Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.7s

Alternatives: 17

Speedup: N/A×

Specification

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

Math

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

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

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

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

\[0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(18.84955596923828 \cdot s\right) \cdot r} \]

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (* 0.125 (/ (exp (/ (- r) s)) (* PI (* s r))))
 (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* 18.84955596923828 s) r))))

C

float code(float s, float r) {
	return (0.125f * (expf((-r / s)) / (((float) M_PI) * (s * r)))) + ((0.75f * expf((-r / (3.0f * s)))) / ((18.84955596923828f * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(pi) * Float32(s * r)))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(18.84955596923828) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp((-r / s)) / (single(pi) * (s * r)))) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / ((single(18.84955596923828) * s) * r));
end

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 99.6%

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

3. Applied rewrites 99.6%

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

4. Evaluated real constant 99.6%

   \[\leadsto 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\color{blue}{18.84955596923828} \cdot s\right) \cdot r} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.1× speedup

Math

\[\frac{\frac{e^{\frac{-r}{s}}}{\pi} \cdot 0.125 + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125}{s \cdot r} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-r}{s}}}{\pi} \cdot 0.125 + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125}{s \cdot r}} \]
4. Add Preprocessing

Alternative 3: 99.5% accurate, 1.1× speedup

Math

\[\frac{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125}{s \cdot r} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-r}{s}}}{\pi} \cdot 0.125 + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125}{s \cdot r}} \]
4. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. distribute-rgt-out N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125}}{s \cdot r} \]
6. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

6. Applied rewrites 99.5%

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

Alternative 5: 11.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(s * r) * Float32(pi))) * Float32(0.125)) + Float32(Float32(Float32(0.125) / Float32(r * Float32(s * Float32(pi)))) - Float32(Float32(Float32(Float32(0.041666666666666664) / Float32(pi)) - Float32(Float32(r / Float32(Float32(pi) * s)) * Float32(0.006944444444444444))) / Float32(s * s))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) / ((s * r) * single(pi))) * single(0.125)) + ((single(0.125) / (r * (s * single(pi)))) - (((single(0.041666666666666664) / single(pi)) - ((r / (single(pi) * s)) * single(0.006944444444444444))) / (s * s)));
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{144} \cdot \frac{r}{s \cdot \pi} - \frac{1}{24} \cdot \frac{1}{\pi}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \pi}}{s}} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

4. Applied rewrites 11.1%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.006944444444444444 \cdot \frac{r}{s \cdot \pi} - 0.041666666666666664 \cdot \frac{1}{\pi}}{s} - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Applied rewrites 11.1%

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

7. Taylor expanded in s around 0

   \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot \pi} \cdot 0.125 + \left(\frac{\frac{1}{8}}{r \cdot \left(s \cdot \pi\right)} - \frac{\color{blue}{\frac{0.041666666666666664}{\pi} - \frac{r}{\pi \cdot s} \cdot 0.006944444444444444}}{s \cdot s}\right) \]
8. Step-by-step derivation

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 11.1%

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

9. Applied rewrites 11.1%

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

Alternative 6: 11.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \left(s \cdot r\right) \cdot \pi\\ \frac{e^{\frac{-r}{s}}}{t\_0} \cdot 0.125 + \left(\frac{0.125}{t\_0} - \frac{0.013262911699712276 - \frac{r}{\pi \cdot s} \cdot 0.006944444444444444}{s \cdot s}\right) \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (let* ((t_0 (* (* s r) PI)))
  (+
   (* (/ (exp (/ (- r) s)) t_0) 0.125)
   (-
    (/ 0.125 t_0)
    (/
     (- 0.013262911699712276 (* (/ r (* PI s)) 0.006944444444444444))
     (* s s))))))

C

float code(float s, float r) {
	float t_0 = (s * r) * ((float) M_PI);
	return ((expf((-r / s)) / t_0) * 0.125f) + ((0.125f / t_0) - ((0.013262911699712276f - ((r / (((float) M_PI) * s)) * 0.006944444444444444f)) / (s * s)));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(s * r) * Float32(pi))
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) / t_0) * Float32(0.125)) + Float32(Float32(Float32(0.125) / t_0) - Float32(Float32(Float32(0.013262911699712276) - Float32(Float32(r / Float32(Float32(pi) * s)) * Float32(0.006944444444444444))) / Float32(s * s))))
end

