Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.0%

Time: 19.1s

Alternatives: 20

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{0.3333333333333333 \cdot \left(r \cdot \frac{-0.3333333333333333}{s}\right)} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (exp (/ r (- s))) r)
   (/
    (*
     (exp (* 0.3333333333333333 (* r (/ -0.3333333333333333 s))))
     (cbrt (pow (exp -0.6666666666666666) (/ r s))))
    r))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + ((expf((0.3333333333333333f * (r * (-0.3333333333333333f / s)))) * cbrtf(powf(expf(-0.6666666666666666f), (r / s)))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(exp(Float32(Float32(0.3333333333333333) * Float32(r * Float32(Float32(-0.3333333333333333) / s)))) * cbrt((exp(Float32(-0.6666666666666666)) ^ Float32(r / s)))) / r)))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

   1. add-cube-cbrt 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\left(\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}\right) \cdot \sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   2. associate-*l* 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \left(\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}\right)}}{r}\right) \]

   3. cbrt-unprod 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \color{blue}{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)} \cdot {\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   4. pow-prod-down 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{\color{blue}{{\left(e^{-0.3333333333333333} \cdot e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   5. prod-exp 99.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\color{blue}{\left(e^{-0.3333333333333333 + -0.3333333333333333}\right)}}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   6. metadata-eval 99.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]
7. Step-by-step derivation

   1. exp-prod 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   2. *-commutative 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   3. associate-*l/ 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   4. associate-/l* 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{e^{\color{blue}{r \cdot \frac{-0.3333333333333333}{s}}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

8. Simplified 99.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt[3]{e^{r \cdot \frac{-0.3333333333333333}{s}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]
9. Step-by-step derivation

   1. pow1/3 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{r \cdot \frac{-0.3333333333333333}{s}}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   2. pow-to-exp 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{\log \left(e^{r \cdot \frac{-0.3333333333333333}{s}}\right) \cdot 0.3333333333333333}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   3. add-log-exp 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\left(r \cdot \frac{-0.3333333333333333}{s}\right)} \cdot 0.3333333333333333} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   4. *-commutative 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\left(\frac{-0.3333333333333333}{s} \cdot r\right)} \cdot 0.3333333333333333} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

10. Applied egg-rr 99.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{\left(\frac{-0.3333333333333333}{s} \cdot r\right) \cdot 0.3333333333333333}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

11. Final simplification 99.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{0.3333333333333333 \cdot \left(r \cdot \frac{-0.3333333333333333}{s}\right)} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]
12. Add Preprocessing

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{0.3333333333333333 \cdot \left(r \cdot \frac{-0.3333333333333333}{s}\right)} \cdot \sqrt[3]{e^{\frac{r \cdot -0.6666666666666666}{s}}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (exp (/ r (- s))) r)
   (/
    (*
     (exp (* 0.3333333333333333 (* r (/ -0.3333333333333333 s))))
     (cbrt (exp (/ (* r -0.6666666666666666) s))))
    r))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + ((expf((0.3333333333333333f * (r * (-0.3333333333333333f / s)))) * cbrtf(expf(((r * -0.6666666666666666f) / s)))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(exp(Float32(Float32(0.3333333333333333) * Float32(r * Float32(Float32(-0.3333333333333333) / s)))) * cbrt(exp(Float32(Float32(r * Float32(-0.6666666666666666)) / 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. Simplified 99.4%

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

   1. add-cube-cbrt 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\left(\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}\right) \cdot \sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   2. associate-*l* 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \left(\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}\right)}}{r}\right) \]

   3. cbrt-unprod 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \color{blue}{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)} \cdot {\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   4. pow-prod-down 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{\color{blue}{{\left(e^{-0.3333333333333333} \cdot e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   5. prod-exp 99.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\color{blue}{\left(e^{-0.3333333333333333 + -0.3333333333333333}\right)}}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   6. metadata-eval 99.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]
7. Step-by-step derivation

   1. exp-prod 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   2. *-commutative 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   3. associate-*l/ 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   4. associate-/l* 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{e^{\color{blue}{r \cdot \frac{-0.3333333333333333}{s}}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

8. Simplified 99.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt[3]{e^{r \cdot \frac{-0.3333333333333333}{s}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]
9. Step-by-step derivation

