Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 15.2s

Alternatives: 11

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp((r / -s)) * single(0.125)) / (s * (r * single(pi)))) + ((single(0.75) * exp((r / (s * single(-3.0))))) / ((r * single(6.0)) * (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. Add Preprocessing
3. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.7

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.7%

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

   1. lift-*.f32 N/A

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

   2. distribute-frac-neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. lower-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.7

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

8. Applied rewrites 99.7%

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

Alternative 2: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

Alternative 3: 10.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{r}{-s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (/ r (- s))) 0.25) (* r (* s (* PI 2.0))))
  (/
   (+
    (/ 0.125 (* r PI))
    (fma
     r
     (/ 0.006944444444444444 (* s (* s PI)))
     (/ -0.041666666666666664 (* s PI))))
   s)))

C

float code(float s, float r) {
	return ((expf((r / -s)) * 0.25f) / (r * (s * (((float) M_PI) * 2.0f)))) + (((0.125f / (r * ((float) M_PI))) + fmaf(r, (0.006944444444444444f / (s * (s * ((float) M_PI)))), (-0.041666666666666664f / (s * ((float) M_PI))))) / s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) * Float32(0.25)) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(Float32(0.125) / Float32(r * Float32(pi))) + fma(r, Float32(Float32(0.006944444444444444) / Float32(s * Float32(s * Float32(pi)))), Float32(Float32(-0.041666666666666664) / Float32(s * Float32(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. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{24}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 10.0%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]

6. Final simplification 10.0%

   \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s} \]
7. Add Preprocessing

Alternative 4: 10.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.75}{r} + \frac{\mathsf{fma}\left(\frac{r}{s}, 0.041666666666666664, -0.25\right)}{s}, \frac{0.16666666666666666}{s \cdot \pi}, e^{\frac{r}{-s}} \cdot \frac{0.125}{\pi \cdot \left(r \cdot s\right)}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (+ (/ 0.75 r) (/ (fma (/ r s) 0.041666666666666664 -0.25) s))
  (/ 0.16666666666666666 (* s PI))
  (* (exp (/ r (- s))) (/ 0.125 (* PI (* r s))))))

C

float code(float s, float r) {
	return fmaf(((0.75f / r) + (fmaf((r / s), 0.041666666666666664f, -0.25f) / s)), (0.16666666666666666f / (s * ((float) M_PI))), (expf((r / -s)) * (0.125f / (((float) M_PI) * (r * s)))));
}

Julia

function code(s, r)
	return fma(Float32(Float32(Float32(0.75) / r) + Float32(fma(Float32(r / s), Float32(0.041666666666666664), Float32(-0.25)) / s)), Float32(Float32(0.16666666666666666) / Float32(s * Float32(pi))), Float32(exp(Float32(r / Float32(-s))) * Float32(Float32(0.125) / Float32(Float32(pi) * Float32(r * s)))))
end

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in s around -inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \frac{\frac{1}{4} + \frac{-1}{24} \cdot \frac{r}{s}}{s} + \frac{3}{4} \cdot \frac{1}{r}}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{3}{4} \cdot \frac{1}{r} + -1 \cdot \frac{\frac{1}{4} + \frac{-1}{24} \cdot \frac{r}{s}}{s}}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   2. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{3}{4} \cdot \frac{1}{r} + -1 \cdot \frac{\frac{1}{4} + \frac{-1}{24} \cdot \frac{r}{s}}{s}}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{3}{4} \cdot 1}{r}} + -1 \cdot \frac{\frac{1}{4} + \frac{-1}{24} \cdot \frac{r}{s}}{s}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{3}{4}}}{r} + -1 \cdot \frac{\frac{1}{4} + \frac{-1}{24} \cdot \frac{r}{s}}{s}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   5. lower-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{3}{4}}{r}} + -1 \cdot \frac{\frac{1}{4} + \frac{-1}{24} \cdot \frac{r}{s}}{s}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4}}{r} + \color{blue}{\frac{-1 \cdot \left(\frac{1}{4} + \frac{-1}{24} \cdot \frac{r}{s}\right)}{s}}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   7. lower-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4}}{r} + \color{blue}{\frac{-1 \cdot \left(\frac{1}{4} + \frac{-1}{24} \cdot \frac{r}{s}\right)}{s}}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4}}{r} + \frac{-1 \cdot \color{blue}{\left(\frac{-1}{24} \cdot \frac{r}{s} + \frac{1}{4}\right)}}{s}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   9. distribute-rgt-in N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4}}{r} + \frac{\color{blue}{\left(\frac{-1}{24} \cdot \frac{r}{s}\right) \cdot -1 + \frac{1}{4} \cdot -1}}{s}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4}}{r} + \frac{\color{blue}{\left(\frac{r}{s} \cdot \frac{-1}{24}\right)} \cdot -1 + \frac{1}{4} \cdot -1}{s}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4}}{r} + \frac{\color{blue}{\frac{r}{s} \cdot \left(\frac{-1}{24} \cdot -1\right)} + \frac{1}{4} \cdot -1}{s}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4}}{r} + \frac{\frac{r}{s} \cdot \color{blue}{\frac{1}{24}} + \frac{1}{4} \cdot -1}{s}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4}}{r} + \frac{\frac{r}{s} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{4}}}{s}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   14. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4}}{r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{r}{s}, \frac{1}{24}, \frac{-1}{4}\right)}}{s}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   15. lower-/.f32 9.9

       \[\leadsto \mathsf{fma}\left(\frac{0.75}{r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{r}{s}}, 0.041666666666666664, -0.25\right)}{s}, \frac{0.16666666666666666}{s \cdot \pi}, e^{\frac{r}{-s}} \cdot \frac{0.125}{\pi \cdot \left(r \cdot s\right)}\right) \]

