Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.3%

Time: 16.7s

Alternatives: 14

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(e^{-\log \left(\left(r \cdot \left(\pi \cdot 6\right)\right) \cdot e^{0 - \frac{r}{s \cdot -3}}\right)}, 0.75, \frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\frac{r}{s}}}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   (exp (- (log (* (* r (* PI 6.0)) (exp (- 0.0 (/ r (* s -3.0))))))))
   0.75
   (/ 0.125 (* (* r PI) (exp (/ r s)))))
  s))

C

float code(float s, float r) {
	return fmaf(expf(-logf(((r * (((float) M_PI) * 6.0f)) * expf((0.0f - (r / (s * -3.0f))))))), 0.75f, (0.125f / ((r * ((float) M_PI)) * expf((r / s))))) / s;
}

Julia

function code(s, r)
	return Float32(fma(exp(Float32(-log(Float32(Float32(r * Float32(Float32(pi) * Float32(6.0))) * exp(Float32(Float32(0.0) - Float32(r / Float32(s * Float32(-3.0))))))))), Float32(0.75), Float32(Float32(0.125) / Float32(Float32(r * Float32(pi)) * exp(Float32(r / s))))) / s)
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around 0

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

   1. /-lowering-/.f32 N/A

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

5. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{0 - \frac{r}{s}}}{r \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \pi}\right)}{s}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \color{blue}{\left(\frac{e^{\frac{r}{s} \cdot \frac{-1}{3}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}\right)}}{s} \]

   2. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{r}{s} \cdot \frac{-1}{3}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}}{s} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{s} \cdot \color{blue}{\frac{1}{-3}}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   4. div-inv N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{\frac{\frac{r}{s}}{-3}}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{\frac{r}{s \cdot -3}}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{\frac{3}{4}}{6}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   7. times-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{r}{s \cdot -3}} \cdot \frac{3}{4}}{\left(r \cdot \mathsf{PI}\left(\right)\right) \cdot 6}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{3}{4} \cdot e^{\frac{r}{s \cdot -3}}}}{\left(r \cdot \mathsf{PI}\left(\right)\right) \cdot 6} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   9. associate-*r* N/A

      \[\leadsto \frac{\frac{\frac{3}{4} \cdot e^{\frac{r}{s \cdot -3}}}{\color{blue}{r \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   10. associate-/l* N/A

       \[\leadsto \frac{\color{blue}{\frac{3}{4} \cdot \frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   11. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)} \cdot \frac{3}{4}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

7. Applied egg-rr 99.5%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. exp-lowering-exp.f32 N/A

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

   5. *-lowering-*.f32 N/A

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

9. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{e^{\log \left(\left(r \cdot \left(\pi \cdot 6\right)\right) \cdot e^{-\frac{r}{s \cdot -3}}\right) \cdot -1}}, 0.75, \frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\frac{r}{s}}}\right)}{s} \]

10. Final simplification 99.5%

    \[\leadsto \frac{\mathsf{fma}\left(e^{-\log \left(\left(r \cdot \left(\pi \cdot 6\right)\right) \cdot e^{0 - \frac{r}{s \cdot -3}}\right)}, 0.75, \frac{0.125}{\left(r \cdot \pi\right) \cdot e^{\frac{r}{s}}}\right)}{s} \]
11. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around 0

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

   1. /-lowering-/.f32 N/A

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

5. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{0 - \frac{r}{s}}}{r \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \pi}\right)}{s}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \color{blue}{\left(\frac{e^{\frac{r}{s} \cdot \frac{-1}{3}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}\right)}}{s} \]

   2. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{r}{s} \cdot \frac{-1}{3}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}}{s} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{s} \cdot \color{blue}{\frac{1}{-3}}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   4. div-inv N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{\frac{\frac{r}{s}}{-3}}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{\frac{r}{s \cdot -3}}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{\frac{3}{4}}{6}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   7. times-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{r}{s \cdot -3}} \cdot \frac{3}{4}}{\left(r \cdot \mathsf{PI}\left(\right)\right) \cdot 6}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{3}{4} \cdot e^{\frac{r}{s \cdot -3}}}}{\left(r \cdot \mathsf{PI}\left(\right)\right) \cdot 6} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   9. associate-*r* N/A

      \[\leadsto \frac{\frac{\frac{3}{4} \cdot e^{\frac{r}{s \cdot -3}}}{\color{blue}{r \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   10. associate-/l* N/A

       \[\leadsto \frac{\color{blue}{\frac{3}{4} \cdot \frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   11. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)} \cdot \frac{3}{4}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

