Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 15.3s

Alternatives: 15

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/ (* 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 * (2.0f * ((float) M_PI))))) + ((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(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + 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(2.0) * single(pi))))) + ((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. Final simplification 99.6%

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

Alternative 2: 99.5% 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(Float32(-r) / 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 98.9%

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

5. Taylor expanded in r around inf 99.5%

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

   1. mul-1-neg 99.5%

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

   2. exp-neg 99.5%

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

7. Applied egg-rr 99.5%

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

   1. rec-exp 99.5%

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

   2. distribute-neg-frac 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((-r / s)) + exp(((r / s) / single(-3.0)))) / (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 98.9%

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

5. Taylor expanded in r around inf 99.5%

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

   1. metadata-eval 99.5%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-1}{3}} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   2. times-frac 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-1 \cdot r}{3 \cdot s}}}}{r \cdot \left(s \cdot \pi\right)} \]

   3. neg-mul-1 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\color{blue}{-r}}{3 \cdot s}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. *-commutative 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-r}{\color{blue}{s \cdot 3}}}}{r \cdot \left(s \cdot \pi\right)} \]

   5. associate-/r* 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{\frac{-r}{s}}{3}}}}{r \cdot \left(s \cdot \pi\right)} \]

   6. frac-2neg 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-\frac{-r}{s}}{-3}}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. add-sqr-sqrt -0.0%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{-r} \cdot \sqrt{-r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   8. sqrt-unprod 10.0%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   9. sqr-neg 10.0%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\sqrt{\color{blue}{r \cdot r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   10. sqrt-unprod 10.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{r} \cdot \sqrt{r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   11. add-sqr-sqrt 10.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{r}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   12. distribute-frac-neg 10.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\color{blue}{\frac{-r}{s}}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   13. add-sqr-sqrt -0.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{\sqrt{-r} \cdot \sqrt{-r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   14. sqrt-unprod 99.6%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   15. sqr-neg 99.6%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\sqrt{\color{blue}{r \cdot r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   16. sqrt-unprod 99.5%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{\sqrt{r} \cdot \sqrt{r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   17. add-sqr-sqrt 99.6%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{r}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   18. metadata-eval 99.6%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{r}{s}}{\color{blue}{-3}}}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.6%

   \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{\frac{r}{s}}{-3}}}}{r \cdot \left(s \cdot \pi\right)} \]
8. Step-by-step derivation

   1. mul-1-neg 99.5%

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

   2. exp-neg 99.5%

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

9. Applied egg-rr 99.6%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{\frac{1}{e^{\frac{r}{s}}}} + e^{\frac{\frac{r}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]
10. Step-by-step derivation

    1. rec-exp 99.5%

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

    2. distribute-neg-frac 99.5%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 4: 11.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(s * Float32(r * Float32(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 98.9%

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

5. Taylor expanded in s around inf 11.1%

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

   1. add-sqr-sqrt 11.1%

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

   2. sqrt-unprod 11.1%

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

   3. sqr-neg 11.1%

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

   4. sqrt-unprod -0.0%

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

   5. add-sqr-sqrt 4.4%

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

   6. distribute-lft-neg-in 4.4%

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

   7. log1p-expm1-u 7.8%

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

   8. distribute-lft-neg-in 7.8%

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

   9. add-sqr-sqrt -0.0%

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

   10. sqrt-unprod 13.4%

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

   11. sqr-neg 13.4%

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

   12. sqrt-unprod 13.4%

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

   13. add-sqr-sqrt 13.4%

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

   14. *-commutative 13.4%

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

   15. associate-*l* 13.4%

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

7. Applied egg-rr 13.4%

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

8. Final simplification 13.4%

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

Alternative 5: 43.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / Float32(s * log1p(expm1(Float32(r * Float32(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 98.9%

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

5. Taylor expanded in s around inf 11.1%

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

   1. expm1-log1p-u 11.1%

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

   2. expm1-udef 9.4%

      \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)} - 1}} \]

   3. *-commutative 9.4%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{\left(s \cdot \pi\right) \cdot r}\right)} - 1} \]

   4. associate-*l* 9.4%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{s \cdot \left(\pi \cdot r\right)}\right)} - 1} \]