   1. pow1/3 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{r \cdot \frac{-0.3333333333333333}{s}}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   2. pow-to-exp 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{\log \left(e^{r \cdot \frac{-0.3333333333333333}{s}}\right) \cdot 0.3333333333333333}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   3. add-log-exp 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\left(r \cdot \frac{-0.3333333333333333}{s}\right)} \cdot 0.3333333333333333} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   4. *-commutative 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\left(\frac{-0.3333333333333333}{s} \cdot r\right)} \cdot 0.3333333333333333} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

10. Applied egg-rr 99.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{\left(\frac{-0.3333333333333333}{s} \cdot r\right) \cdot 0.3333333333333333}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

11. Taylor expanded in r around inf 99.6%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\left(\frac{-0.3333333333333333}{s} \cdot r\right) \cdot 0.3333333333333333} \cdot \sqrt[3]{\color{blue}{e^{-0.6666666666666666 \cdot \frac{r}{s}}}}}{r}\right) \]
12. Step-by-step derivation

    1. *-commutative 99.6%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\left(\frac{-0.3333333333333333}{s} \cdot r\right) \cdot 0.3333333333333333} \cdot \sqrt[3]{e^{\color{blue}{\frac{r}{s} \cdot -0.6666666666666666}}}}{r}\right) \]

    2. associate-*l/ 99.7%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\left(\frac{-0.3333333333333333}{s} \cdot r\right) \cdot 0.3333333333333333} \cdot \sqrt[3]{e^{\color{blue}{\frac{r \cdot -0.6666666666666666}{s}}}}}{r}\right) \]

13. Simplified 99.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\left(\frac{-0.3333333333333333}{s} \cdot r\right) \cdot 0.3333333333333333} \cdot \sqrt[3]{\color{blue}{e^{\frac{r \cdot -0.6666666666666666}{s}}}}}{r}\right) \]

14. Final simplification 99.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{0.3333333333333333 \cdot \left(r \cdot \frac{-0.3333333333333333}{s}\right)} \cdot \sqrt[3]{e^{\frac{r \cdot -0.6666666666666666}{s}}}}{r}\right) \]
15. Add Preprocessing

Alternative 3: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{e^{\frac{r \cdot -0.6666666666666666}{s}}} \cdot e^{\frac{r \cdot -0.1111111111111111}{s}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (exp (/ r (- s))) r)
   (/
    (*
     (cbrt (exp (/ (* r -0.6666666666666666) s)))
     (exp (/ (* r -0.1111111111111111) s)))
    r))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + ((cbrtf(expf(((r * -0.6666666666666666f) / s))) * expf(((r * -0.1111111111111111f) / s))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(cbrt(exp(Float32(Float32(r * Float32(-0.6666666666666666)) / s))) * exp(Float32(Float32(r * Float32(-0.1111111111111111)) / 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. Simplified 99.4%

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

   1. add-cube-cbrt 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\left(\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}\right) \cdot \sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   2. associate-*l* 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \left(\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}\right)}}{r}\right) \]

   3. cbrt-unprod 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \color{blue}{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)} \cdot {\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   4. pow-prod-down 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{\color{blue}{{\left(e^{-0.3333333333333333} \cdot e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

   5. prod-exp 99.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\color{blue}{\left(e^{-0.3333333333333333 + -0.3333333333333333}\right)}}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   6. metadata-eval 99.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt[3]{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]
7. Step-by-step derivation

   1. exp-prod 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   2. *-commutative 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   3. associate-*l/ 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   4. associate-/l* 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{e^{\color{blue}{r \cdot \frac{-0.3333333333333333}{s}}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

8. Simplified 99.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt[3]{e^{r \cdot \frac{-0.3333333333333333}{s}}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]
9. Step-by-step derivation

   1. pow1/3 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{r \cdot \frac{-0.3333333333333333}{s}}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   2. pow-to-exp 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{\log \left(e^{r \cdot \frac{-0.3333333333333333}{s}}\right) \cdot 0.3333333333333333}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   3. add-log-exp 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\left(r \cdot \frac{-0.3333333333333333}{s}\right)} \cdot 0.3333333333333333} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