MATLAB

function tmp = code(s, r)
	t_0 = (s * r) * single(pi);
	tmp = ((exp((-r / s)) / t_0) * single(0.125)) + ((single(0.125) / t_0) - ((single(0.013262911699712276) - ((r / (single(pi) * s)) * single(0.006944444444444444))) / (s * s)));
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{144} \cdot \frac{r}{s \cdot \pi} - \frac{1}{24} \cdot \frac{1}{\pi}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \pi}}{s}} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

4. Applied rewrites 11.1%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.006944444444444444 \cdot \frac{r}{s \cdot \pi} - 0.041666666666666664 \cdot \frac{1}{\pi}}{s} - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Applied rewrites 11.1%

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

7. Evaluated real constant 11.1%

   \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot \pi} \cdot 0.125 + \left(\frac{0.125}{\left(s \cdot r\right) \cdot \pi} - \frac{0.013262911699712276 - \frac{r}{\pi \cdot s} \cdot 0.006944444444444444}{s \cdot s}\right) \]
8. Add Preprocessing

Alternative 7: 10.5% accurate, 1.6× speedup

Math

\[\frac{r \cdot \left(0.06944444444444445 \cdot \frac{r}{{s}^{2} \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}\right) + 0.25 \cdot \frac{1}{\pi}}{s \cdot r} \]

FPCore

(FPCore (s r)
  :precision binary32
  (/
 (+
  (*
   r
   (-
    (* 0.06944444444444445 (/ r (* (pow s 2.0) PI)))
    (* 0.16666666666666666 (/ 1.0 (* s PI)))))
  (* 0.25 (/ 1.0 PI)))
 (* s r)))

C

float code(float s, float r) {
	return ((r * ((0.06944444444444445f * (r / (powf(s, 2.0f) * ((float) M_PI)))) - (0.16666666666666666f * (1.0f / (s * ((float) M_PI)))))) + (0.25f * (1.0f / ((float) M_PI)))) / (s * r);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(r * Float32(Float32(Float32(0.06944444444444445) * Float32(r / Float32((s ^ Float32(2.0)) * Float32(pi)))) - Float32(Float32(0.16666666666666666) * Float32(Float32(1.0) / Float32(s * Float32(pi)))))) + Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(pi)))) / Float32(s * r))
end

MATLAB

function tmp = code(s, r)
	tmp = ((r * ((single(0.06944444444444445) * (r / ((s ^ single(2.0)) * single(pi)))) - (single(0.16666666666666666) * (single(1.0) / (s * single(pi)))))) + (single(0.25) * (single(1.0) / single(pi)))) / (s * r);
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-r}{s}}}{\pi} \cdot 0.125 + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125}{s \cdot r}} \]

4. Taylor expanded in r around 0

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

   1. lower-+.f32 N/A

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

6. Applied rewrites 10.5%

   \[\leadsto \frac{\color{blue}{r \cdot \left(0.06944444444444445 \cdot \frac{r}{{s}^{2} \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}\right) + 0.25 \cdot \frac{1}{\pi}}}{s \cdot r} \]
7. Add Preprocessing

Alternative 8: 10.5% accurate, 2.3× speedup

Math

\[-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

FPCore

(FPCore (s r)
  :precision binary32
  (*
 -1.0
 (/
  (-
   (*
    -1.0
    (/
     (-
      (*
       -1.0
       (/
        (+ (* -0.0625 (/ r PI)) (* -0.006944444444444444 (/ r PI)))
        s))
      (* 0.16666666666666666 (/ 1.0 PI)))
     s))
   (* 0.25 (/ 1.0 (* r PI))))
  s)))

C

float code(float s, float r) {
	return -1.0f * (((-1.0f * (((-1.0f * (((-0.0625f * (r / ((float) M_PI))) + (-0.006944444444444444f * (r / ((float) M_PI)))) / s)) - (0.16666666666666666f * (1.0f / ((float) M_PI)))) / s)) - (0.25f * (1.0f / (r * ((float) M_PI))))) / s);
}

Julia

function code(s, r)
	return Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(-0.0625) * Float32(r / Float32(pi))) + Float32(Float32(-0.006944444444444444) * Float32(r / Float32(pi)))) / s)) - Float32(Float32(0.16666666666666666) * Float32(Float32(1.0) / Float32(pi)))) / s)) - Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(r * Float32(pi))))) / s))
end