7. Applied egg-rr 9.4%

   \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)} - 1}} \]
8. Step-by-step derivation

   1. expm1-def 11.1%

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

   2. expm1-log1p 11.1%

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

9. Simplified 11.1%

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

    1. log1p-expm1-u 46.9%

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

    2. *-commutative 46.9%

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

11. Applied egg-rr 46.9%

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

12. Final simplification 46.9%

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

Alternative 6: 9.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp(((r / s) * single(-0.3333333333333333))) + (single(1.0) - (r / s))) / (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 98.9%

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

5. Taylor expanded in r around inf 99.5%

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

6. Taylor expanded in r around 0 11.4%

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

   1. mul-1-neg 11.4%

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

   2. unsub-neg 11.4%

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

8. Simplified 11.4%

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

9. Final simplification 11.4%

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

Alternative 7: 9.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{e^{\frac{\frac{r}{s}}{-3}} + \left(1 - \frac{r}{s}\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* 0.125 (/ (+ (exp (/ (/ r s) -3.0)) (- 1.0 (/ r s))) (* r (* s PI)))))

C

float code(float s, float r) {
	return 0.125f * ((expf(((r / s) / -3.0f)) + (1.0f - (r / s))) / (r * (s * ((float) M_PI))));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(exp(Float32(Float32(r / s) / Float32(-3.0))) + Float32(Float32(1.0) - Float32(r / s))) / Float32(r * Float32(s * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp(((r / s) / single(-3.0))) + (single(1.0) - (r / s))) / (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 98.9%

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

5. Taylor expanded in r around inf 99.5%

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

   1. metadata-eval 99.5%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-1}{3}} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   2. times-frac 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-1 \cdot r}{3 \cdot s}}}}{r \cdot \left(s \cdot \pi\right)} \]

   3. neg-mul-1 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\color{blue}{-r}}{3 \cdot s}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. *-commutative 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-r}{\color{blue}{s \cdot 3}}}}{r \cdot \left(s \cdot \pi\right)} \]

   5. associate-/r* 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{\frac{-r}{s}}{3}}}}{r \cdot \left(s \cdot \pi\right)} \]

   6. frac-2neg 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-\frac{-r}{s}}{-3}}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. add-sqr-sqrt -0.0%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{-r} \cdot \sqrt{-r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   8. sqrt-unprod 10.0%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   9. sqr-neg 10.0%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\sqrt{\color{blue}{r \cdot r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   10. sqrt-unprod 10.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{r} \cdot \sqrt{r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   11. add-sqr-sqrt 10.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{r}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   12. distribute-frac-neg 10.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\color{blue}{\frac{-r}{s}}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   13. add-sqr-sqrt -0.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{\sqrt{-r} \cdot \sqrt{-r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   14. sqrt-unprod 99.6%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   15. sqr-neg 99.6%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\sqrt{\color{blue}{r \cdot r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   16. sqrt-unprod 99.5%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{\sqrt{r} \cdot \sqrt{r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   17. add-sqr-sqrt 99.6%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{r}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   18. metadata-eval 99.6%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{r}{s}}{\color{blue}{-3}}}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.6%

   \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{\frac{r}{s}}{-3}}}}{r \cdot \left(s \cdot \pi\right)} \]

8. Taylor expanded in r around 0 11.4%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(1 + -1 \cdot \frac{r}{s}\right)} + e^{\frac{\frac{r}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]
9. Step-by-step derivation

   1. mul-1-neg 11.4%

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

   2. unsub-neg 11.4%

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

10. Simplified 11.4%

    \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(1 - \frac{r}{s}\right)} + e^{\frac{\frac{r}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

11. Final simplification 11.4%

    \[\leadsto 0.125 \cdot \frac{e^{\frac{\frac{r}{s}}{-3}} + \left(1 - \frac{r}{s}\right)}{r \cdot \left(s \cdot \pi\right)} \]
12. Add Preprocessing

Alternative 8: 9.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{e^{\frac{\frac{r}{s}}{-3}} + \left(1 - \frac{r}{s}\right)}{s \cdot \left(r \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* 0.125 (/ (+ (exp (/ (/ r s) -3.0)) (- 1.0 (/ r s))) (* s (* r PI)))))

