Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 14.0s

Alternatives: 18

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: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around inf 99.6%

   \[\leadsto \frac{\color{blue}{0.25 \cdot e^{-1 \cdot \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. neg-mul-1 99.6%

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

   2. rec-exp 99.6%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. associate-*r/ 99.6%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*l* 99.6%

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

8. Simplified 99.6%

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

   1. distribute-frac-neg 99.6%

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

   2. *-commutative 99.6%

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

10. Applied egg-rr 99.6%

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

    1. distribute-neg-frac2 99.6%

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

    2. distribute-rgt-neg-in 99.6%

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

    3. metadata-eval 99.6%

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

12. Simplified 99.6%

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

    1. +-commutative 99.6%

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

    2. times-frac 99.5%

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

    3. fma-define 99.6%

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

    4. *-commutative 99.6%

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

    5. *-commutative 99.6%

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

    6. associate-/r* 99.6%

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

    7. associate-*r* 99.6%

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

14. Applied egg-rr 99.6%

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

15. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / exp((r / s))) / (r * (s * (single(pi) * single(2.0))))) + ((single(0.75) * exp(-(r / (s * single(3.0))))) / (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 inf 99.6%

   \[\leadsto \frac{\color{blue}{0.25 \cdot e^{-1 \cdot \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. neg-mul-1 99.6%

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

   2. rec-exp 99.6%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. associate-*r/ 99.6%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * (r * (single(pi) * exp((r / s)))))) + ((single(0.75) * exp((r / (s * single(-3.0))))) / (single(pi) * (single(6.0) * (s * r))));
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around inf 99.6%

   \[\leadsto \frac{\color{blue}{0.25 \cdot e^{-1 \cdot \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. neg-mul-1 99.6%

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

   2. rec-exp 99.6%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. associate-*r/ 99.6%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*l* 99.6%

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

8. Simplified 99.6%

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

   1. distribute-frac-neg 99.6%

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

   2. *-commutative 99.6%

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

10. Applied egg-rr 99.6%

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

    1. distribute-neg-frac2 99.6%

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

    2. distribute-rgt-neg-in 99.6%

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

    3. metadata-eval 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in s around 0 99.5%

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

    1. associate-*r* 99.5%

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

    2. *-commutative 99.5%

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

    3. *-commutative 99.5%

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

    4. associate-*r* 99.6%

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

    5. *-commutative 99.6%

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

    6. associate-*l* 99.6%

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

15. Simplified 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.5%

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

   3. fma-define 99.6%

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

   4. associate-*l* 99.5%

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

   5. associate-/r* 99.5%

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

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.4%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.4%

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

5. Taylor expanded in s around 0 99.4%

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

   1. associate-*r/ 99.5%

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

7. Applied egg-rr 99.5%

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

   1. associate-*r/ 99.5%

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

   2. mul-1-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. *-commutative 99.5%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (exp(((r / s) * single(-0.3333333333333333))) / 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 99.2%

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

   1. pow-exp 99.4%

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

   2. *-commutative 99.4%

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

6. Applied egg-rr 99.4%

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

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

5. Taylor expanded in r around inf 99.4%

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

6. Final simplification 99.4%

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

Alternative 7: 50.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 28:\\ \;\;\;\;\frac{0.125}{r \cdot \left(s \cdot \pi + r \cdot \left(\pi + 0.5 \cdot \frac{\pi \cdot r}{s}\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 28.0)
   (+
    (/ 0.125 (* r (+ (* s PI) (* r (+ PI (* 0.5 (/ (* PI r) s)))))))
    (/ (* 0.75 (exp (/ r (* s -3.0)))) (* r (* s (* PI 6.0)))))
   (/ 0.25 (* s (log1p (expm1 (* PI r)))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 28.0f) {
		tmp = (0.125f / (r * ((s * ((float) M_PI)) + (r * (((float) M_PI) + (0.5f * ((((float) M_PI) * r) / s))))))) + ((0.75f * expf((r / (s * -3.0f)))) / (r * (s * (((float) M_PI) * 6.0f))));
	} else {
		tmp = 0.25f / (s * log1pf(expm1f((((float) M_PI) * r))));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(28.0))
		tmp = Float32(Float32(Float32(0.125) / Float32(r * Float32(Float32(s * Float32(pi)) + Float32(r * Float32(Float32(pi) + Float32(Float32(0.5) * Float32(Float32(Float32(pi) * r) / s))))))) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-3.0))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))));
	else
		tmp = Float32(Float32(0.25) / Float32(s * log1p(expm1(Float32(Float32(pi) * r)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 28