   4. *-commutative 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\left(\frac{-0.3333333333333333}{s} \cdot r\right)} \cdot 0.3333333333333333} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

10. Applied egg-rr 99.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{\left(\frac{-0.3333333333333333}{s} \cdot r\right) \cdot 0.3333333333333333}} \cdot \sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

11. Taylor expanded in r around inf 99.6%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\left(\frac{-0.3333333333333333}{s} \cdot r\right) \cdot 0.3333333333333333} \cdot \sqrt[3]{\color{blue}{e^{-0.6666666666666666 \cdot \frac{r}{s}}}}}{r}\right) \]
12. Step-by-step derivation

    1. *-commutative 99.6%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\left(\frac{-0.3333333333333333}{s} \cdot r\right) \cdot 0.3333333333333333} \cdot \sqrt[3]{e^{\color{blue}{\frac{r}{s} \cdot -0.6666666666666666}}}}{r}\right) \]

    2. associate-*l/ 99.7%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\left(\frac{-0.3333333333333333}{s} \cdot r\right) \cdot 0.3333333333333333} \cdot \sqrt[3]{e^{\color{blue}{\frac{r \cdot -0.6666666666666666}{s}}}}}{r}\right) \]

13. Simplified 99.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\left(\frac{-0.3333333333333333}{s} \cdot r\right) \cdot 0.3333333333333333} \cdot \sqrt[3]{\color{blue}{e^{\frac{r \cdot -0.6666666666666666}{s}}}}}{r}\right) \]

14. Taylor expanded in s around 0 99.6%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{-0.1111111111111111 \cdot \frac{r}{s}}} \cdot \sqrt[3]{e^{\frac{r \cdot -0.6666666666666666}{s}}}}{r}\right) \]
15. Step-by-step derivation

    1. associate-*r/ 99.6%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{-0.1111111111111111 \cdot r}{s}}} \cdot \sqrt[3]{e^{\frac{r \cdot -0.6666666666666666}{s}}}}{r}\right) \]

16. Simplified 99.6%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{-0.1111111111111111 \cdot r}{s}}} \cdot \sqrt[3]{e^{\frac{r \cdot -0.6666666666666666}{s}}}}{r}\right) \]

17. Final simplification 99.6%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt[3]{e^{\frac{r \cdot -0.6666666666666666}{s}}} \cdot e^{\frac{r \cdot -0.1111111111111111}{s}}}{r}\right) \]
18. Add Preprocessing

Alternative 4: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (exp (/ r (- s))) r)
   (/ (sqrt (pow (exp -0.6666666666666666) (/ r s))) r))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + (sqrtf(powf(expf(-0.6666666666666666f), (r / s))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(sqrt((exp(Float32(-0.6666666666666666)) ^ Float32(r / s))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (sqrt((exp(single(-0.6666666666666666)) ^ (r / s))) / r));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 99.4%

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

   3. pow-prod-down 99.4%

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

   4. prod-exp 99.6%

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

   5. metadata-eval 99.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

7. Final simplification 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]
8. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{r}{-s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (/ r (- s))) 0.25) (* r (* s (* PI 2.0))))
  (/ (* 0.75 (exp (/ -0.3333333333333333 (/ s r)))) (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	return ((expf((r / -s)) * 0.25f) / (r * (s * (((float) M_PI) * 2.0f)))) + ((0.75f * expf((-0.3333333333333333f / (s / r)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) * Float32(0.25)) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-0.3333333333333333) / Float32(s / r)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((r / -s)) * single(0.25)) / (r * (s * (single(pi) * single(2.0))))) + ((single(0.75) * exp((single(-0.3333333333333333) / (s / r)))) / (r * (s * (single(pi) * single(6.0)))));
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around 0 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. associate-*r/ 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

   \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
7. Add Preprocessing

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around inf 99.6%

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

6. Final simplification 99.6%

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

Alternative 7: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (exp((r * (single(-0.3333333333333333) / 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. Simplified 99.4%

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

5. Taylor expanded in r around inf 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/l* 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 8: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (exp(((r * single(-0.3333333333333333)) / 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. Simplified 99.4%

