Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.6s

Alternatives: 19

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. lower-*.f32 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. mul-1-neg N/A

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

   3. distribute-frac-neg N/A

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

   4. lower-/.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 99.6

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

10. Applied rewrites 99.6%

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

11. Taylor expanded in s around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. lift-/.f32 99.6

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

13. Applied rewrites 99.6%

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

Alternative 2: 10.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\mathsf{fma}\left(0.0007716049382716049, \frac{\frac{r \cdot r}{s}}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} + \frac{0.041666666666666664}{\pi}}{s} - \frac{0.125}{\pi \cdot r}}{-s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/
   (-
    (/
     (+
      (/
       (fma
        0.0007716049382716049
        (/ (/ (* r r) s) PI)
        (* -0.006944444444444444 (/ r PI)))
       s)
      (/ 0.041666666666666664 PI))
     s)
    (/ 0.125 (* PI r)))
   (- s))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + (((((fmaf(0.0007716049382716049f, (((r * r) / s) / ((float) M_PI)), (-0.006944444444444444f * (r / ((float) M_PI)))) / s) + (0.041666666666666664f / ((float) M_PI))) / s) - (0.125f / (((float) M_PI) * r))) / -s);
}

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(Float32(Float32(fma(Float32(0.0007716049382716049), Float32(Float32(Float32(r * r) / s) / Float32(pi)), Float32(Float32(-0.006944444444444444) * Float32(r / Float32(pi)))) / s) + Float32(Float32(0.041666666666666664) / Float32(pi))) / s) - Float32(Float32(0.125) / Float32(Float32(pi) * r))) / Float32(-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

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

5. Applied rewrites 10.6%

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

6. Final simplification 10.6%

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

Alternative 3: 10.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\mathsf{fma}\left(-0.006944444444444444, \frac{r}{\pi}, \frac{0.0007716049382716049 \cdot \left(r \cdot r\right)}{s \cdot \pi}\right)}{-s} - \frac{0.041666666666666664}{\pi}}{s} + \frac{0.125}{r \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/
   (+
    (/
     (-
      (/
       (fma
        -0.006944444444444444
        (/ r PI)
        (/ (* 0.0007716049382716049 (* r r)) (* s PI)))
       (- s))
      (/ 0.041666666666666664 PI))
     s)
    (/ 0.125 (* r PI)))
   s)))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + (((((fmaf(-0.006944444444444444f, (r / ((float) M_PI)), ((0.0007716049382716049f * (r * r)) / (s * ((float) M_PI)))) / -s) - (0.041666666666666664f / ((float) M_PI))) / s) + (0.125f / (r * ((float) M_PI)))) / s);
}

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(Float32(Float32(fma(Float32(-0.006944444444444444), Float32(r / Float32(pi)), Float32(Float32(Float32(0.0007716049382716049) * Float32(r * r)) / Float32(s * Float32(pi)))) / Float32(-s)) - Float32(Float32(0.041666666666666664) / Float32(pi))) / s) + Float32(Float32(0.125) / Float32(r * 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

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 9.8

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

5. Applied rewrites 9.8%

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

6. Taylor expanded in s around -inf

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

8. Applied rewrites 10.6%

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

9. Final simplification 10.6%

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

Alternative 4: 10.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.75, \frac{\frac{r \cdot \mathsf{fma}\left(-0.00102880658436214, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.009259259259259259}{s \cdot \pi}\right) - \frac{0.05555555555555555}{\pi}}{s} + \frac{0.16666666666666666}{\pi \cdot r}}{s}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.75
  (/
   (+
    (/
     (-
      (*
       r
       (fma
        -0.00102880658436214
        (/ r (* (* s s) PI))
        (/ 0.009259259259259259 (* s PI))))
      (/ 0.05555555555555555 PI))
     s)
    (/ 0.16666666666666666 (* PI r)))
   s)
  (* 0.25 (/ (exp (/ (- r) s)) (* (* (* PI 2.0) s) r)))))

C

float code(float s, float r) {
	return fmaf(0.75f, (((((r * fmaf(-0.00102880658436214f, (r / ((s * s) * ((float) M_PI))), (0.009259259259259259f / (s * ((float) M_PI))))) - (0.05555555555555555f / ((float) M_PI))) / s) + (0.16666666666666666f / (((float) M_PI) * r))) / s), (0.25f * (expf((-r / s)) / (((((float) M_PI) * 2.0f) * s) * r))));
}

