Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 21.4s

Alternatives: 17

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return fma(Float32(Float32(Float32(0.125) / Float32(pi)) / s), Float32((Float32(exp(1)) ^ Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / r), Float32(Float32(Float32(0.125) / Float32(Float32(pi) * s)) * Float32(exp(Float32(-Float32(r / s))) / r)))
end

Derivation

1. Initial program 99.5%

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

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. *-un-lft-identity 99.5%

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

   2. exp-prod 99.6%

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

   3. rem-log-exp 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-*l/ 99.4%

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

   6. rem-log-exp 99.6%

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

6. Applied egg-rr 99.6%

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

   1. clear-num 99.6%

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

   2. inv-pow 99.6%

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

   3. associate-/l* 99.6%

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

8. Applied egg-rr 99.6%

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

   1. unpow-1 99.6%

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

   2. metadata-eval 99.6%

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

   3. associate-*r/ 99.6%

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

   4. associate-/r/ 99.6%

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

   5. associate-*r/ 99.6%

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

   6. metadata-eval 99.6%

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

10. Simplified 99.6%

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

11. Taylor expanded in s around 0 99.6%

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

    1. associate-/l/ 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

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 99.6%

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

   5. metadata-eval 99.6%

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

   6. div-inv 99.6%

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

   7. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{\pi \cdot s}\\ \mathsf{fma}\left(t_0, \frac{{e}^{\left(\frac{r \cdot -0.3333333333333333}{s}\right)}}{r}, t_0 \cdot \frac{e^{-\frac{r}{s}}}{r}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* PI s))))
   (fma
    t_0
    (/ (pow E (/ (* r -0.3333333333333333) s)) r)
    (* t_0 (/ (exp (- (/ r s))) r)))))

C

float code(float s, float r) {
	float t_0 = 0.125f / (((float) M_PI) * s);
	return fmaf(t_0, (powf(((float) M_E), ((r * -0.3333333333333333f) / s)) / r), (t_0 * (expf(-(r / s)) / r)));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(Float32(pi) * s))
	return fma(t_0, Float32((Float32(exp(1)) ^ Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / r), Float32(t_0 * Float32(exp(Float32(-Float32(r / s))) / r)))
end

Derivation

1. Initial program 99.5%

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

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. *-un-lft-identity 99.5%

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

   2. exp-prod 99.6%

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

   3. rem-log-exp 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-*l/ 99.4%

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

   6. rem-log-exp 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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%

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

Alternative 5: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 99.4%

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

   2. associate-*r/ 99.3%

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

   3. *-commutative 99.3%

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

   4. rem-log-exp 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. times-frac 99.6%

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

   3. /-rgt-identity 99.6%

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 6: 11.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

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 9.3%

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

6. Taylor expanded in s around inf 8.9%

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

   1. log1p-expm1-u 11.7%

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

8. Applied egg-rr 11.7%

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

9. Final simplification 11.7%

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

Alternative 7: 43.2% 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 99.5%

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

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 9.3%

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

6. Taylor expanded in s around inf 8.9%

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

   1. expm1-log1p-u 8.9%

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

   2. expm1-udef 7.2%

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

8. Applied egg-rr 7.2%

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

   1. expm1-def 8.9%

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

   2. expm1-log1p 8.9%

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

   3. *-commutative 8.9%

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

   4. associate-*l* 8.9%

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

10. Simplified 8.9%

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

    1. log1p-expm1-u 44.8%

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

12. Applied egg-rr 44.8%

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

13. Final simplification 44.8%

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

Alternative 8: 9.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.7%

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

6. Final simplification 9.7%

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

Alternative 9: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.3%

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

6. Taylor expanded in r around -inf 9.3%

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

   1. associate-*r/ 9.3%

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

   2. associate-*r* 9.3%

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

   3. times-frac 9.3%

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

   4. sub-neg 9.3%

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

   5. metadata-eval 9.3%

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

   6. +-commutative 9.3%

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

   7. mul-1-neg 9.3%

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

   8. rec-exp 9.3%

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

   9. associate-*r/ 9.3%

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

   10. metadata-eval 9.3%

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

8. Simplified 9.3%

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

   1. expm1-log1p-u 1.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.125}{r \cdot s}\right)\right)} \cdot \frac{-1 + \frac{-1}{e^{\frac{r}{s}}}}{\pi} \]

