Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 4.2s

Alternatives: 15

Speedup: 1.1×

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, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{r \cdot 6}, \frac{0.75}{\pi \cdot s}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ r (* -3.0 s))) (* r 6.0))
  (/ 0.75 (* PI s))
  (/ (/ 0.125 (* (* PI s) (exp (/ r s)))) r)))

C

float code(float s, float r) {
	return fmaf((expf((r / (-3.0f * s))) / (r * 6.0f)), (0.75f / (((float) M_PI) * s)), ((0.125f / ((((float) M_PI) * s) * expf((r / s)))) / r));
}

Julia

function code(s, r)
	return fma(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(r * Float32(6.0))), Float32(Float32(0.75) / Float32(Float32(pi) * s)), Float32(Float32(Float32(0.125) / Float32(Float32(Float32(pi) * s) * exp(Float32(r / s)))) / r))
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{6 \cdot r}, \frac{0.75}{\pi \cdot s}, \frac{0.125}{\left(\pi \cdot s\right) \cdot \left(e^{\frac{r}{s}} \cdot r\right)}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ r (* s -3.0))) (* 6.0 r))
  (/ 0.75 (* PI s))
  (/ 0.125 (* (* PI s) (* (exp (/ r s)) r)))))

C

float code(float s, float r) {
	return fmaf((expf((r / (s * -3.0f))) / (6.0f * r)), (0.75f / (((float) M_PI) * s)), (0.125f / ((((float) M_PI) * s) * (expf((r / s)) * r))));
}

Julia

function code(s, r)
	return fma(Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / Float32(Float32(6.0) * r)), Float32(Float32(0.75) / Float32(Float32(pi) * s)), Float32(Float32(0.125) / Float32(Float32(Float32(pi) * s) * Float32(exp(Float32(r / s)) * r))))
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.5

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 99.5

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

   7. lift-/.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. associate-/l/ N/A

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

   10. lower-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 99.5

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

5. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{s}, \frac{0.75}{\left(6 \cdot \pi\right) \cdot r}, \frac{0.125}{\left(\pi \cdot s\right) \cdot \left(e^{\frac{r}{s}} \cdot r\right)}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ r (* -3.0 s))) s)
  (/ 0.75 (* (* 6.0 PI) r))
  (/ 0.125 (* (* PI s) (* (exp (/ r s)) r)))))

C

float code(float s, float r) {
	return fmaf((expf((r / (-3.0f * s))) / s), (0.75f / ((6.0f * ((float) M_PI)) * r)), (0.125f / ((((float) M_PI) * s) * (expf((r / s)) * r))));
}

Julia

function code(s, r)
	return fma(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / s), Float32(Float32(0.75) / Float32(Float32(Float32(6.0) * Float32(pi)) * r)), Float32(Float32(0.125) / Float32(Float32(Float32(pi) * s) * Float32(exp(Float32(r / 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%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.6

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

5. Applied rewrites 99.6%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.6%

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

Alternative 5: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi}, 0.125 \cdot \frac{1}{\pi \cdot e^{\frac{r}{s}}}\right)}{r \cdot s}} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-*.f32 N/A

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

   6. times-frac N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

8. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi}, 0.125 \cdot \frac{1}{\pi \cdot e^{\frac{r}{s}}}\right)}{r \cdot s}} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. distribute-lft-out N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

8. Applied rewrites 99.5%

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

Alternative 7: 46.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 28:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{e^{\frac{r}{-3 \cdot s}}}{\left(\pi \cdot s\right) \cdot r}, \frac{\frac{0.125}{s \cdot \mathsf{fma}\left(\frac{\pi}{s}, r, \pi\right)}}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s \cdot \log \left(e^{\pi \cdot r}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 28.0)
   (fma
    0.125
    (/ (exp (/ r (* -3.0 s))) (* (* PI s) r))
    (/ (/ 0.125 (* s (fma (/ PI s) r PI))) r))
   (/ 0.25 (* s (log (exp (* PI r)))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 28.0f) {
		tmp = fmaf(0.125f, (expf((r / (-3.0f * s))) / ((((float) M_PI) * s) * r)), ((0.125f / (s * fmaf((((float) M_PI) / s), r, ((float) M_PI)))) / r));
	} else {
		tmp = 0.25f / (s * logf(expf((((float) M_PI) * r))));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(28.0))
		tmp = fma(Float32(0.125), Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(Float32(Float32(pi) * s) * r)), Float32(Float32(Float32(0.125) / Float32(s * fma(Float32(Float32(pi) / s), r, Float32(pi)))) / r));
	else
		tmp = Float32(Float32(0.25) / Float32(s * log(exp(Float32(Float32(pi) * r)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 28