Julia

function code(s, r)
	return fma(Float32(0.75), Float32(Float32(Float32(Float32(Float32(r * fma(Float32(-0.00102880658436214), Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(Float32(0.009259259259259259) / Float32(s * Float32(pi))))) - Float32(Float32(0.05555555555555555) / Float32(pi))) / s) + Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * r))) / s), Float32(Float32(0.25) * Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(Float32(pi) * Float32(2.0)) * s) * r))))
end

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around -inf

   \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{108} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{972} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{18} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

7. Applied rewrites 10.7%

   \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{\frac{r \cdot r}{s}}{\pi}, 0.00102880658436214, -0.009259259259259259 \cdot \frac{r}{\pi}\right)}{s}\right) - \frac{0.05555555555555555}{\pi}}{s}\right) - \frac{0.16666666666666666}{\pi \cdot r}}{s}}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

8. Taylor expanded in r around 0

   \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \left(\frac{-1}{972} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{108} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\pi}}{s}\right) - \frac{\frac{1}{6}}{\pi \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \left(\frac{-1}{972} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{108} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\pi}}{s}\right) - \frac{\frac{1}{6}}{\pi \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   2. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-1}{972}, \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}, \frac{1}{108} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\pi}}{s}\right) - \frac{\frac{1}{6}}{\pi \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   3. lower-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-1}{972}, \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}, \frac{1}{108} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\pi}}{s}\right) - \frac{\frac{1}{6}}{\pi \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-1}{972}, \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}, \frac{1}{108} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\pi}}{s}\right) - \frac{\frac{1}{6}}{\pi \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-1}{972}, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{108} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\pi}}{s}\right) - \frac{\frac{1}{6}}{\pi \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-1}{972}, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{108} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\pi}}{s}\right) - \frac{\frac{1}{6}}{\pi \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   7. lift-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-1}{972}, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{1}{108} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\pi}}{s}\right) - \frac{\frac{1}{6}}{\pi \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-1}{972}, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{\frac{1}{108} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\pi}}{s}\right) - \frac{\frac{1}{6}}{\pi \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-1}{972}, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{\frac{1}{108}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\pi}}{s}\right) - \frac{\frac{1}{6}}{\pi \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   10. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-1}{972}, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{\frac{1}{108}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\pi}}{s}\right) - \frac{\frac{1}{6}}{\pi \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-1}{972}, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{\frac{1}{108}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\pi}}{s}\right) - \frac{\frac{1}{6}}{\pi \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   12. lift-PI.f32 10.7

       \[\leadsto \mathsf{fma}\left(0.75, -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(-0.00102880658436214, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.009259259259259259}{s \cdot \pi}\right) - \frac{0.05555555555555555}{\pi}}{s}\right) - \frac{0.16666666666666666}{\pi \cdot r}}{s}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

10. Applied rewrites 10.7%

    \[\leadsto \mathsf{fma}\left(0.75, -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(-0.00102880658436214, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.009259259259259259}{s \cdot \pi}\right) - \frac{0.05555555555555555}{\pi}}{s}\right) - \frac{0.16666666666666666}{\pi \cdot r}}{s}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

11. Final simplification 10.7%

    \[\leadsto \mathsf{fma}\left(0.75, \frac{\frac{r \cdot \mathsf{fma}\left(-0.00102880658436214, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.009259259259259259}{s \cdot \pi}\right) - \frac{0.05555555555555555}{\pi}}{s} + \frac{0.16666666666666666}{\pi \cdot r}}{s}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]
12. Add Preprocessing

Alternative 5: 10.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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(Float32(Float32(Float32(Float32(r / s) / Float32(pi)) * Float32(0.006944444444444444)) - Float32(Float32(0.041666666666666664) / Float32(pi))) / s) + Float32(Float32(0.125) / Float32(Float32(pi) * r))) / s))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + (((((((r / s) / single(pi)) * single(0.006944444444444444)) - (single(0.041666666666666664) / single(pi))) / s) + (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

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-/.f32 N/A

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

5. Applied rewrites 10.8%

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

6. Final simplification 10.8%

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

Alternative 6: 10.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(fma(Float32(Float32(Float32(r / Float32(s * s)) * Float32(0.041666666666666664)) - Float32(Float32(0.25) / s)), r, Float32(0.75)) / Float32(Float32(Float32(Float32(pi) * s) * r) * Float32(6.0))))
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 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. pow2 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 10.8