   2. expm1-udef 2.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.125}{r \cdot s}\right)} - 1\right)} \cdot \frac{-1 + \frac{-1}{e^{\frac{r}{s}}}}{\pi} \]

   3. *-commutative 2.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{-0.125}{\color{blue}{s \cdot r}}\right)} - 1\right) \cdot \frac{-1 + \frac{-1}{e^{\frac{r}{s}}}}{\pi} \]

   4. associate-/r* 2.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{-0.125}{s}}{r}}\right)} - 1\right) \cdot \frac{-1 + \frac{-1}{e^{\frac{r}{s}}}}{\pi} \]

10. Applied egg-rr 2.0%

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

    1. expm1-def 1.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-0.125}{s}}{r}\right)\right)} \cdot \frac{-1 + \frac{-1}{e^{\frac{r}{s}}}}{\pi} \]

    2. expm1-log1p 9.3%

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

    3. associate-/r* 9.3%

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

    4. associate-/l/ 9.3%

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

12. Simplified 9.3%

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

13. Final simplification 9.3%

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

Alternative 10: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.3%

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

6. Taylor expanded in r around -inf 9.3%

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

   1. associate-*r/ 9.3%

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

   2. associate-*r* 9.3%

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

   3. times-frac 9.3%

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

   4. sub-neg 9.3%

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

   5. metadata-eval 9.3%

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

   6. +-commutative 9.3%

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

   7. mul-1-neg 9.3%

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

   8. rec-exp 9.3%

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

   9. associate-*r/ 9.3%

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

   10. metadata-eval 9.3%

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

8. Simplified 9.3%

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

9. Taylor expanded in r around inf 9.3%

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

    1. rec-exp 9.3%

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

    2. distribute-frac-neg 9.3%

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

    3. *-commutative 9.3%

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

11. Simplified 9.3%

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

12. Final simplification 9.3%

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

Alternative 11: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.3%

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

6. Taylor expanded in r around inf 9.3%

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

   1. associate-*r/ 9.3%

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

   2. *-commutative 9.3%

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

   3. times-frac 9.3%

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

   4. mul-1-neg 9.3%

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

   5. distribute-neg-frac 9.3%

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

8. Simplified 9.3%

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

9. Final simplification 9.3%

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

Alternative 12: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.3%

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

6. Taylor expanded in r around -inf 9.3%

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

   1. associate-*r/ 9.3%

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

   2. associate-*r* 9.3%

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

   3. times-frac 9.3%

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

   4. sub-neg 9.3%

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

   5. metadata-eval 9.3%

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

   6. +-commutative 9.3%

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

   7. mul-1-neg 9.3%

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

   8. rec-exp 9.3%

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

   9. associate-*r/ 9.3%

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

   10. metadata-eval 9.3%

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

8. Simplified 9.3%

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

9. Taylor expanded in r around inf 9.3%

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

    1. rec-exp 9.3%

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

    2. associate-*r/ 9.3%

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

    3. *-commutative 9.3%

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

    4. distribute-lft-in 9.3%

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

    5. metadata-eval 9.3%

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

    6. rec-exp 9.3%

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

    7. associate-*r/ 9.3%

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

    8. metadata-eval 9.3%

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

    9. associate-*l* 9.3%

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

11. Simplified 9.3%

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

12. Final simplification 9.3%

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

Alternative 13: 9.4% accurate, 13.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.3%

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

6. Taylor expanded in r around -inf 9.3%

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

   1. associate-*r/ 9.3%

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

   2. associate-*r* 9.3%

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

   3. times-frac 9.3%

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

   4. sub-neg 9.3%

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

   5. metadata-eval 9.3%

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

   6. +-commutative 9.3%

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

   7. mul-1-neg 9.3%

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

   8. rec-exp 9.3%

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

   9. associate-*r/ 9.3%

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

   10. metadata-eval 9.3%

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

8. Simplified 9.3%

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

9. Taylor expanded in r around 0 9.3%

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

10. Final simplification 9.3%

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

Alternative 14: 9.4% accurate, 13.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.3%