   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%

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

   4. Taylor expanded in s around inf

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-PI.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 16.1

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

   6. Applied rewrites 16.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-/.f32 16.4

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

   8. Applied rewrites 16.4%

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

   if 28 < r

   1. Initial program 99.6%

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

   2. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 9.1

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

   4. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 9.1

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

   6. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 9.1

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

      7. rem-log-exp N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f32 N/A

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

      13. exp-prod N/A

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

      14. log-pow N/A

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

      15. lower-unsound-*.f32 N/A

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

      16. lower-unsound-log.f32 N/A

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

      17. lower-exp.f32 43.2

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f32 43.2

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

   8. Applied rewrites 43.2%

      \[\leadsto \frac{0.25}{s \cdot \color{blue}{\log \left(e^{\pi \cdot r}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 46.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 28:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot r}, \frac{0.125}{\pi}, \frac{0.125}{\left(\left(s + r\right) \cdot \pi\right) \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s \cdot \log \left(e^{\pi \cdot r}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 28.0)
   (fma
    (/ (exp (* -0.3333333333333333 (/ r s))) (* s r))
    (/ 0.125 PI)
    (/ 0.125 (* (* (+ s r) PI) r)))
   (/ 0.25 (* s (log (exp (* PI r)))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 28.0f) {
		tmp = fmaf((expf((-0.3333333333333333f * (r / s))) / (s * r)), (0.125f / ((float) M_PI)), (0.125f / (((s + r) * ((float) M_PI)) * r)));
	} else {
		tmp = 0.25f / (s * logf(expf((((float) M_PI) * r))));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(28.0))
		tmp = fma(Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) / Float32(s * r)), Float32(Float32(0.125) / Float32(pi)), Float32(Float32(0.125) / Float32(Float32(Float32(s + r) * Float32(pi)) * r)));
	else
		tmp = Float32(Float32(0.25) / Float32(s * log(exp(Float32(Float32(pi) * r)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 28

   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%

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

   4. Taylor expanded in r around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-PI.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 12.4

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

   6. Applied rewrites 12.4%

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

      1. lift-fma.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 12.4

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

   8. Applied rewrites 12.4%

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

      1. lift-fma.f32 N/A

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

   10. Applied rewrites 12.4%

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

   if 28 < r

   1. Initial program 99.6%

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

   2. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 9.1

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

   4. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 9.1

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

   6. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 9.1

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

      7. rem-log-exp N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f32 N/A

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

      13. exp-prod N/A

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

      14. log-pow N/A

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

      15. lower-unsound-*.f32 N/A

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

      16. lower-unsound-log.f32 N/A

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

      17. lower-exp.f32 43.2

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f32 43.2

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

   8. Applied rewrites 43.2%

      \[\leadsto \frac{0.25}{s \cdot \color{blue}{\log \left(e^{\pi \cdot r}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 46.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 28:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\pi \cdot s}, 0.125, \frac{0.125}{\mathsf{fma}\left(r, \pi, s \cdot \pi\right)}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s \cdot \log \left(e^{\pi \cdot r}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 28.0)
   (/
    (fma
     (/ (exp (/ r (* -3.0 s))) (* PI s))
     0.125
     (/ 0.125 (fma r PI (* s PI))))
    r)
   (/ 0.25 (* s (log (exp (* PI r)))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 28.0f) {
		tmp = fmaf((expf((r / (-3.0f * s))) / (((float) M_PI) * s)), 0.125f, (0.125f / fmaf(r, ((float) M_PI), (s * ((float) M_PI))))) / r;
	} else {
		tmp = 0.25f / (s * logf(expf((((float) M_PI) * r))));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(28.0))
		tmp = Float32(fma(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(Float32(pi) * s)), Float32(0.125), Float32(Float32(0.125) / fma(r, Float32(pi), Float32(s * Float32(pi))))) / r);
	else
		tmp = Float32(Float32(0.25) / Float32(s * log(exp(Float32(Float32(pi) * r)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 28

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in r around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-PI.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 12.4