MATLAB

function tmp = code(s, r)
	tmp = single(-1.0) * (((single(-1.0) * (((single(-1.0) * (((single(-0.0625) * (r / single(pi))) + (single(-0.006944444444444444) * (r / single(pi)))) / s)) - (single(0.16666666666666666) * (single(1.0) / single(pi)))) / s)) - (single(0.25) * (single(1.0) / (r * single(pi))))) / s);
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\pi} + \frac{-1}{144} \cdot \frac{r}{\pi}}{s} - \frac{1}{6} \cdot \frac{1}{\pi}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \pi}}{s}} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

4. Applied rewrites 10.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
5. Add Preprocessing

Alternative 9: 10.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{r}{s \cdot \pi}\\ -1 \cdot \frac{-1 \cdot \frac{\left(0.006944444444444444 \cdot t\_0 + 0.0625 \cdot t\_0\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (let* ((t_0 (/ r (* s PI))))
  (*
   -1.0
   (/
    (-
     (*
      -1.0
      (/
       (-
        (+ (* 0.006944444444444444 t_0) (* 0.0625 t_0))
        (* 0.16666666666666666 (/ 1.0 PI)))
       s))
     (* 0.25 (/ 1.0 (* r PI))))
    s))))

C

float code(float s, float r) {
	float t_0 = r / (s * ((float) M_PI));
	return -1.0f * (((-1.0f * ((((0.006944444444444444f * t_0) + (0.0625f * t_0)) - (0.16666666666666666f * (1.0f / ((float) M_PI)))) / s)) - (0.25f * (1.0f / (r * ((float) M_PI))))) / s);
}

Julia

function code(s, r)
	t_0 = Float32(r / Float32(s * Float32(pi)))
	return Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(0.006944444444444444) * t_0) + Float32(Float32(0.0625) * t_0)) - Float32(Float32(0.16666666666666666) * Float32(Float32(1.0) / Float32(pi)))) / s)) - Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(r * Float32(pi))))) / s))
end

MATLAB

function tmp = code(s, r)
	t_0 = r / (s * single(pi));
	tmp = single(-1.0) * (((single(-1.0) * ((((single(0.006944444444444444) * t_0) + (single(0.0625) * t_0)) - (single(0.16666666666666666) * (single(1.0) / single(pi)))) / s)) - (single(0.25) * (single(1.0) / (r * single(pi))))) / s);
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]

5. Taylor expanded in s around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

7. Applied rewrites 10.5%

   \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\left(0.006944444444444444 \cdot \frac{r}{s \cdot \pi} + 0.0625 \cdot \frac{r}{s \cdot \pi}\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
8. Add Preprocessing

Alternative 10: 9.4% accurate, 3.3× speedup

Math

\[\frac{-1 \cdot \frac{0.013262911890762306 + 0.125 \cdot \frac{1}{\pi}}{s} + \left(0.03978873567228692 \cdot \frac{1}{r} + 0.125 \cdot \frac{1}{r \cdot \pi}\right)}{s} \]

FPCore

(FPCore (s r)
  :precision binary32
  (/
 (+
  (* -1.0 (/ (+ 0.013262911890762306 (* 0.125 (/ 1.0 PI))) s))
  (+ (* 0.03978873567228692 (/ 1.0 r)) (* 0.125 (/ 1.0 (* r PI)))))
 s))

C

float code(float s, float r) {
	return ((-1.0f * ((0.013262911890762306f + (0.125f * (1.0f / ((float) M_PI)))) / s)) + ((0.03978873567228692f * (1.0f / r)) + (0.125f * (1.0f / (r * ((float) M_PI)))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(-1.0) * Float32(Float32(Float32(0.013262911890762306) + Float32(Float32(0.125) * Float32(Float32(1.0) / Float32(pi)))) / s)) + Float32(Float32(Float32(0.03978873567228692) * Float32(Float32(1.0) / r)) + Float32(Float32(0.125) * Float32(Float32(1.0) / Float32(r * Float32(pi)))))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(-1.0) * ((single(0.013262911890762306) + (single(0.125) * (single(1.0) / single(pi)))) / s)) + ((single(0.03978873567228692) * (single(1.0) / r)) + (single(0.125) * (single(1.0) / (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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 99.6%

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

3. Applied rewrites 99.6%

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

4. Evaluated real constant 99.6%

   \[\leadsto 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\pi \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\color{blue}{18.84955596923828} \cdot s\right) \cdot r} \]

5. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{\frac{32768}{2470649} + \frac{1}{8} \cdot \frac{1}{\pi}}{s} + \left(\frac{98304}{2470649} \cdot \frac{1}{r} + \frac{1}{8} \cdot \frac{1}{r \cdot \pi}\right)}{s}} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{-1 \cdot \frac{\frac{32768}{2470649} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{98304}{2470649} \cdot \frac{1}{r} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{\color{blue}{s}} \]

