Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 21.0s

Alternatives: 16

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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\frac{0.125}{s}}{\pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (/ 0.125 s) PI)
  (/ (pow (exp -0.6666666666666666) (/ (* r 0.5) s)) r)
  (* (/ 0.125 (* s PI)) (/ (exp (/ (- r) s)) r))))

C

float code(float s, float r) {
	return fmaf(((0.125f / s) / ((float) M_PI)), (powf(expf(-0.6666666666666666f), ((r * 0.5f) / s)) / r), ((0.125f / (s * ((float) M_PI))) * (expf((-r / s)) / r)));
}

Julia

function code(s, r)
	return fma(Float32(Float32(Float32(0.125) / s) / Float32(pi)), Float32((exp(Float32(-0.6666666666666666)) ^ Float32(Float32(r * Float32(0.5)) / s)) / r), Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(exp(Float32(Float32(-r) / s)) / r)))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.5%

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

   1. pow-exp 99.3%

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

   2. sqr-pow 99.3%

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

   3. pow-prod-down 99.3%

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

   4. prod-exp 99.7%

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

   5. metadata-eval 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   6. div-inv 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r}{s} \cdot \frac{1}{2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   7. metadata-eval 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot \color{blue}{0.5}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

8. Simplified 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
9. Step-by-step derivation

   1. expm1-log1p-u 99.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{s \cdot \pi}\right)\right)}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

10. Applied egg-rr 99.8%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{s \cdot \pi}\right)\right)}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

11. Taylor expanded in s around 0 99.7%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.125}{s \cdot \pi}}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
12. Step-by-step derivation

    1. associate-/r* 99.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.125}{s}}{\pi}}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

13. Simplified 99.8%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.125}{s}}{\pi}}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

14. Final simplification 99.8%

    \[\leadsto \mathsf{fma}\left(\frac{\frac{0.125}{s}}{\pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
15. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ \mathsf{fma}\left(t_0, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, t_0 \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* s PI))))
   (fma
    t_0
    (/ (pow (exp -0.6666666666666666) (/ (* r 0.5) s)) r)
    (* t_0 (/ (exp (/ (- r) s)) r)))))

C

float code(float s, float r) {
	float t_0 = 0.125f / (s * ((float) M_PI));
	return fmaf(t_0, (powf(expf(-0.6666666666666666f), ((r * 0.5f) / s)) / r), (t_0 * (expf((-r / s)) / r)));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(s * Float32(pi)))
	return fma(t_0, Float32((exp(Float32(-0.6666666666666666)) ^ Float32(Float32(r * Float32(0.5)) / s)) / r), Float32(t_0 * Float32(exp(Float32(Float32(-r) / s)) / r)))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.5%

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

   1. pow-exp 99.3%

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

   2. sqr-pow 99.3%

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

   3. pow-prod-down 99.3%

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

   4. prod-exp 99.7%

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

   5. metadata-eval 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   6. div-inv 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r}{s} \cdot \frac{1}{2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

   7. metadata-eval 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot \color{blue}{0.5}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

8. Simplified 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]

9. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \]
10. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) * single(0.25)) / (r * (s * (single(pi) * single(2.0))))) + ((single(0.75) * exp(((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. Step-by-step derivation

   1. frac-2neg 99.6%

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

   2. div-inv 99.6%

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

   3. remove-double-neg 99.6%

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

   4. *-commutative 99.6%

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

   5. distribute-rgt-neg-in 99.6%

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

   6. metadata-eval 99.6%

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

4. Applied egg-rr 99.6%

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

   1. div-inv 99.6%

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

   2. associate-/r* 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. fma-def 99.6%

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

   3. associate-*l* 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. metadata-eval 99.6%

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

   7. associate-/r* 99.6%

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

   8. associate-*l* 99.6%

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

   9. /-rgt-identity 99.6%

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

   10. fma-def 99.6%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around 0 99.5%

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

6. Taylor expanded in s around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*r* 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
10. 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 99.3%

