Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 4.7s

Alternatives: 11

Speedup: N/A×

Specification

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

Math

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

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: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (((expf((-r / s)) + expf((-0.3333333333333333f * (r / 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(Float32(-0.3333333333333333) * Float32(r / s)))) / Float32(pi)) * Float32(0.125)) / Float32(s * r))
end

MATLAB

function tmp = code(s, r)
	tmp = (((exp((-r / s)) + exp((single(-0.3333333333333333) * (r / 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%

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

   1. lift-fma.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

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. 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%

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

Alternative 2: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

   1. lift-fma.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

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. 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%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lower-/.f32 N/A

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

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{\frac{r}{s} \cdot -0.3333333333333333} + e^{\frac{-r}{s}}\right)}{\left(s \cdot r\right) \cdot \pi}} \]
8. Add Preprocessing

Alternative 3: 97.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lift-fma.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

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. 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%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lower-/.f32 N/A

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

7. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lower-/.f32 97.4%

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 97.4%

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

9. Applied rewrites 97.4%

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

Alternative 4: 45.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;r \leq 25:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi \cdot \left(s \cdot r\right)}, \frac{0.125}{r \cdot \mathsf{fma}\left(r, \pi, s \cdot \pi\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left(e^{\pi \cdot r}\right) \cdot s}\\ \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 25.0)
   (fma
    0.125
    (/ (exp (/ r (* -3.0 s))) (* PI (* s r)))
    (/ 0.125 (* r (fma r PI (* s PI)))))
   (/ 0.25 (* (log (exp (* PI r))) s))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 25.0f) {
		tmp = fmaf(0.125f, (expf((r / (-3.0f * s))) / (((float) M_PI) * (s * r))), (0.125f / (r * fmaf(r, ((float) M_PI), (s * ((float) M_PI))))));
	} else {
		tmp = 0.25f / (logf(expf((((float) M_PI) * r))) * s);
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(25.0))
		tmp = fma(Float32(0.125), Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(Float32(pi) * Float32(s * r))), Float32(Float32(0.125) / Float32(r * fma(r, Float32(pi), Float32(s * Float32(pi))))));
	else
		tmp = Float32(Float32(0.25) / Float32(log(exp(Float32(Float32(pi) * r))) * s));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 25

   1. Initial program 99.6%

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

   3. Applied rewrites 99.5%

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

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-*.f32 99.5%

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f32 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in r around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-PI.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 12.3%

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

   8. Applied rewrites 12.3%

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

   if 25 < r

   1. Initial program 99.6%

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

   2. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
   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 8.9%

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

   4. Applied rewrites 8.9%

      \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
   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 8.9%

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

   6. Applied rewrites 8.9%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 8.9%

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

   8. Applied rewrites 8.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f32 N/A

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

      4. add-log-exp N/A

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

      5. log-pow-rev N/A

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

      6. lower-log.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. pow-exp N/A

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

      9. lift-*.f32 N/A

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

      10. lower-exp.f32 43.0%

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

   10. Applied rewrites 43.0%

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

Alternative 5: 43.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

4. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

6. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 8.9%

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

8. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-exp N/A

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

   9. lift-*.f32 N/A

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

   10. lower-exp.f32 43.0%

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

10. Applied rewrites 43.0%

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

Alternative 6: 8.9% accurate, 2.9× speedup

Math

\[\frac{\frac{2 + -1.3333333333333333 \cdot \frac{r}{s}}{\pi} \cdot 0.125}{s \cdot r} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (* (/ (+ 2.0 (* -1.3333333333333333 (/ r s))) PI) 0.125) (* s r)))

C

float code(float s, float r) {
	return (((2.0f + (-1.3333333333333333f * (r / s))) / ((float) M_PI)) * 0.125f) / (s * r);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(-1.3333333333333333) * Float32(r / s))) / Float32(pi)) * Float32(0.125)) / Float32(s * r))
end

MATLAB

function tmp = code(s, r)
	tmp = (((single(2.0) + (single(-1.3333333333333333) * (r / 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%

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

   1. lift-fma.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

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. 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%

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

6. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{\color{blue}{2 + \frac{-4}{3} \cdot \frac{r}{s}}}{\pi} \cdot 0.125}{s \cdot r} \]
7. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{\frac{2 + \color{blue}{\frac{-4}{3} \cdot \frac{r}{s}}}{\pi} \cdot \frac{1}{8}}{s \cdot r} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{2 + \frac{-4}{3} \cdot \color{blue}{\frac{r}{s}}}{\pi} \cdot \frac{1}{8}}{s \cdot r} \]

   3. lower-/.f32 8.9%

      \[\leadsto \frac{\frac{2 + -1.3333333333333333 \cdot \frac{r}{\color{blue}{s}}}{\pi} \cdot 0.125}{s \cdot r} \]

8. Applied rewrites 8.9%

   \[\leadsto \frac{\frac{\color{blue}{2 + -1.3333333333333333 \cdot \frac{r}{s}}}{\pi} \cdot 0.125}{s \cdot r} \]
9. Add Preprocessing

Alternative 7: 8.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

   1. lift-fma.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

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. 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%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lower-/.f32 N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in r around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 8.9%

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

10. Applied rewrites 8.9%

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

Alternative 8: 8.9% accurate, 6.0× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

4. Applied rewrites 8.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 8.9%

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 8.9%

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

6. Applied rewrites 8.9%

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

Alternative 9: 8.9% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

4. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

6. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 8.9%

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

8. Applied rewrites 8.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 8.9%

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

10. Applied rewrites 8.9%

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

Alternative 10: 8.9% accurate, 6.5× speedup

Math

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

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(0.25) / Float32(Float32(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 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 8.9%

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

4. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

6. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 8.9%

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

8. Applied rewrites 8.9%

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

Alternative 11: 8.9% accurate, 6.5× 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 8.9%

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

4. Applied rewrites 8.9%

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

Reproduce

herbie shell --seed 2025209 
(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))))