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

5. Taylor expanded in r around inf 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/l* 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in r around 0 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-commutative 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 9: 11.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(\pi \cdot r\right)\right)\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (log1p (expm1 (* s (* PI r))))))

C

float code(float s, float r) {
	return 0.25f / log1pf(expm1f((s * (((float) M_PI) * r))));
}

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(s * Float32(Float32(pi) * r)))))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.5%

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

6. Taylor expanded in s around inf 9.9%

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

   1. log1p-expm1-u 12.1%

      \[\leadsto \frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)}} \]

   2. *-commutative 12.1%

      \[\leadsto \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \color{blue}{\left(\pi \cdot s\right)}\right)\right)} \]

   3. associate-*r* 12.1%

      \[\leadsto \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(r \cdot \pi\right) \cdot s}\right)\right)} \]

8. Applied egg-rr 12.1%

   \[\leadsto \frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(r \cdot \pi\right) \cdot s\right)\right)}} \]

9. Final simplification 12.1%

   \[\leadsto \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(\pi \cdot r\right)\right)\right)} \]
10. Add Preprocessing

Alternative 10: 9.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{r \cdot \frac{-0.3333333333333333}{s}}}{r} + \left(\frac{-1 - \frac{r}{s} \cdot -0.5}{s} + \frac{1}{r}\right)\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (exp (* r (/ -0.3333333333333333 s))) r)
   (+ (/ (- -1.0 (* (/ r s) -0.5)) s) (/ 1.0 r)))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r * (-0.3333333333333333f / s))) / r) + (((-1.0f - ((r / s) * -0.5f)) / s) + (1.0f / r)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r * Float32(Float32(-0.3333333333333333) / s))) / r) + Float32(Float32(Float32(Float32(-1.0) - Float32(Float32(r / s) * Float32(-0.5))) / s) + Float32(Float32(1.0) / r))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r * (single(-0.3333333333333333) / s))) / r) + (((single(-1.0) - ((r / s) * single(-0.5))) / s) + (single(1.0) / 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. Simplified 99.4%

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

5. Taylor expanded in r around inf 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/l* 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in s around -inf 11.0%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\left(-1 \cdot \frac{1 + -0.5 \cdot \frac{r}{s}}{s} + \frac{1}{r}\right)} + \frac{e^{r \cdot \frac{-0.3333333333333333}{s}}}{r}\right) \]

9. Final simplification 11.0%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{r \cdot \frac{-0.3333333333333333}{s}}}{r} + \left(\frac{-1 - \frac{r}{s} \cdot -0.5}{s} + \frac{1}{r}\right)\right) \]
10. Add Preprocessing

Alternative 11: 9.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{r \cdot \frac{-0.3333333333333333}{s} + 1}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+ (/ (exp (/ r (- s))) r) (/ (+ (* r (/ -0.3333333333333333 s)) 1.0) r))))

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(Float32(r * Float32(Float32(-0.3333333333333333) / s)) + Float32(1.0)) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (((r * (single(-0.3333333333333333) / s)) + single(1.0)) / 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. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.8%

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

   1. *-commutative 10.8%

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

   2. associate-*l/ 10.8%

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

   3. associate-/l* 10.8%

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

7. Simplified 10.8%

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

8. Final simplification 10.8%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{r \cdot \frac{-0.3333333333333333}{s} + 1}{r}\right) \]
9. Add Preprocessing

Alternative 12: 9.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (single(1.0) / 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. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.5%

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

6. Final simplification 10.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \]
7. Add Preprocessing

Alternative 13: 9.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{e^{\frac{r}{-s}} + 1}{\left(s \cdot \pi\right) \cdot r} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((r / -s)) + single(1.0)) / ((s * single(pi)) * r));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.5%

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

6. Taylor expanded in r around inf 10.5%

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

   1. mul-1-neg 10.5%

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

8. Simplified 10.5%

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

9. Final simplification 10.5%

   \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + 1}{\left(s \cdot \pi\right) \cdot r} \]
10. Add Preprocessing

Alternative 14: 9.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{e^{\frac{r}{-s}} + 1}{s \cdot \left(\pi \cdot r\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((r / -s)) + single(1.0)) / (s * (single(pi) * r)));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.5%