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

5. Taylor expanded in r around inf 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

   1. pow-to-exp 99.3%

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

   2. rem-log-exp 99.5%

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

   3. metadata-eval 99.5%

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

   4. times-frac 99.5%

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

   5. neg-mul-1 99.5%

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

   6. div-inv 99.5%

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

   7. div-inv 99.5%

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

   8. frac-2neg 99.5%

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

   9. remove-double-neg 99.5%

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

   10. *-commutative 99.5%

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

   11. distribute-rgt-neg-in 99.5%

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

   12. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 7: 12.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(r * Float32(s * 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 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 r around 0 8.7%

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

6. Taylor expanded in s around inf 8.3%

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

   1. log1p-expm1-u 12.0%

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

   2. *-commutative 12.0%

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

8. Applied egg-rr 12.0%

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

9. Final simplification 12.0%

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

Alternative 8: 43.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in s around 0 8.7%

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

   1. associate-*r/ 8.7%

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

   2. times-frac 8.7%

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

   3. +-commutative 8.7%

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

   4. associate-*r/ 8.7%

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

   5. mul-1-neg 8.7%

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

8. Simplified 8.7%

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

9. Taylor expanded in s around inf 8.3%

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

    1. *-commutative 8.3%

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

    2. associate-*r* 8.3%

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

11. Simplified 8.3%

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

    1. log1p-expm1-u 44.8%

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

13. Applied egg-rr 44.8%

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

14. Final simplification 44.8%

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

Alternative 9: 9.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-r}{s}} \cdot 0.25}{\pi \cdot \left(r \cdot \left(s \cdot 2\right)\right)} + \frac{0.75 \cdot \left(1 + \frac{r}{s} \cdot -0.3333333333333333\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (/ (- r) s)) 0.25) (* PI (* r (* s 2.0))))
  (/ (* 0.75 (+ 1.0 (* (/ r s) -0.3333333333333333))) (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	return ((expf((-r / s)) * 0.25f) / (((float) M_PI) * (r * (s * 2.0f)))) + ((0.75f * (1.0f + ((r / s) * -0.3333333333333333f))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) * Float32(0.25)) / Float32(Float32(pi) * Float32(r * Float32(s * Float32(2.0))))) + Float32(Float32(Float32(0.75) * Float32(Float32(1.0) + Float32(Float32(r / s) * Float32(-0.3333333333333333)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) * single(0.25)) / (single(pi) * (r * (s * single(2.0))))) + ((single(0.75) * (single(1.0) + ((r / s) * single(-0.3333333333333333)))) / (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 s around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*r* 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*r* 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0 8.9%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(r \cdot \left(s \cdot 2\right)\right) \cdot \pi} + \frac{0.75 \cdot \color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
7. Step-by-step derivation

   1. *-commutative 8.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(r \cdot \left(s \cdot 2\right)\right) \cdot \pi} + \frac{0.75 \cdot \left(1 + \color{blue}{\frac{r}{s} \cdot -0.3333333333333333}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Simplified 8.9%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(r \cdot \left(s \cdot 2\right)\right) \cdot \pi} + \frac{0.75 \cdot \color{blue}{\left(1 + \frac{r}{s} \cdot -0.3333333333333333\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

9. Final simplification 8.9%

   \[\leadsto \frac{e^{\frac{-r}{s}} \cdot 0.25}{\pi \cdot \left(r \cdot \left(s \cdot 2\right)\right)} + \frac{0.75 \cdot \left(1 + \frac{r}{s} \cdot -0.3333333333333333\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
10. Add Preprocessing

Alternative 10: 9.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.9%

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

   1. associate-*r/ 8.9%

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

7. Simplified 8.9%

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

8. Final simplification 8.9%

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

Alternative 11: 9.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / s) * (single(0.5) / single(pi))) * ((exp((r / -s)) / r) + (single(1.0) / r));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.7%

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

   1. metadata-eval 8.7%

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

   2. associate-/r* 8.7%

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

   3. *-commutative 8.7%

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

   4. associate-*l* 8.7%

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

   5. associate-/l/ 8.7%

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

   6. div-inv 8.7%

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

   7. *-commutative 8.7%

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

7. Applied egg-rr 8.7%

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

   1. *-commutative 8.7%

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

   2. associate-/r* 8.7%

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

   3. metadata-eval 8.7%

      \[\leadsto \left(\frac{0.25}{s} \cdot \frac{\color{blue}{0.5}}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \]