7. Applied egg-rr 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(e^{0 - \frac{r}{s}} \cdot \frac{0.125}{s \cdot \pi}, s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right), r \cdot \left(0.75 \cdot e^{\frac{r}{s \cdot -3}}\right)\right)}{r}}{s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)}} \]

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{\frac{r}{s} \cdot -0.3333333333333333} + e^{0 - \frac{r}{s}}\right)}{r \cdot \left(s \cdot \pi\right)}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(e^{\frac{r}{s} \cdot \frac{-1}{3}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(e^{\frac{r}{s} \cdot \frac{-1}{3}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\left(e^{\frac{r}{s} \cdot \color{blue}{\frac{1}{-3}}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   4. div-inv N/A

      \[\leadsto \frac{\left(e^{\color{blue}{\frac{\frac{r}{s}}{-3}}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\left(e^{\color{blue}{\frac{r}{s \cdot -3}}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(e^{\frac{r}{s \cdot -3}} + e^{0 - \frac{r}{s}}\right)} \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\left(e^{\color{blue}{\frac{\frac{r}{s}}{-3}}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   8. div-inv N/A

      \[\leadsto \frac{\left(e^{\color{blue}{\frac{r}{s} \cdot \frac{1}{-3}}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\left(e^{\frac{r}{s} \cdot \color{blue}{\frac{-1}{3}}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   10. exp-lowering-exp.f32 N/A

       \[\leadsto \frac{\left(\color{blue}{e^{\frac{r}{s} \cdot \frac{-1}{3}}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\left(e^{\frac{r}{s} \cdot \color{blue}{\frac{1}{-3}}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   12. div-inv N/A

       \[\leadsto \frac{\left(e^{\color{blue}{\frac{\frac{r}{s}}{-3}}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   13. associate-/r* N/A

       \[\leadsto \frac{\left(e^{\color{blue}{\frac{r}{s \cdot -3}}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \frac{\left(e^{\color{blue}{\frac{r}{s \cdot -3}}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \frac{\left(e^{\frac{r}{\color{blue}{s \cdot -3}}} + e^{0 - \frac{r}{s}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   16. exp-lowering-exp.f32 N/A

       \[\leadsto \frac{\left(e^{\frac{r}{s \cdot -3}} + \color{blue}{e^{0 - \frac{r}{s}}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   17. --lowering--.f32 N/A

       \[\leadsto \frac{\left(e^{\frac{r}{s \cdot -3}} + e^{\color{blue}{0 - \frac{r}{s}}}\right) \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   18. /-lowering-/.f32 99.4

       \[\leadsto \frac{\left(e^{\frac{r}{s \cdot -3}} + e^{0 - \color{blue}{\frac{r}{s}}}\right) \cdot 0.125}{r \cdot \left(s \cdot \pi\right)} \]

9. Applied egg-rr 99.4%

   \[\leadsto \frac{\color{blue}{\left(e^{\frac{r}{s \cdot -3}} + e^{0 - \frac{r}{s}}\right) \cdot 0.125}}{r \cdot \left(s \cdot \pi\right)} \]

10. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(e^{0 - \frac{r}{s}} \cdot \frac{0.125}{s \cdot \pi}, s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right), r \cdot \left(0.75 \cdot e^{\frac{r}{s \cdot -3}}\right)\right)}{r}}{s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)}} \]

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{\frac{r}{s} \cdot -0.3333333333333333} + e^{0 - \frac{r}{s}}\right)}{r \cdot \left(s \cdot \pi\right)}} \]

8. Final simplification 99.4%

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

Alternative 5: 93.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around 0

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

   1. /-lowering-/.f32 N/A

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

5. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{0 - \frac{r}{s}}}{r \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \pi}\right)}{s}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \color{blue}{\left(\frac{e^{\frac{r}{s} \cdot \frac{-1}{3}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}\right)}}{s} \]

   2. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{r}{s} \cdot \frac{-1}{3}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}}{s} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{s} \cdot \color{blue}{\frac{1}{-3}}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   4. div-inv N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{\frac{\frac{r}{s}}{-3}}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{\frac{r}{s \cdot -3}}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{\frac{3}{4}}{6}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   7. times-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{r}{s \cdot -3}} \cdot \frac{3}{4}}{\left(r \cdot \mathsf{PI}\left(\right)\right) \cdot 6}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{3}{4} \cdot e^{\frac{r}{s \cdot -3}}}}{\left(r \cdot \mathsf{PI}\left(\right)\right) \cdot 6} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   9. associate-*r* N/A

      \[\leadsto \frac{\frac{\frac{3}{4} \cdot e^{\frac{r}{s \cdot -3}}}{\color{blue}{r \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   10. associate-/l* N/A

       \[\leadsto \frac{\color{blue}{\frac{3}{4} \cdot \frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