   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 inf 99.5%

      \[\leadsto \frac{\color{blue}{0.25 \cdot e^{-1 \cdot \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   4. Step-by-step derivation

      1. neg-mul-1 99.5%

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

      2. rec-exp 99.6%

         \[\leadsto \frac{0.25 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

      3. associate-*r/ 99.6%

         \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

      4. metadata-eval 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in r around inf 99.6%

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

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.5%

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

   8. Simplified 99.5%

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

      1. distribute-frac-neg 99.5%

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

      2. *-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. distribute-neg-frac2 99.5%

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

       2. distribute-rgt-neg-in 99.5%

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

       3. metadata-eval 99.5%

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

   12. Simplified 99.5%

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

   13. Taylor expanded in r around 0 18.7%

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

   if 28 < r

   1. Initial program 99.6%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in s around inf 5.7%

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

      1. pow1 5.7%

         \[\leadsto \frac{0.25}{\color{blue}{{\left(r \cdot \left(s \cdot \pi\right)\right)}^{1}}} \]

      2. associate-*r* 5.7%

         \[\leadsto \frac{0.25}{{\color{blue}{\left(\left(r \cdot s\right) \cdot \pi\right)}}^{1}} \]

   7. Applied egg-rr 5.7%

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

      1. unpow1 5.7%

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

      2. associate-*r* 5.7%

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

      3. *-commutative 5.7%

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

      4. associate-*l* 5.7%

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

   9. Simplified 5.7%

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

       1. *-commutative 5.7%

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

       2. log1p-expm1-u 94.6%

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

   11. Applied egg-rr 94.6%

       \[\leadsto \frac{0.25}{s \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 28:\\ \;\;\;\;\frac{0.125}{r \cdot \left(s \cdot \pi + r \cdot \left(\pi + 0.5 \cdot \frac{\pi \cdot r}{s}\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{r \cdot \left(s \cdot \pi + r \cdot \left(\pi + 0.5 \cdot \frac{\pi \cdot r}{s}\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ 0.125 (* r (+ (* s PI) (* r (+ PI (* 0.5 (/ (* PI r) s)))))))
  (/ (* 0.75 (exp (/ r (* s -3.0)))) (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	return (0.125f / (r * ((s * ((float) M_PI)) + (r * (((float) M_PI) + (0.5f * ((((float) M_PI) * r) / s))))))) + ((0.75f * expf((r / (s * -3.0f)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(r * Float32(Float32(s * Float32(pi)) + Float32(r * Float32(Float32(pi) + Float32(Float32(0.5) * Float32(Float32(Float32(pi) * r) / s))))))) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-3.0))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (r * ((s * single(pi)) + (r * (single(pi) + (single(0.5) * ((single(pi) * r) / s))))))) + ((single(0.75) * exp((r / (s * single(-3.0))))) / (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 inf 99.6%

   \[\leadsto \frac{\color{blue}{0.25 \cdot e^{-1 \cdot \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. neg-mul-1 99.6%

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

   2. rec-exp 99.6%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. associate-*r/ 99.6%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*l* 99.6%

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

8. Simplified 99.6%

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

   1. distribute-frac-neg 99.6%

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

   2. *-commutative 99.6%

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

10. Applied egg-rr 99.6%

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

    1. distribute-neg-frac2 99.6%

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

    2. distribute-rgt-neg-in 99.6%

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

    3. metadata-eval 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in r around 0 25.5%

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

14. Final simplification 25.5%

    \[\leadsto \frac{0.125}{r \cdot \left(s \cdot \pi + r \cdot \left(\pi + 0.5 \cdot \frac{\pi \cdot r}{s}\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
15. Add Preprocessing