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

8. Applied rewrites 10.8%

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

Alternative 7: 10.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.75, \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.75
  (/
   (fma
    (- (* (/ r (* s s)) 0.05555555555555555) (/ 0.3333333333333333 s))
    r
    1.0)
   (* (* (* PI 6.0) s) r))
  (* 0.125 (/ (exp (/ (- r) s)) (* r (* s PI))))))

C

float code(float s, float r) {
	return fmaf(0.75f, (fmaf((((r / (s * s)) * 0.05555555555555555f) - (0.3333333333333333f / s)), r, 1.0f) / (((((float) M_PI) * 6.0f) * s) * r)), (0.125f * (expf((-r / s)) / (r * (s * ((float) M_PI))))));
}

Julia

function code(s, r)
	return fma(Float32(0.75), Float32(fma(Float32(Float32(Float32(r / Float32(s * s)) * Float32(0.05555555555555555)) - Float32(Float32(0.3333333333333333) / s)), r, Float32(1.0)) / Float32(Float32(Float32(Float32(pi) * Float32(6.0)) * s) * r)), Float32(Float32(0.125) * Float32(exp(Float32(Float32(-r) / s)) / 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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. lower-*.f32 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. mul-1-neg N/A

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

   3. distribute-frac-neg N/A

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

   4. lower-/.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 99.6

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

10. Applied rewrites 99.6%

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

11. Taylor expanded in r around 0

    \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{\color{blue}{1 + r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right)}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) + \color{blue}{1}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) \cdot r + 1}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]

    3. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{\mathsf{fma}\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, \color{blue}{r}, 1\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]

    4. lower--.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{\mathsf{fma}\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]

    6. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]

    7. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]

    8. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]

    9. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]

    10. associate-*r/ N/A

        \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3} \cdot 1}{s}, r, 1\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3}}{s}, r, 1\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]

    12. lower-/.f32 10.8

        \[\leadsto \mathsf{fma}\left(0.75, \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]

13. Applied rewrites 10.8%

    \[\leadsto \mathsf{fma}\left(0.75, \frac{\color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \pi\right)}\right) \]
14. Add Preprocessing

Alternative 8: 9.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, \frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{-s}\right)}{s} + \frac{0.16666666666666666}{\pi}}{s} - \frac{\frac{0.25}{\pi}}{r}}{-s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/
    (+
     (/
      (fma
       (/ r PI)
       -0.06944444444444445
       (/ (* (/ (* r r) PI) -0.021604938271604937) (- s)))
      s)
     (/ 0.16666666666666666 PI))
    s)
   (/ (/ 0.25 PI) r))
  (- s)))

C

float code(float s, float r) {
	return ((((fmaf((r / ((float) M_PI)), -0.06944444444444445f, ((((r * r) / ((float) M_PI)) * -0.021604938271604937f) / -s)) / s) + (0.16666666666666666f / ((float) M_PI))) / s) - ((0.25f / ((float) M_PI)) / r)) / -s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(fma(Float32(r / Float32(pi)), Float32(-0.06944444444444445), Float32(Float32(Float32(Float32(r * r) / Float32(pi)) * Float32(-0.021604938271604937)) / Float32(-s))) / s) + Float32(Float32(0.16666666666666666) / Float32(pi))) / s) - Float32(Float32(Float32(0.25) / Float32(pi)) / r)) / Float32(-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

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

5. Applied rewrites 9.9%

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

   1. lift-/.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   2. lift-PI.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   3. lift-*.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   4. associate-/r* N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{\mathsf{PI}\left(\right)}}{r}}{s} \]

   5. metadata-eval N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4} \cdot 1}{\mathsf{PI}\left(\right)}}{r}}{s} \]

   6. associate-*r/ N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{s} \]

   7. lower-/.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{s} \]

   8. associate-*r/ N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4} \cdot 1}{\mathsf{PI}\left(\right)}}{r}}{s} \]

   9. metadata-eval N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{\mathsf{PI}\left(\right)}}{r}}{s} \]

   10. lower-/.f32 N/A

       \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{\mathsf{PI}\left(\right)}}{r}}{s} \]