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

   6. Applied rewrites 12.4%

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

   if 28 < r

   1. Initial program 99.6%

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

   2. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 9.1

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

   4. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 9.1

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

   6. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 9.1

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

      7. rem-log-exp N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f32 N/A

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

      13. exp-prod N/A

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

      14. log-pow N/A

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

      15. lower-unsound-*.f32 N/A

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

      16. lower-unsound-log.f32 N/A

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

      17. lower-exp.f32 43.2

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f32 43.2

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

   8. Applied rewrites 43.2%

      \[\leadsto \frac{0.25}{s \cdot \color{blue}{\log \left(e^{\pi \cdot r}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 46.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 28:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\pi \cdot r\right) \cdot s}, 0.125, \frac{0.125}{\left(\left(s + r\right) \cdot \pi\right) \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s \cdot \log \left(e^{\pi \cdot r}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 28.0)
   (fma
    (/ (exp (* -0.3333333333333333 (/ r s))) (* (* PI r) s))
    0.125
    (/ 0.125 (* (* (+ s r) PI) r)))
   (/ 0.25 (* s (log (exp (* PI r)))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 28.0f) {
		tmp = fmaf((expf((-0.3333333333333333f * (r / s))) / ((((float) M_PI) * r) * s)), 0.125f, (0.125f / (((s + r) * ((float) M_PI)) * r)));
	} else {
		tmp = 0.25f / (s * logf(expf((((float) M_PI) * r))));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(28.0))
		tmp = fma(Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) / Float32(Float32(Float32(pi) * r) * s)), Float32(0.125), Float32(Float32(0.125) / Float32(Float32(Float32(s + r) * Float32(pi)) * r)));
	else
		tmp = Float32(Float32(0.25) / Float32(s * log(exp(Float32(Float32(pi) * r)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 28

   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%

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

   4. Taylor expanded in r around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-PI.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 12.4

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

   6. Applied rewrites 12.4%

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

      1. lift-fma.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 12.4

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

   8. Applied rewrites 12.4%

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

   if 28 < r

   1. Initial program 99.6%

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

   2. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 9.1

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

   4. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 9.1

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

   6. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 9.1

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

      7. rem-log-exp N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f32 N/A

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

      13. exp-prod N/A

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

      14. log-pow N/A

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

      15. lower-unsound-*.f32 N/A

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

      16. lower-unsound-log.f32 N/A

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

      17. lower-exp.f32 43.2

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f32 43.2

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

   8. Applied rewrites 43.2%

      \[\leadsto \frac{0.25}{s \cdot \color{blue}{\log \left(e^{\pi \cdot r}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 43.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 9.1

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

6. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 9.1

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

   7. rem-log-exp N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. exp-prod N/A

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

   14. log-pow N/A

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

   15. lower-unsound-*.f32 N/A

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

   16. lower-unsound-log.f32 N/A

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

   17. lower-exp.f32 43.2

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

   18. lift-*.f32 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f32 43.2

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

8. Applied rewrites 43.2%

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

Alternative 12: 10.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-PI.f32 N/A

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

   5. add-log-exp N/A

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

   6. log-pow-rev N/A

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

   7. lower-log.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. pow-exp N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lower-exp.f32 10.2

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

6. Applied rewrites 10.2%

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

Alternative 13: 9.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (- (* 0.25 (/ 1.0 (* r PI))) (* 0.16666666666666666 (/ 1.0 (* s PI)))) s))

C

float code(float s, float r) {
	return ((0.25f * (1.0f / (r * ((float) M_PI)))) - (0.16666666666666666f * (1.0f / (s * ((float) M_PI))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(r * Float32(pi)))) - Float32(Float32(0.16666666666666666) * Float32(Float32(1.0) / Float32(s * Float32(pi))))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * (single(1.0) / (r * single(pi)))) - (single(0.16666666666666666) * (single(1.0) / (s * single(pi))))) / s;
end

Derivation

1. Initial program 99.6%

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

2. 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

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.1

       \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s} \]

4. Applied rewrites 9.1%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s}} \]
5. Add Preprocessing

Alternative 14: 9.1% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (/ (fma -0.16666666666666666 (/ r (* s PI)) (* 0.25 (/ 1.0 PI))) (* r s)))

C

float code(float s, float r) {
	return fmaf(-0.16666666666666666f, (r / (s * ((float) M_PI))), (0.25f * (1.0f / ((float) M_PI)))) / (r * s);
}

Julia

function code(s, r)
	return Float32(fma(Float32(-0.16666666666666666), Float32(r / Float32(s * Float32(pi))), Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(pi)))) / Float32(r * s))
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi}, 0.125 \cdot \frac{1}{\pi \cdot e^{\frac{r}{s}}}\right)}{r \cdot s}} \]

7. Taylor expanded in r around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-PI.f32 9.1

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

9. Applied rewrites 9.1%

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

Alternative 15: 9.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.1

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

4. Applied rewrites 9.1%

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

Reproduce

herbie shell --seed 2025162 
(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))))