7. Applied rewrites 9.4%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{0.013262911890762306 + 0.125 \cdot \frac{1}{\pi}}{s} + \left(0.03978873567228692 \cdot \frac{1}{r} + 0.125 \cdot \frac{1}{r \cdot \pi}\right)}{s}} \]
8. Add Preprocessing

Alternative 11: 9.4% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-r}{s}}}{\pi} \cdot 0.125 + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125}{s \cdot r}} \]

4. Taylor expanded in r around 0

   \[\leadsto \frac{\color{blue}{\frac{-1}{6} \cdot \frac{r}{s \cdot \pi} + \frac{1}{4} \cdot \frac{1}{\pi}}}{s \cdot r} \]
5. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-PI.f32 9.4%

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

6. Applied rewrites 9.4%

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

Alternative 12: 9.4% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \pi} - \frac{1}{6} \cdot \frac{1}{s \cdot \pi}}{s}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.4%

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

4. Applied rewrites 9.4%

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

Alternative 13: 9.3% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]

5. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{\frac{1}{8}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 9.5%

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

7. Applied rewrites 9.5%

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

8. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{\frac{1}{4}}{r \cdot \pi}}{s} \]
9. 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. lower-PI.f32 9.3%

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

10. Applied rewrites 9.3%

    \[\leadsto \frac{\frac{0.25}{r \cdot \pi}}{s} \]
11. Step-by-step derivation

    1. lift-/.f32 N/A

       \[\leadsto \frac{\frac{\frac{1}{4}}{r \cdot \pi}}{s} \]

    2. lift-*.f32 N/A

       \[\leadsto \frac{\frac{\frac{1}{4}}{r \cdot \pi}}{s} \]

    3. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

    4. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{4}}{\pi}}{r}}{s} \]

    5. lower-/.f32 N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{4}}{\pi}}{r}}{s} \]

    6. lower-/.f32 9.3%

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

12. Applied rewrites 9.3%

    \[\leadsto \frac{\frac{\frac{0.25}{\pi}}{r}}{s} \]
13. Add Preprocessing

Alternative 14: 9.3% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]

5. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{\frac{1}{8}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 9.5%

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

7. Applied rewrites 9.5%

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

8. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{\frac{1}{4}}{r \cdot \pi}}{s} \]
9. 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. lower-PI.f32 9.3%

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

10. Applied rewrites 9.3%

    \[\leadsto \frac{\frac{0.25}{r \cdot \pi}}{s} \]
11. Step-by-step derivation

    1. lift-/.f32 N/A

       \[\leadsto \frac{\frac{\frac{1}{4}}{r \cdot \pi}}{\color{blue}{s}} \]

    2. mult-flip N/A

       \[\leadsto \frac{\frac{1}{4}}{r \cdot \pi} \cdot \color{blue}{\frac{1}{s}} \]

    3. *-commutative N/A

       \[\leadsto \frac{1}{s} \cdot \color{blue}{\frac{\frac{1}{4}}{r \cdot \pi}} \]

    4. lower-*.f32 N/A

       \[\leadsto \frac{1}{s} \cdot \color{blue}{\frac{\frac{1}{4}}{r \cdot \pi}} \]

    5. lower-/.f32 9.3%

       \[\leadsto \frac{1}{s} \cdot \frac{\color{blue}{0.25}}{r \cdot \pi} \]

    6. lift-*.f32 N/A

       \[\leadsto \frac{1}{s} \cdot \frac{\frac{1}{4}}{r \cdot \pi} \]

    7. *-commutative N/A

       \[\leadsto \frac{1}{s} \cdot \frac{\frac{1}{4}}{\pi \cdot r} \]

    8. lower-*.f32 9.3%

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

12. Applied rewrites 9.3%

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

Alternative 15: 9.3% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.3%

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

4. Applied rewrites 9.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 9.3%

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

6. Applied rewrites 9.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi}}{\color{blue}{s \cdot r}} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi}}{\color{blue}{s \cdot r}} \]

   6. lower-/.f32 9.3%

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

8. Applied rewrites 9.3%

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

Alternative 16: 9.3% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.3%

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

4. Applied rewrites 9.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 9.3%

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

6. Applied rewrites 9.3%

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

Alternative 17: 9.3% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.3%

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

4. Applied rewrites 9.3%

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

Reproduce

herbie shell --seed 2025258 
(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))))