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

6. Taylor expanded in r around -inf 9.3%

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

   1. associate-*r/ 9.3%

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

   2. associate-*r* 9.3%

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

   3. times-frac 9.3%

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

   4. sub-neg 9.3%

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

   5. metadata-eval 9.3%

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

   6. +-commutative 9.3%

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

   7. mul-1-neg 9.3%

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

   8. rec-exp 9.3%

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

   9. associate-*r/ 9.3%

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

   10. metadata-eval 9.3%

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

8. Simplified 9.3%

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

   1. expm1-log1p-u 1.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.125}{r \cdot s}\right)\right)} \cdot \frac{-1 + \frac{-1}{e^{\frac{r}{s}}}}{\pi} \]

   2. expm1-udef 2.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.125}{r \cdot s}\right)} - 1\right)} \cdot \frac{-1 + \frac{-1}{e^{\frac{r}{s}}}}{\pi} \]

   3. *-commutative 2.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{-0.125}{\color{blue}{s \cdot r}}\right)} - 1\right) \cdot \frac{-1 + \frac{-1}{e^{\frac{r}{s}}}}{\pi} \]

   4. associate-/r* 2.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{-0.125}{s}}{r}}\right)} - 1\right) \cdot \frac{-1 + \frac{-1}{e^{\frac{r}{s}}}}{\pi} \]

10. Applied egg-rr 2.0%

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

    1. expm1-def 1.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-0.125}{s}}{r}\right)\right)} \cdot \frac{-1 + \frac{-1}{e^{\frac{r}{s}}}}{\pi} \]

    2. expm1-log1p 9.3%

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

    3. associate-/r* 9.3%

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

    4. associate-/l/ 9.3%

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

12. Simplified 9.3%

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

13. Taylor expanded in r around 0 9.3%

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

14. Final simplification 9.3%

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

Alternative 15: 9.0% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.3%

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

6. Taylor expanded in r around -inf 9.3%

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

   1. associate-*r/ 9.3%

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

   2. associate-*r* 9.3%

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

   3. times-frac 9.3%

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

   4. sub-neg 9.3%

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

   5. metadata-eval 9.3%

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

   6. +-commutative 9.3%

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

   7. mul-1-neg 9.3%

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

   8. rec-exp 9.3%

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

   9. associate-*r/ 9.3%

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

   10. metadata-eval 9.3%

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

8. Simplified 9.3%

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

   1. associate-*l/ 9.3%

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

   2. associate-/r* 9.3%

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

   3. +-commutative 9.3%

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

   4. div-inv 9.3%

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

   5. fma-def 9.3%

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

   6. rec-exp 9.3%

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

10. Applied egg-rr 9.3%

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

11. Taylor expanded in r around 0 8.9%

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

12. Final simplification 8.9%

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

Alternative 16: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.3%

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

6. Taylor expanded in s around inf 8.9%

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

7. Final simplification 8.9%

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

Alternative 17: 8.9% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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 9.3%

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

6. Taylor expanded in r around -inf 9.3%

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

   1. associate-*r/ 9.3%

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

   2. associate-*r* 9.3%

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

   3. times-frac 9.3%

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

   4. sub-neg 9.3%

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

   5. metadata-eval 9.3%

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

   6. +-commutative 9.3%

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

   7. mul-1-neg 9.3%

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

   8. rec-exp 9.3%

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

   9. associate-*r/ 9.3%

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

   10. metadata-eval 9.3%

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

8. Simplified 9.3%

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

   1. associate-*l/ 9.3%

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

   2. associate-/r* 9.3%

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

   3. +-commutative 9.3%

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

   4. div-inv 9.3%

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

   5. fma-def 9.3%

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

   6. rec-exp 9.3%

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

10. Applied egg-rr 9.3%

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

11. Taylor expanded in r around 0 8.9%

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

    1. *-commutative 8.9%

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

13. Simplified 8.9%

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

14. Final simplification 8.9%

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

Reproduce

herbie shell --seed 2024017 
(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))))