C

float code(float s, float r) {
	return 0.125f * ((expf(((r / s) / -3.0f)) + (1.0f - (r / s))) / (s * (r * ((float) M_PI))));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(exp(Float32(Float32(r / s) / Float32(-3.0))) + Float32(Float32(1.0) - Float32(r / s))) / Float32(s * Float32(r * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp(((r / s) / single(-3.0))) + (single(1.0) - (r / s))) / (s * (r * single(pi))));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 98.9%

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

5. Taylor expanded in r around inf 99.5%

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

   1. metadata-eval 99.5%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-1}{3}} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   2. times-frac 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-1 \cdot r}{3 \cdot s}}}}{r \cdot \left(s \cdot \pi\right)} \]

   3. neg-mul-1 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\color{blue}{-r}}{3 \cdot s}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. *-commutative 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-r}{\color{blue}{s \cdot 3}}}}{r \cdot \left(s \cdot \pi\right)} \]

   5. associate-/r* 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{\frac{-r}{s}}{3}}}}{r \cdot \left(s \cdot \pi\right)} \]

   6. frac-2neg 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-\frac{-r}{s}}{-3}}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. add-sqr-sqrt -0.0%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{-r} \cdot \sqrt{-r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   8. sqrt-unprod 10.0%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   9. sqr-neg 10.0%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\sqrt{\color{blue}{r \cdot r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   10. sqrt-unprod 10.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{r} \cdot \sqrt{r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   11. add-sqr-sqrt 10.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{r}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   12. distribute-frac-neg 10.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\color{blue}{\frac{-r}{s}}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   13. add-sqr-sqrt -0.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{\sqrt{-r} \cdot \sqrt{-r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   14. sqrt-unprod 99.6%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   15. sqr-neg 99.6%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\sqrt{\color{blue}{r \cdot r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   16. sqrt-unprod 99.5%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{\sqrt{r} \cdot \sqrt{r}}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   17. add-sqr-sqrt 99.6%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{r}}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]

   18. metadata-eval 99.6%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{r}{s}}{\color{blue}{-3}}}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.6%

   \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{\frac{r}{s}}{-3}}}}{r \cdot \left(s \cdot \pi\right)} \]

8. Taylor expanded in r around 0 11.4%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(1 + -1 \cdot \frac{r}{s}\right)} + e^{\frac{\frac{r}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]
9. Step-by-step derivation

   1. mul-1-neg 11.4%

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

   2. unsub-neg 11.4%

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

10. Simplified 11.4%

    \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(1 - \frac{r}{s}\right)} + e^{\frac{\frac{r}{s}}{-3}}}{r \cdot \left(s \cdot \pi\right)} \]
11. Step-by-step derivation

    1. expm1-log1p-u 11.1%

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

    2. expm1-udef 9.4%

       \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)} - 1}} \]

    3. *-commutative 9.4%

       \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{\left(s \cdot \pi\right) \cdot r}\right)} - 1} \]

    4. associate-*l* 9.4%

       \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{s \cdot \left(\pi \cdot r\right)}\right)} - 1} \]

12. Applied egg-rr 9.2%

    \[\leadsto 0.125 \cdot \frac{\left(1 - \frac{r}{s}\right) + e^{\frac{\frac{r}{s}}{-3}}}{\color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)} - 1}} \]
13. Step-by-step derivation

    1. expm1-def 11.1%

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

    2. expm1-log1p 11.1%

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

14. Simplified 11.4%

    \[\leadsto 0.125 \cdot \frac{\left(1 - \frac{r}{s}\right) + e^{\frac{\frac{r}{s}}{-3}}}{\color{blue}{s \cdot \left(\pi \cdot r\right)}} \]

15. Final simplification 11.4%

    \[\leadsto 0.125 \cdot \frac{e^{\frac{\frac{r}{s}}{-3}} + \left(1 - \frac{r}{s}\right)}{s \cdot \left(r \cdot \pi\right)} \]
16. Add Preprocessing

Alternative 9: 9.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp(((r / s) * single(-0.3333333333333333))) + single(1.0)) / (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 98.9%

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

5. Taylor expanded in r around inf 99.5%

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

6. Taylor expanded in r around 0 11.3%

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

7. Final simplification 11.3%

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

Alternative 10: 9.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp(((r / s) * single(-0.3333333333333333))) + single(1.0)) / (s * (r * single(pi))));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 98.9%