9. Simplified 8.7%

   \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \]

10. Final simplification 8.7%

    \[\leadsto \left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \]
11. Add Preprocessing

Alternative 12: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / s) * (((exp((-r / s)) / r) + (single(1.0) / 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 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 r around 0 8.7%

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

6. Taylor expanded in s around 0 8.7%

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

   1. associate-*r/ 8.7%

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

   2. times-frac 8.7%

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

   3. +-commutative 8.7%

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

   4. associate-*r/ 8.7%

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

   5. mul-1-neg 8.7%

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

8. Simplified 8.7%

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

9. Final simplification 8.7%

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

Alternative 13: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in s around 0 8.7%

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

   1. associate-*r/ 8.7%

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

   2. times-frac 8.7%

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

   3. +-commutative 8.7%

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

   4. associate-*r/ 8.7%

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

   5. mul-1-neg 8.7%

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

8. Simplified 8.7%

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

9. Taylor expanded in r around inf 8.7%

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

    1. mul-1-neg 8.7%

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

    2. *-commutative 8.7%

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

    3. associate-*r* 8.7%

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

11. Simplified 8.7%

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

12. Final simplification 8.7%

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

Alternative 14: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) + (single(0.125) * exp((-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 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 r around 0 8.7%

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

6. Taylor expanded in r around inf 8.7%

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

   1. associate-*r/ 8.7%

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

   2. distribute-lft-in 8.7%

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

   3. metadata-eval 8.7%

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

   4. associate-*r/ 8.7%

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

   5. mul-1-neg 8.7%

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

8. Simplified 8.7%

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

9. Final simplification 8.7%

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

Alternative 15: 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 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 r around 0 8.7%

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

6. Taylor expanded in s around inf 8.3%

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

7. Final simplification 8.3%

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

Alternative 16: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in s around 0 8.7%

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

   1. associate-*r/ 8.7%

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

   2. times-frac 8.7%

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

   3. +-commutative 8.7%

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

   4. associate-*r/ 8.7%

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

   5. mul-1-neg 8.7%

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

8. Simplified 8.7%

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

9. Taylor expanded in s around inf 8.3%

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

    1. *-commutative 8.3%

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

    2. associate-*r* 8.3%

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

11. Simplified 8.3%

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

12. Taylor expanded in s around 0 8.3%

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

    1. *-commutative 8.3%

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

    2. associate-*r* 8.3%

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

    3. associate-/r* 8.3%

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

    4. rem-3cbrt-lft 8.3%

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{0.25}{s}} \cdot \sqrt[3]{\frac{0.25}{s}}\right) \cdot \sqrt[3]{\frac{0.25}{s}}}}{\pi \cdot r} \]

    5. unpow2 8.3%

       \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{0.25}{s}}\right)}^{2}} \cdot \sqrt[3]{\frac{0.25}{s}}}{\pi \cdot r} \]

    6. *-commutative 8.3%

       \[\leadsto \frac{{\left(\sqrt[3]{\frac{0.25}{s}}\right)}^{2} \cdot \sqrt[3]{\frac{0.25}{s}}}{\color{blue}{r \cdot \pi}} \]

    7. times-frac 8.3%

       \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\frac{0.25}{s}}\right)}^{2}}{r} \cdot \frac{\sqrt[3]{\frac{0.25}{s}}}{\pi}} \]

    8. associate-*l/ 8.3%

       \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\frac{0.25}{s}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{0.25}{s}}}{\pi}}{r}} \]

    9. associate-*r/ 8.3%

       \[\leadsto \frac{\color{blue}{\frac{{\left(\sqrt[3]{\frac{0.25}{s}}\right)}^{2} \cdot \sqrt[3]{\frac{0.25}{s}}}{\pi}}}{r} \]

    10. unpow2 8.3%

        \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{0.25}{s}} \cdot \sqrt[3]{\frac{0.25}{s}}\right)} \cdot \sqrt[3]{\frac{0.25}{s}}}{\pi}}{r} \]

    11. rem-3cbrt-lft 8.3%

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

14. Simplified 8.3%

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

15. Final simplification 8.3%

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

Reproduce

herbie shell --seed 2024024 
(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))))