   11. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)} \cdot \frac{3}{4}} + \frac{e^{0 - \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8}}{s} \]

7. Applied egg-rr 99.5%

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

8. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. accelerator-lowering-fma.f32 N/A

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

   5. /-lowering-/.f32 N/A

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f32 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f32 N/A

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

   16. /-lowering-/.f32 74.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(\pi \cdot 6\right)}, 0.75, \frac{0.125}{\left(r \cdot \pi\right) \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(0.16666666666666666, \frac{r}{s \cdot \left(s \cdot s\right)}, \frac{0.5}{s \cdot s}\right), \color{blue}{\frac{1}{s}}\right), 1\right)}\right)}{s} \]

10. Simplified 74.4%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(\pi \cdot 6\right)}, 0.75, \frac{0.125}{\left(r \cdot \pi\right) \cdot \color{blue}{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(0.16666666666666666, \frac{r}{s \cdot \left(s \cdot s\right)}, \frac{0.5}{s \cdot s}\right), \frac{1}{s}\right), 1\right)}}\right)}{s} \]

11. Taylor expanded in r around inf

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

    1. *-lowering-*.f32 N/A

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

    2. /-lowering-/.f32 N/A

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

    3. exp-lowering-exp.f32 N/A

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

    4. associate-*r/ N/A

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

    5. /-lowering-/.f32 N/A

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

    6. *-commutative N/A

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

    7. *-lowering-*.f32 N/A

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

    8. *-lowering-*.f32 N/A

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

    9. *-commutative N/A

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

    10. *-lowering-*.f32 N/A

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

    11. PI-lowering-PI.f32 92.3

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

13. Simplified 92.3%

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

Alternative 6: 10.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{1.5 + \frac{\frac{0.4166666666666667 \cdot \left(r \cdot r\right) - \left(r \cdot \left(r \cdot r\right)\right) \cdot \frac{0.12962962962962962}{s}}{s} - r}{s}}{\left(r \cdot \left(\pi \cdot 6\right)\right) \cdot s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   1.5
   (/
    (-
     (/
      (-
       (* 0.4166666666666667 (* r r))
       (* (* r (* r r)) (/ 0.12962962962962962 s)))
      s)
     r)
    s))
  (* (* r (* PI 6.0)) s)))

C

float code(float s, float r) {
	return (1.5f + (((((0.4166666666666667f * (r * r)) - ((r * (r * r)) * (0.12962962962962962f / s))) / s) - r) / s)) / ((r * (((float) M_PI) * 6.0f)) * s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(1.5) + Float32(Float32(Float32(Float32(Float32(Float32(0.4166666666666667) * Float32(r * r)) - Float32(Float32(r * Float32(r * r)) * Float32(Float32(0.12962962962962962) / s))) / s) - r) / s)) / Float32(Float32(r * Float32(Float32(pi) * Float32(6.0))) * s))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(1.5) + (((((single(0.4166666666666667) * (r * r)) - ((r * (r * r)) * (single(0.12962962962962962) / s))) / s) - r) / s)) / ((r * (single(pi) * single(6.0))) * s);
end

Derivation

1. Initial program 99.5%

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

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(e^{0 - \frac{r}{s}} \cdot \frac{0.125}{s \cdot \pi}, s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right), r \cdot \left(0.75 \cdot e^{\frac{r}{s \cdot -3}}\right)\right)}{r}}{s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)}} \]

5. Taylor expanded in s around -inf

   \[\leadsto \frac{\color{blue}{\frac{3}{2} + -1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{216} \cdot {r}^{3} + \frac{1}{8} \cdot {r}^{3}}{s} + \left(\frac{1}{24} \cdot {r}^{2} + \frac{3}{8} \cdot {r}^{2}\right)}{s} + \left(\frac{1}{4} \cdot r + \frac{3}{4} \cdot r\right)}{s}}}{s \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot r\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\frac{3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{216} \cdot {r}^{3} + \frac{1}{8} \cdot {r}^{3}}{s} + \left(\frac{1}{24} \cdot {r}^{2} + \frac{3}{8} \cdot {r}^{2}\right)}{s} + \left(\frac{1}{4} \cdot r + \frac{3}{4} \cdot r\right)}{s}\right)\right)}}{s \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot r\right)} \]

   2. unsub-neg N/A

      \[\leadsto \frac{\color{blue}{\frac{3}{2} - \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{216} \cdot {r}^{3} + \frac{1}{8} \cdot {r}^{3}}{s} + \left(\frac{1}{24} \cdot {r}^{2} + \frac{3}{8} \cdot {r}^{2}\right)}{s} + \left(\frac{1}{4} \cdot r + \frac{3}{4} \cdot r\right)}{s}}}{s \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot r\right)} \]

