
    (fma (fma -0.25 r (* s 0.75)) s (* (* r r) 0.041666666666666664))
    (* s 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)) + ((fmaf(fmaf(-0.25f, r, (s * 0.75f)), s, ((r * r) * 0.041666666666666664f)) / (s * 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(fma(fma(Float32(-0.25), r, Float32(s * Float32(0.75))), s, Float32(Float32(r * r) * Float32(0.041666666666666664))) / Float32(s * s)) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) + \color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r + \frac{3}{4}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, \color{blue}{r}, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4} \cdot 1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   12. lower-/.f32 10.7

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 10.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{24} \cdot {r}^{2} + s \cdot \left(\frac{-1}{4} \cdot r + \frac{3}{4} \cdot s\right)}{\color{blue}{{s}^{2}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{24} \cdot {r}^{2} + s \cdot \left(\frac{-1}{4} \cdot r + \frac{3}{4} \cdot s\right)}{{s}^{\color{blue}{2}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{s \cdot \left(\frac{-1}{4} \cdot r + \frac{3}{4} \cdot s\right) + \frac{1}{24} \cdot {r}^{2}}{{s}^{2}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\left(\frac{-1}{4} \cdot r + \frac{3}{4} \cdot s\right) \cdot s + \frac{1}{24} \cdot {r}^{2}}{{s}^{2}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\frac{-1}{4} \cdot r + \frac{3}{4} \cdot s, s, \frac{1}{24} \cdot {r}^{2}\right)}{{s}^{2}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, r, \frac{3}{4} \cdot s\right), s, \frac{1}{24} \cdot {r}^{2}\right)}{{s}^{2}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, r, s \cdot \frac{3}{4}\right), s, \frac{1}{24} \cdot {r}^{2}\right)}{{s}^{2}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, r, s \cdot \frac{3}{4}\right), s, \frac{1}{24} \cdot {r}^{2}\right)}{{s}^{2}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, r, s \cdot \frac{3}{4}\right), s, {r}^{2} \cdot \frac{1}{24}\right)}{{s}^{2}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, r, s \cdot \frac{3}{4}\right), s, {r}^{2} \cdot \frac{1}{24}\right)}{{s}^{2}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. unpow2 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, r, s \cdot \frac{3}{4}\right), s, \left(r \cdot r\right) \cdot \frac{1}{24}\right)}{{s}^{2}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, r, s \cdot \frac{3}{4}\right), s, \left(r \cdot r\right) \cdot \frac{1}{24}\right)}{{s}^{2}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, r, s \cdot \frac{3}{4}\right), s, \left(r \cdot r\right) \cdot \frac{1}{24}\right)}{s \cdot s}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   13. lift-*.f32 10.7

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, r, s \cdot 0.75\right), s, \left(r \cdot r\right) \cdot 0.041666666666666664\right)}{s \cdot s}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Applied rewrites 10.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, r, s \cdot 0.75\right), s, \left(r \cdot r\right) \cdot 0.041666666666666664\right)}{\color{blue}{s \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
8. Add Preprocessing

Alternative 3: 10.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}, 0.25, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r}{s}, -0.041666666666666664, 0.25\right)}{-s}, r, 0.75\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ (- r) s)) (* (* (* 2.0 PI) s) r))
  0.25
  (/
   (fma (/ (fma (/ r s) -0.041666666666666664 0.25) (- s)) r 0.75)
   (* (* (* PI 6.0) s) r))))

C

float code(float s, float r) {
	return fmaf((expf((-r / s)) / (((2.0f * ((float) M_PI)) * s) * r)), 0.25f, (fmaf((fmaf((r / s), -0.041666666666666664f, 0.25f) / -s), r, 0.75f) / (((((float) M_PI) * 6.0f) * s) * r)));
}

Julia

function code(s, r)
	return fma(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)), Float32(0.25), Float32(fma(Float32(fma(Float32(r / s), Float32(-0.041666666666666664), Float32(0.25)) / Float32(-s)), r, Float32(0.75)) / Float32(Float32(Float32(Float32(pi) * Float32(6.0)) * s) * r)))
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) + \color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r + \frac{3}{4}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, \color{blue}{r}, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4} \cdot 1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   12. lower-/.f32 10.7

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 10.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(-1 \cdot \frac{\frac{1}{4} + \frac{-1}{24} \cdot \frac{r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\mathsf{neg}\left(\frac{\frac{1}{4} + \frac{-1}{24} \cdot \frac{r}{s}}{s}\right), r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. lower-neg.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(-\frac{\frac{1}{4} + \frac{-1}{24} \cdot \frac{r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(-\frac{\frac{1}{4} + \frac{-1}{24} \cdot \frac{r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(-\frac{\frac{-1}{24} \cdot \frac{r}{s} + \frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(\frac{-1}{24}, \frac{r}{s}, \frac{1}{4}\right)}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-/.f32 10.7

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(-0.041666666666666664, \frac{r}{s}, 0.25\right)}{s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Applied rewrites 10.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(-0.041666666666666664, \frac{r}{s}, 0.25\right)}{s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

9. Applied rewrites 10.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}, 0.25, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r}{s}, -0.041666666666666664, 0.25\right)}{-s}, r, 0.75\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}\right)} \]
10. Add Preprocessing