Alternative 9: 19.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp(-(r / (s * single(3.0))))) / (r * (s * (single(pi) * single(6.0))))) + (single(0.125) / (s * (r * (single(pi) + (r * (single(pi) / s))))));
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around inf 99.6%

   \[\leadsto \frac{\color{blue}{0.25 \cdot e^{-1 \cdot \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. neg-mul-1 99.6%

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

   2. rec-exp 99.6%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. associate-*r/ 99.6%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*l* 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in r around 0 19.5%

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

    1. associate-/l* 19.8%

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

11. Simplified 19.8%

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

12. Final simplification 19.8%

    \[\leadsto \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{s \cdot \left(r \cdot \left(\pi + r \cdot \frac{\pi}{s}\right)\right)} \]
13. Add Preprocessing

Alternative 10: 12.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp((r / (s * single(-3.0))))) / (r * (s * (single(pi) * single(6.0))))) + (single(0.125) / (r * ((single(pi) * r) + (s * single(pi)))));
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around inf 99.6%

   \[\leadsto \frac{\color{blue}{0.25 \cdot e^{-1 \cdot \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. neg-mul-1 99.6%

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

   2. rec-exp 99.6%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. associate-*r/ 99.6%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*l* 99.6%

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

8. Simplified 99.6%

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

   1. distribute-frac-neg 99.6%

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

   2. *-commutative 99.6%

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

10. Applied egg-rr 99.6%

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

    1. distribute-neg-frac2 99.6%

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

    2. distribute-rgt-neg-in 99.6%

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

    3. metadata-eval 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in r around 0 12.3%

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

14. Final simplification 12.3%

    \[\leadsto \frac{0.75 \cdot e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{r \cdot \left(\pi \cdot r + s \cdot \pi\right)} \]
15. Add Preprocessing

Alternative 11: 12.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp((r / (s * single(-3.0))))) / (r * (s * (single(pi) * single(6.0))))) + (single(0.125) / (r * (single(pi) * (s + r))));
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around inf 99.6%

   \[\leadsto \frac{\color{blue}{0.25 \cdot e^{-1 \cdot \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. neg-mul-1 99.6%

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

   2. rec-exp 99.6%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. associate-*r/ 99.6%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*l* 99.6%

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

8. Simplified 99.6%

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

   1. distribute-frac-neg 99.6%

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

   2. *-commutative 99.6%

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

10. Applied egg-rr 99.6%

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

    1. distribute-neg-frac2 99.6%

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

    2. distribute-rgt-neg-in 99.6%

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

    3. metadata-eval 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in r around 0 12.3%

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

    1. distribute-rgt-out 12.3%

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

15. Simplified 12.3%

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

16. Final simplification 12.3%

    \[\leadsto \frac{0.75 \cdot e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{r \cdot \left(\pi \cdot \left(s + r\right)\right)} \]
17. Add Preprocessing

Alternative 12: 10.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.5%

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

   3. fma-define 99.6%

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

   4. associate-*l* 99.5%

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

   5. associate-/r* 99.5%

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

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.4%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.4%

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

5. Taylor expanded in s around inf 9.9%

   \[\leadsto \color{blue}{\frac{\left(0.006944444444444444 \cdot \frac{r}{{s}^{2} \cdot \pi} + \left(0.0625 \cdot \frac{r}{{s}^{2} \cdot \pi} + 0.25 \cdot \frac{1}{r \cdot \pi}\right)\right) - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]
6. Step-by-step derivation

7. Simplified 9.9%

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

8. Final simplification 9.9%

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

Alternative 13: 10.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} + 0.16666666666666666 \cdot \frac{-1}{\pi}}{s} + 0.25 \cdot \frac{1}{\pi \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/
    (+
     (* 0.125 (/ (+ (* 0.05555555555555555 (/ r PI)) (* 0.5 (/ r PI))) s))
     (* 0.16666666666666666 (/ -1.0 PI)))
    s)
   (* 0.25 (/ 1.0 (* PI r))))
  s))