   3. --lowering--.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{3}{2} - \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{216} \cdot {r}^{3} + \frac{1}{8} \cdot {r}^{3}}{s} + \left(\frac{1}{24} \cdot {r}^{2} + \frac{3}{8} \cdot {r}^{2}\right)}{s} + \left(\frac{1}{4} \cdot r + \frac{3}{4} \cdot r\right)}{s}}}{s \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot r\right)} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{3}{2} - \color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{216} \cdot {r}^{3} + \frac{1}{8} \cdot {r}^{3}}{s} + \left(\frac{1}{24} \cdot {r}^{2} + \frac{3}{8} \cdot {r}^{2}\right)}{s} + \left(\frac{1}{4} \cdot r + \frac{3}{4} \cdot r\right)}{s}}}{s \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot r\right)} \]

7. Simplified 10.3%

   \[\leadsto \frac{\color{blue}{1.5 - \frac{r - \frac{0.4166666666666667 \cdot \left(r \cdot r\right) - \left(r \cdot \left(r \cdot r\right)\right) \cdot \frac{0.12962962962962962}{s}}{s}}{s}}}{s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)} \]

8. Final simplification 10.3%

   \[\leadsto \frac{1.5 + \frac{\frac{0.4166666666666667 \cdot \left(r \cdot r\right) - \left(r \cdot \left(r \cdot r\right)\right) \cdot \frac{0.12962962962962962}{s}}{s} - r}{s}}{\left(r \cdot \left(\pi \cdot 6\right)\right) \cdot s} \]
9. Add Preprocessing

Alternative 7: 10.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{1.5 + \frac{\frac{0.4166666666666667 \cdot \left(r \cdot r\right)}{s} - r}{s}}{\left(r \cdot \left(\pi \cdot 6\right)\right) \cdot s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+ 1.5 (/ (- (/ (* 0.4166666666666667 (* r r)) s) r) s))
  (* (* r (* PI 6.0)) s)))

C

float code(float s, float r) {
	return (1.5f + ((((0.4166666666666667f * (r * r)) / s) - r) / s)) / ((r * (((float) M_PI) * 6.0f)) * s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(1.5) + Float32(Float32(Float32(Float32(Float32(0.4166666666666667) * Float32(r * r)) / s) - r) / s)) / Float32(Float32(r * Float32(Float32(pi) * Float32(6.0))) * s))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(1.5) + ((((single(0.4166666666666667) * (r * r)) / s) - r) / s)) / ((r * (single(pi) * single(6.0))) * s);
end

Derivation

1. Initial program 99.5%

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

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(e^{0 - \frac{r}{s}} \cdot \frac{0.125}{s \cdot \pi}, s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right), r \cdot \left(0.75 \cdot e^{\frac{r}{s \cdot -3}}\right)\right)}{r}}{s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)}} \]

5. Taylor expanded in s around -inf

   \[\leadsto \frac{\color{blue}{\frac{3}{2} + -1 \cdot \frac{-1 \cdot \frac{\frac{1}{24} \cdot {r}^{2} + \frac{3}{8} \cdot {r}^{2}}{s} + \left(\frac{1}{4} \cdot r + \frac{3}{4} \cdot r\right)}{s}}}{s \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot r\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\frac{3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{24} \cdot {r}^{2} + \frac{3}{8} \cdot {r}^{2}}{s} + \left(\frac{1}{4} \cdot r + \frac{3}{4} \cdot r\right)}{s}\right)\right)}}{s \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot r\right)} \]

   2. unsub-neg N/A

      \[\leadsto \frac{\color{blue}{\frac{3}{2} - \frac{-1 \cdot \frac{\frac{1}{24} \cdot {r}^{2} + \frac{3}{8} \cdot {r}^{2}}{s} + \left(\frac{1}{4} \cdot r + \frac{3}{4} \cdot r\right)}{s}}}{s \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot r\right)} \]

   3. --lowering--.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{3}{2} - \frac{-1 \cdot \frac{\frac{1}{24} \cdot {r}^{2} + \frac{3}{8} \cdot {r}^{2}}{s} + \left(\frac{1}{4} \cdot r + \frac{3}{4} \cdot r\right)}{s}}}{s \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot r\right)} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{3}{2} - \color{blue}{\frac{-1 \cdot \frac{\frac{1}{24} \cdot {r}^{2} + \frac{3}{8} \cdot {r}^{2}}{s} + \left(\frac{1}{4} \cdot r + \frac{3}{4} \cdot r\right)}{s}}}{s \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot r\right)} \]