Alternative 4: 10.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, s, 0.041666666666666664 \cdot r\right)}{s \cdot s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/
   (fma (/ (fma -0.25 s (* 0.041666666666666664 r)) (* s s)) r 0.75)
   (* (* (* 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)) + (fmaf((fmaf(-0.25f, s, (0.041666666666666664f * r)) / (s * s)), r, 0.75f) / (((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(fma(Float32(fma(Float32(-0.25), s, Float32(Float32(0.041666666666666664) * r)) / Float32(s * s)), r, Float32(0.75)) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) + \color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r + \frac{3}{4}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, \color{blue}{r}, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4} \cdot 1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   12. lower-/.f32 10.7

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 10.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot s + \frac{1}{24} \cdot r}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} \cdot s + \frac{1}{24} \cdot r}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-1}{4}, s, \frac{1}{24} \cdot r\right)}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-1}{4}, s, \frac{1}{24} \cdot r\right)}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. pow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-1}{4}, s, \frac{1}{24} \cdot r\right)}{s \cdot s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lift-*.f32 10.7

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, s, 0.041666666666666664 \cdot r\right)}{s \cdot s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Applied rewrites 10.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, s, 0.041666666666666664 \cdot r\right)}{s \cdot s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
8. Add Preprocessing

Alternative 5: 10.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \frac{r \cdot r}{s}, r\right)}{s}, -1, 1\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/
   (* 0.25 (fma (/ (fma -0.5 (/ (* r r) s) r) s) -1.0 1.0))
   (* (* (* 2.0 PI) s) r))
  (/
   (fma (- (* (/ r (* s s)) 0.041666666666666664) (/ 0.25 s)) r 0.75)
   (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * fmaf((fmaf(-0.5f, ((r * r) / s), r) / s), -1.0f, 1.0f)) / (((2.0f * ((float) M_PI)) * s) * r)) + (fmaf((((r / (s * s)) * 0.041666666666666664f) - (0.25f / s)), r, 0.75f) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * fma(Float32(fma(Float32(-0.5), Float32(Float32(r * r) / s), r) / s), Float32(-1.0), Float32(1.0))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(fma(Float32(Float32(Float32(r / Float32(s * s)) * Float32(0.041666666666666664)) - Float32(Float32(0.25) / s)), r, Float32(0.75)) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) + \color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r + \frac{3}{4}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, \color{blue}{r}, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4} \cdot 1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   12. lower-/.f32 10.7

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 10.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(1 + -1 \cdot \frac{r + \frac{-1}{2} \cdot \frac{{r}^{2}}{s}}{s}\right)}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(-1 \cdot \frac{r + \frac{-1}{2} \cdot \frac{{r}^{2}}{s}}{s} + \color{blue}{1}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\frac{r + \frac{-1}{2} \cdot \frac{{r}^{2}}{s}}{s} \cdot -1 + 1\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \mathsf{fma}\left(\frac{r + \frac{-1}{2} \cdot \frac{{r}^{2}}{s}}{s}, \color{blue}{-1}, 1\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \mathsf{fma}\left(\frac{r + \frac{-1}{2} \cdot \frac{{r}^{2}}{s}}{s}, -1, 1\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \mathsf{fma}\left(\frac{\frac{-1}{2} \cdot \frac{{r}^{2}}{s} + r}{s}, -1, 1\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{{r}^{2}}{s}, r\right)}{s}, -1, 1\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{{r}^{2}}{s}, r\right)}{s}, -1, 1\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{r \cdot r}{s}, r\right)}{s}, -1, 1\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 10.1

      \[\leadsto \frac{0.25 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \frac{r \cdot r}{s}, r\right)}{s}, -1, 1\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Applied rewrites 10.1%

   \[\leadsto \frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \frac{r \cdot r}{s}, r\right)}{s}, -1, 1\right)}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
8. Add Preprocessing

Alternative 6: 10.1% accurate, 3.3× 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 10.1%

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

Alternative 7: 10.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return -(((fmaf(((r / s) / ((float) M_PI)), -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(Float32(r / s) / Float32(pi)), 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

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

4. Applied rewrites 9.8%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]

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. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 10.1

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

7. Applied rewrites 10.1%

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

Alternative 8: 9.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\pi \cdot r} - \frac{0.16666666666666666}{\pi \cdot s}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (single(pi) * r)) - (single(0.16666666666666666) / (single(pi) * s))) / 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

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 9.0%

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

Alternative 9: 9.0% accurate, 10.6× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. 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 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lift-PI.f32 9.0

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

6. Applied rewrites 9.0%

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

Alternative 10: 9.0% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / r) / Float32(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. *-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 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 9.0

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

6. Applied rewrites 9.0%

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

Alternative 11: 9.0% accurate, 13.5× 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 9.0

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

4. Applied rewrites 9.0%

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

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

6. Applied rewrites 9.0%

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

Alternative 12: 9.0% accurate, 13.5× 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 9.0

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

4. Applied rewrites 9.0%

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

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

6. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   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 9.0

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

8. Applied rewrites 9.0%

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

Reproduce

herbie shell --seed 2025102 
(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))))