C

float code(float s, float r) {
	return ((((0.125f * (((0.05555555555555555f * (r / ((float) M_PI))) + (0.5f * (r / ((float) M_PI)))) / s)) + (0.16666666666666666f * (-1.0f / ((float) M_PI)))) / s) + (0.25f * (1.0f / (((float) M_PI) * r)))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(Float32(0.05555555555555555) * Float32(r / Float32(pi))) + Float32(Float32(0.5) * Float32(r / Float32(pi)))) / s)) + Float32(Float32(0.16666666666666666) * Float32(Float32(-1.0) / Float32(pi)))) / s) + Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(Float32(pi) * r)))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((((single(0.125) * (((single(0.05555555555555555) * (r / single(pi))) + (single(0.5) * (r / single(pi)))) / s)) + (single(0.16666666666666666) * (single(-1.0) / single(pi)))) / s) + (single(0.25) * (single(1.0) / (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 99.2%

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

5. Taylor expanded in s around -inf 9.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Final simplification 9.9%

   \[\leadsto \frac{\frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} + 0.16666666666666666 \cdot \frac{-1}{\pi}}{s} + 0.25 \cdot \frac{1}{\pi \cdot r}}{s} \]
7. Add Preprocessing

Alternative 14: 10.3% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.5%

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

   3. fma-define 99.6%

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

   4. associate-*l* 99.5%

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

   5. associate-/r* 99.5%

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

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.4%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.4%

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

5. Taylor expanded in s around -inf 9.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. mul-1-neg 9.9%

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

7. Simplified 9.9%

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

8. Final simplification 9.9%

   \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s} \]
9. Add Preprocessing

Alternative 15: 10.2% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\pi \cdot r} + \frac{\frac{r \cdot 0.0625}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

5. Taylor expanded in r around 0 9.3%

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

6. Taylor expanded in r around inf 9.3%

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

   1. associate-*r/ 9.3%

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

   2. metadata-eval 9.3%

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

8. Simplified 9.3%

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

9. Taylor expanded in s around -inf 9.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
10. Step-by-step derivation

    1. mul-1-neg 9.6%

       \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

    2. mul-1-neg 9.6%

       \[\leadsto -\frac{\color{blue}{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}\right)} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

    3. associate-*r/ 9.6%

       \[\leadsto -\frac{\left(-\frac{\color{blue}{\frac{0.0625 \cdot r}{s \cdot \pi}} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

    4. associate-*r/ 9.6%

       \[\leadsto -\frac{\left(-\frac{\frac{0.0625 \cdot r}{s \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{\pi}}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

    5. metadata-eval 9.6%

       \[\leadsto -\frac{\left(-\frac{\frac{0.0625 \cdot r}{s \cdot \pi} - \frac{\color{blue}{0.16666666666666666}}{\pi}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

    6. associate-*r/ 9.6%

       \[\leadsto -\frac{\left(-\frac{\frac{0.0625 \cdot r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \color{blue}{\frac{0.25 \cdot 1}{r \cdot \pi}}}{s} \]

    7. metadata-eval 9.6%

       \[\leadsto -\frac{\left(-\frac{\frac{0.0625 \cdot r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\color{blue}{0.25}}{r \cdot \pi}}{s} \]

11. Simplified 9.6%

    \[\leadsto \color{blue}{-\frac{\left(-\frac{\frac{0.0625 \cdot r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{r \cdot \pi}}{s}} \]

12. Final simplification 9.6%

    \[\leadsto \frac{\frac{0.25}{\pi \cdot r} + \frac{\frac{r \cdot 0.0625}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}}{s} \]
13. Add Preprocessing

Alternative 16: 9.2% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 98.9%

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

   2. sqrt-unprod 99.2%

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

   3. prod-exp 99.3%

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

   4. metadata-eval 99.3%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in s around inf 8.8%

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

   1. associate-/r* 8.8%

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

9. Simplified 8.8%

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

Alternative 17: 9.2% 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 99.2%

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

5. Taylor expanded in s around inf 8.8%

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

   1. *-commutative 8.8%

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

   2. associate-*l* 8.8%

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

   3. *-commutative 8.8%

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

7. Simplified 8.8%

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

8. Final simplification 8.8%

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

Alternative 18: 9.2% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

5. Taylor expanded in s around inf 8.8%

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

Reproduce

herbie shell --seed 2024119 
(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))))