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

5. Taylor expanded in r around inf 99.5%

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

6. Taylor expanded in r around 0 11.3%

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

   1. expm1-log1p-u 11.1%

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

   2. expm1-udef 9.4%

      \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)} - 1}} \]

   3. *-commutative 9.4%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{\left(s \cdot \pi\right) \cdot r}\right)} - 1} \]

   4. associate-*l* 9.4%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{s \cdot \left(\pi \cdot r\right)}\right)} - 1} \]

8. Applied egg-rr 9.5%

   \[\leadsto 0.125 \cdot \frac{1 + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)} - 1}} \]
9. Step-by-step derivation

   1. expm1-def 11.1%

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

   2. expm1-log1p 11.1%

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

10. Simplified 11.3%

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

11. Final simplification 11.3%

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

Alternative 11: 8.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 98.9%

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

5. Taylor expanded in r around inf 99.5%

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

   1. associate-*r/ 99.5%

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

   2. *-commutative 99.5%

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

   3. times-frac 99.1%

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

   4. mul-1-neg 99.1%

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

   5. distribute-neg-frac 99.1%

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

   6. associate-*r/ 99.1%

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

   7. associate-/l* 99.2%

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

7. Simplified 99.2%

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

8. Taylor expanded in s around inf 11.2%

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

   1. sub-neg 11.2%

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

   2. associate-*r/ 11.2%

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

   3. metadata-eval 11.2%

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

   4. associate-/r* 11.2%

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

   5. associate-/r* 11.2%

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

   6. associate-*r* 11.2%

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

   7. *-commutative 11.2%

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

   8. associate-*r/ 11.2%

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

   9. metadata-eval 11.2%

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

   10. distribute-neg-frac 11.2%

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

   11. metadata-eval 11.2%

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

   12. *-commutative 11.2%

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

10. Simplified 11.2%

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

11. Final simplification 11.2%

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

Alternative 12: 8.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 11.2%

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

   1. associate-*r/ 11.2%

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

   2. metadata-eval 11.2%

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

   3. associate-*r/ 11.2%

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

   4. metadata-eval 11.2%

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

   5. *-commutative 11.2%

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

7. Simplified 11.2%

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

8. Final simplification 11.2%

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

Alternative 13: 9.0% 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 98.9%

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

5. Taylor expanded in s around inf 11.1%

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

6. Final simplification 11.1%

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

Alternative 14: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 11.1%

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

   1. expm1-log1p-u 11.1%

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

   2. expm1-udef 9.4%

      \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)} - 1}} \]

   3. *-commutative 9.4%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{\left(s \cdot \pi\right) \cdot r}\right)} - 1} \]

   4. associate-*l* 9.4%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{s \cdot \left(\pi \cdot r\right)}\right)} - 1} \]

7. Applied egg-rr 9.4%

   \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)} - 1}} \]
8. Step-by-step derivation

   1. expm1-def 11.1%

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

   2. expm1-log1p 11.1%

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

9. Simplified 11.1%

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

10. Taylor expanded in s around 0 11.1%

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

    1. associate-*r* 11.1%

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

    2. *-commutative 11.1%

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

12. Simplified 11.1%

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

13. Final simplification 11.1%

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

Alternative 15: 9.0% 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 98.9%

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

5. Taylor expanded in s around inf 11.1%

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

   1. expm1-log1p-u 11.1%

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

   2. expm1-udef 9.4%

      \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)} - 1}} \]

   3. *-commutative 9.4%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{\left(s \cdot \pi\right) \cdot r}\right)} - 1} \]

   4. associate-*l* 9.4%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{s \cdot \left(\pi \cdot r\right)}\right)} - 1} \]

7. Applied egg-rr 9.4%

   \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)} - 1}} \]
8. Step-by-step derivation

   1. expm1-def 11.1%

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

   2. expm1-log1p 11.1%

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

9. Simplified 11.1%

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

10. Taylor expanded in s around 0 11.1%

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

    1. associate-/r* 11.1%

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

12. Simplified 11.1%

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

13. Final simplification 11.1%

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

Reproduce

herbie shell --seed 2024040 
(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))))