7. Simplified 10.2%

   \[\leadsto \frac{\color{blue}{1.5 - \frac{r - \frac{0.4166666666666667 \cdot \left(r \cdot r\right)}{s}}{s}}}{s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)} \]

8. Final simplification 10.2%

   \[\leadsto \frac{1.5 + \frac{\frac{0.4166666666666667 \cdot \left(r \cdot r\right)}{s} - r}{s}}{\left(r \cdot \left(\pi \cdot 6\right)\right) \cdot s} \]
9. Add Preprocessing

Alternative 8: 10.2% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(e^{0 - \frac{r}{s}} \cdot \frac{0.125}{s \cdot \pi}, s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right), r \cdot \left(0.75 \cdot e^{\frac{r}{s \cdot -3}}\right)\right)}{r}}{s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)}} \]

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{\frac{r}{s} \cdot -0.3333333333333333} + e^{0 - \frac{r}{s}}\right)}{r \cdot \left(s \cdot \pi\right)}} \]

8. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. accelerator-lowering-fma.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. /-lowering-/.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 N/A

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. distribute-neg-frac N/A

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

   18. metadata-eval N/A

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

   19. /-lowering-/.f32 10.2

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

10. Simplified 10.2%

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

11. Final simplification 10.2%

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

Alternative 9: 9.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. /-lowering-/.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. PI-lowering-PI.f32 9.3

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

5. Simplified 9.3%

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

   1. add-sqr-sqrt N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. sqrt-lowering-sqrt.f32 N/A

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

   6. PI-lowering-PI.f32 N/A

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

   7. sqrt-lowering-sqrt.f32 N/A

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

   8. PI-lowering-PI.f32 9.3

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

7. Applied egg-rr 9.3%

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

8. Final simplification 9.3%

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

Alternative 10: 9.2% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. /-lowering-/.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. PI-lowering-PI.f32 9.3

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

5. Simplified 9.3%

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

   1. associate-/r* N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. /-lowering-/.f32 N/A

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

   6. PI-lowering-PI.f32 9.3

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

7. Applied egg-rr 9.3%

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

Alternative 11: 9.2% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. /-lowering-/.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. PI-lowering-PI.f32 9.3

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

5. Simplified 9.3%

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. PI-lowering-PI.f32 9.3

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

7. Applied egg-rr 9.3%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

   7. inv-pow N/A

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

   8. clear-num N/A

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

   9. *-lowering-*.f32 N/A

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

   10. /-lowering-/.f32 N/A

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

   11. PI-lowering-PI.f32 N/A

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

   12. /-lowering-/.f32 N/A

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

   13. *-lowering-*.f32 9.3

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

9. Applied egg-rr 9.3%

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

Alternative 12: 9.2% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{1.5}{\left(r \cdot \left(\pi \cdot 6\right)\right) \cdot s} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 1.5 (* (* r (* PI 6.0)) s)))

C

float code(float s, float r) {
	return 1.5f / ((r * (((float) M_PI) * 6.0f)) * s);
}

Julia

function code(s, r)
	return Float32(Float32(1.5) / Float32(Float32(r * Float32(Float32(pi) * Float32(6.0))) * s))
end

MATLAB

function tmp = code(s, r)
	tmp = single(1.5) / ((r * (single(pi) * single(6.0))) * s);
end

Derivation

1. Initial program 99.5%

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

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(e^{0 - \frac{r}{s}} \cdot \frac{0.125}{s \cdot \pi}, s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right), r \cdot \left(0.75 \cdot e^{\frac{r}{s \cdot -3}}\right)\right)}{r}}{s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)}} \]

5. Taylor expanded in r around 0

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

7. Simplified 9.3%

   \[\leadsto \frac{\color{blue}{1.5}}{s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)} \]

8. Final simplification 9.3%

   \[\leadsto \frac{1.5}{\left(r \cdot \left(\pi \cdot 6\right)\right) \cdot s} \]
9. Add Preprocessing

Alternative 13: 9.2% accurate, 13.5× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. /-lowering-/.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. PI-lowering-PI.f32 9.3

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

5. Simplified 9.3%

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. PI-lowering-PI.f32 9.3

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

7. Applied egg-rr 9.3%

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

8. Final simplification 9.3%

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

Alternative 14: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. /-lowering-/.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. PI-lowering-PI.f32 9.3

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

5. Simplified 9.3%

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

6. Final simplification 9.3%

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

Reproduce

herbie shell --seed 2024196 
(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))))