   11. lift-PI.f32 9.9

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

7. Applied rewrites 9.9%

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

8. Final simplification 9.9%

   \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, \frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{-s}\right)}{s} + \frac{0.16666666666666666}{\pi}}{s} - \frac{\frac{0.25}{\pi}}{r}}{-s} \]
9. Add Preprocessing

Alternative 9: 9.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{r \cdot \mathsf{fma}\left(-0.021604938271604937, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.06944444444444445}{s \cdot \pi}\right) - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/
    (-
     (*
      r
      (fma
       -0.021604938271604937
       (/ r (* (* s s) PI))
       (/ 0.06944444444444445 (* s PI))))
     (/ 0.16666666666666666 PI))
    s)
   (/ 0.25 (* PI r)))
  s))

C

float code(float s, float r) {
	return ((((r * fmaf(-0.021604938271604937f, (r / ((s * s) * ((float) M_PI))), (0.06944444444444445f / (s * ((float) M_PI))))) - (0.16666666666666666f / ((float) M_PI))) / s) + (0.25f / (((float) M_PI) * r))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(r * fma(Float32(-0.021604938271604937), Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(Float32(0.06944444444444445) / Float32(s * Float32(pi))))) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s) + Float32(Float32(0.25) / Float32(Float32(pi) * r))) / s)
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 9.9%

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

6. Taylor expanded in r around 0

   \[\leadsto -\frac{\left(-\frac{r \cdot \left(\frac{-7}{324} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{5}{72} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto -\frac{\left(-\frac{r \cdot \left(\frac{-7}{324} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{5}{72} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   2. lower-fma.f32 N/A

      \[\leadsto -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-7}{324}, \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}, \frac{5}{72} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   3. lower-/.f32 N/A

      \[\leadsto -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-7}{324}, \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}, \frac{5}{72} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   4. lower-*.f32 N/A

      \[\leadsto -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-7}{324}, \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}, \frac{5}{72} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   5. unpow2 N/A

      \[\leadsto -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-7}{324}, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{5}{72} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   6. lower-*.f32 N/A

      \[\leadsto -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-7}{324}, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{5}{72} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   7. lift-PI.f32 N/A

      \[\leadsto -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-7}{324}, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{5}{72} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   8. associate-*r/ N/A

      \[\leadsto -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-7}{324}, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{\frac{5}{72} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   9. metadata-eval N/A

      \[\leadsto -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-7}{324}, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{\frac{5}{72}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   10. lower-/.f32 N/A

       \[\leadsto -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-7}{324}, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{\frac{5}{72}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   11. lower-*.f32 N/A

       \[\leadsto -\frac{\left(-\frac{r \cdot \mathsf{fma}\left(\frac{-7}{324}, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{\frac{5}{72}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   12. lift-PI.f32 9.9

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

8. Applied rewrites 9.9%

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

9. Final simplification 9.9%

   \[\leadsto \frac{\frac{r \cdot \mathsf{fma}\left(-0.021604938271604937, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.06944444444444445}{s \cdot \pi}\right) - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s} \]
10. Add Preprocessing

Alternative 10: 10.2% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 9.9%

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

6. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

8. Applied rewrites 10.2%

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

9. Final simplification 10.2%

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

Alternative 11: 10.2% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 9.9%

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

6. Taylor expanded in r around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 10.2

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

8. Applied rewrites 10.2%

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

9. Final simplification 10.2%

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

Alternative 12: 10.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 9.9%

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

6. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

8. Applied rewrites 10.2%

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

Alternative 13: 9.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 9.9%

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

6. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 9.1

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

8. Applied rewrites 9.1%

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

9. Final simplification 9.1%

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

Alternative 14: 9.1% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 9.1%

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

Alternative 15: 9.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 9.1%

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

Alternative 16: 9.1% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 9.9%

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

6. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 9.1

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

8. Applied rewrites 9.1%

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

9. Final simplification 9.1%

   \[\leadsto \frac{\frac{-0.25}{r \cdot \pi}}{-s} \]
10. Add Preprocessing

Alternative 17: 9.1% accurate, 10.6× speedup

Math

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

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(Float32(0.25) / Float32(r * 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 s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 9.1

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

7. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 9.1

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

9. Applied rewrites 9.1%

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

Alternative 18: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

Alternative 19: 9.1% accurate, 13.5× speedup

Math

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

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(Float32(r * 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 s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 9.1

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

7. Applied rewrites 9.1%

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

Reproduce

herbie shell --seed 2025086 
(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))))