7. Applied rewrites 9.9%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.75}{r} + \frac{\mathsf{fma}\left(\frac{r}{s}, 0.041666666666666664, -0.25\right)}{s}}, \frac{0.16666666666666666}{s \cdot \pi}, e^{\frac{r}{-s}} \cdot \frac{0.125}{\pi \cdot \left(r \cdot s\right)}\right) \]
8. Add Preprocessing

Alternative 5: 9.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

5. Applied rewrites 9.5%

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

Alternative 6: 9.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(r / Float32(pi)), Float32(Float32(-0.06944444444444445) / s), Float32(Float32(0.16666666666666666) / Float32(pi))) / Float32(-Float32(s * s))) - Float32(Float32(-0.25) / Float32(s * Float32(r * Float32(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. Add Preprocessing
3. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.7

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. lower-+.f32 N/A

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

9. Applied rewrites 9.4%

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

10. Final simplification 9.4%

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

Alternative 7: 9.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(0.06944444444444445), Float32(r / Float32(s * Float32(pi))), Float32(Float32(-0.16666666666666666) / Float32(pi))) / Float32(s * s)) + Float32(Float32(0.25) / Float32(r * Float32(s * Float32(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. Add Preprocessing

3. Taylor expanded in s around inf

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

5. Applied rewrites 9.4%

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

Alternative 8: 8.8% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / (r * (sqrt(single(pi)) * (s * sqrt(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. Add Preprocessing

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.5

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

5. Applied rewrites 8.5%

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

   1. add-sqr-sqrt N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-sqrt.f32 8.5

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

7. Applied rewrites 8.5%

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

8. Final simplification 8.5%

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

Alternative 9: 8.8% accurate, 13.5× 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. Add Preprocessing

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.5

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

5. Applied rewrites 8.5%

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

Alternative 10: 7.1% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{\pi \cdot \left(r \cdot s\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.125 (* PI (* r s))))

C

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

Julia

function code(s, r)
	return Float32(Float32(0.125) / Float32(Float32(pi) * Float32(r * s)))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) / (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. Add Preprocessing

3. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{1296} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 9.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.0007716049382716049, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.006944444444444444}{\pi}\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \pi} + \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]

6. Taylor expanded in r around inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{{r}^{2} \cdot \left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)}}{s} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{{r}^{2} \cdot \left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)}}{s} \]

   2. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

   4. sub-neg N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \color{blue}{\left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}}{s} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\color{blue}{\frac{\frac{1}{144} \cdot 1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\color{blue}{\frac{1}{144}}}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   7. lower-+.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \color{blue}{\left(\frac{\frac{1}{144}}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}}{s} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\color{blue}{\frac{\frac{1}{144}}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{\color{blue}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   10. unpow2 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   11. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \color{blue}{\left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \color{blue}{\left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \left(s \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   14. lower-PI.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \left(s \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

8. Applied rewrites 4.2%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(r \cdot r\right) \cdot \left(\frac{0.006944444444444444}{r \cdot \left(s \cdot \left(s \cdot \pi\right)\right)} + \frac{-0.0007716049382716049}{s \cdot \left(s \cdot \left(s \cdot \pi\right)\right)}\right)}}{s} \]

9. Taylor expanded in r around 0

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

    1. lower-/.f32 N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lower-PI.f32 6.9

       \[\leadsto \frac{0.125}{s \cdot \left(r \cdot \color{blue}{\pi}\right)} \]

11. Applied rewrites 6.9%

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

    1. lift-PI.f32 N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. lift-*.f32 N/A

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

    5. lower-*.f32 6.9

       \[\leadsto \frac{0.125}{\color{blue}{\left(r \cdot s\right) \cdot \pi}} \]

13. Applied rewrites 6.9%

    \[\leadsto \frac{0.125}{\color{blue}{\left(r \cdot s\right) \cdot \pi}} \]

14. Final simplification 6.9%

    \[\leadsto \frac{0.125}{\pi \cdot \left(r \cdot s\right)} \]
15. Add Preprocessing

Alternative 11: 7.1% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \left(r \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.125 (* s (* r PI))))

C

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

Julia

function code(s, r)
	return Float32(Float32(0.125) / Float32(s * Float32(r * Float32(pi))))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{1296} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 9.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.0007716049382716049, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.006944444444444444}{\pi}\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \pi} + \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]

6. Taylor expanded in r around inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{{r}^{2} \cdot \left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)}}{s} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{{r}^{2} \cdot \left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)}}{s} \]

   2. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

   4. sub-neg N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \color{blue}{\left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}}{s} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\color{blue}{\frac{\frac{1}{144} \cdot 1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\color{blue}{\frac{1}{144}}}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   7. lower-+.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \color{blue}{\left(\frac{\frac{1}{144}}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}}{s} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\color{blue}{\frac{\frac{1}{144}}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{\color{blue}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   10. unpow2 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   11. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \color{blue}{\left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \color{blue}{\left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \left(s \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   14. lower-PI.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \left(s \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

8. Applied rewrites 4.2%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(r \cdot r\right) \cdot \left(\frac{0.006944444444444444}{r \cdot \left(s \cdot \left(s \cdot \pi\right)\right)} + \frac{-0.0007716049382716049}{s \cdot \left(s \cdot \left(s \cdot \pi\right)\right)}\right)}}{s} \]

9. Taylor expanded in r around 0

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

    1. lower-/.f32 N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lower-PI.f32 6.9

       \[\leadsto \frac{0.125}{s \cdot \left(r \cdot \color{blue}{\pi}\right)} \]

11. Applied rewrites 6.9%

    \[\leadsto \color{blue}{\frac{0.125}{s \cdot \left(r \cdot \pi\right)}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))