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

   1. associate-/r* 10.5%

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

   2. div-inv 10.5%

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

7. Applied egg-rr 10.5%

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

8. Taylor expanded in r around inf 10.5%

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

   1. mul-1-neg 10.5%

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

   2. distribute-neg-frac2 10.5%

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

   3. *-commutative 10.5%

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

   4. associate-*l* 10.5%

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

   5. *-commutative 10.5%

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

10. Simplified 10.5%

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

11. Final simplification 10.5%

    \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + 1}{s \cdot \left(\pi \cdot r\right)} \]
12. Add Preprocessing

Alternative 15: 9.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{r} \cdot \frac{e^{\frac{r}{-s}} + 1}{s \cdot \pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / r) * ((exp((r / -s)) + single(1.0)) / (s * single(pi)));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.5%

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

6. Taylor expanded in r around inf 10.5%

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

   1. associate-*r/ 10.5%

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

   2. times-frac 10.5%

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

   3. mul-1-neg 10.5%

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

   4. distribute-neg-frac2 10.5%

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

8. Simplified 10.5%

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

9. Final simplification 10.5%

   \[\leadsto \frac{0.125}{r} \cdot \frac{e^{\frac{r}{-s}} + 1}{s \cdot \pi} \]
10. Add Preprocessing

Alternative 16: 8.8% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{\pi} \cdot \frac{\frac{2}{r}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / single(pi)) * ((single(2.0) / r) / s);
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.5%

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

6. Taylor expanded in s around 0 10.5%

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

   1. associate-*r/ 10.5%

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

   2. *-commutative 10.5%

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

   3. times-frac 10.5%

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

   4. mul-1-neg 10.5%

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

   5. distribute-neg-frac2 10.5%

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

8. Simplified 10.5%

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

9. Taylor expanded in r around 0 9.3%

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

    1. mul-1-neg 9.3%

       \[\leadsto \frac{0.125}{\pi} \cdot \frac{\frac{1}{r} + \frac{1 + \color{blue}{\left(-\frac{r}{s}\right)}}{r}}{s} \]

    2. unsub-neg 9.3%

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

11. Simplified 9.3%

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

12. Taylor expanded in r around 0 9.9%

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

    1. associate-/r* 9.9%

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

14. Simplified 9.9%

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

15. Final simplification 9.9%

    \[\leadsto \frac{0.125}{\pi} \cdot \frac{\frac{2}{r}}{s} \]
16. Add Preprocessing

Alternative 17: 8.8% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.5%

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

6. Taylor expanded in s around inf 9.9%

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

   1. associate-/r* 9.9%

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

   2. div-inv 9.9%

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

8. Applied egg-rr 9.9%

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

9. Final simplification 9.9%

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

Alternative 18: 8.8% accurate, 33.0× speedup

Math

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

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(0.25) / Float32(Float32(s * 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%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.5%

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

6. Taylor expanded in s around inf 9.9%

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

7. Final simplification 9.9%

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

Alternative 19: 8.8% accurate, 33.0× speedup

Math

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

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(0.25) / Float32(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%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.5%

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

6. Taylor expanded in s around 0 10.5%

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

   1. associate-*r/ 10.5%

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

   2. *-commutative 10.5%

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

   3. times-frac 10.5%

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

   4. mul-1-neg 10.5%

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

   5. distribute-neg-frac2 10.5%

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

8. Simplified 10.5%

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

9. Taylor expanded in r around 0 9.9%

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

    1. associate-*r* 9.9%

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

11. Simplified 9.9%

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

12. Taylor expanded in r around 0 9.9%

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

    1. *-commutative 9.9%

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

    2. associate-*l* 9.9%

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

    3. *-commutative 9.9%

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

14. Simplified 9.9%

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

15. Final simplification 9.9%

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

Alternative 20: 8.8% accurate, 33.0× speedup

Math

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

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(Float32(0.25) / 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%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.5%

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

6. Taylor expanded in s around inf 9.9%

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

   1. *-commutative 9.9%

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

   2. associate-*l* 9.9%

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

   3. *-commutative 9.9%

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

   4. associate-/l/ 9.9%

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

8. Simplified 9.9%

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

9. Final simplification 9.9%

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

Reproduce

herbie shell --seed 2024075 
(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))))