Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 16.2s

Alternatives: 13

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0 99.6%

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

   1. associate-*r* 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*r* 99.7%

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

5. Simplified 99.7%

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

   1. neg-mul-1 99.7%

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

   2. times-frac 99.7%

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

   3. metadata-eval 99.7%

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

   4. *-commutative 99.7%

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

7. Applied egg-rr 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. pow2 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(r \cdot \left(s \cdot 6\right)\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ r (- s)))) (* r (* s (* PI 2.0))))
  (/
   (* 0.75 (exp (* (/ r s) -0.3333333333333333)))
   (* PI (expm1 (log1p (* r (* s 6.0))))))))

C

float code(float s, float r) {
	return ((0.25f * expf((r / -s))) / (r * (s * (((float) M_PI) * 2.0f)))) + ((0.75f * expf(((r / s) * -0.3333333333333333f))) / (((float) M_PI) * expm1f(log1pf((r * (s * 6.0f))))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(r / Float32(-s)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(r / s) * Float32(-0.3333333333333333)))) / Float32(Float32(pi) * expm1(log1p(Float32(r * Float32(s * Float32(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%

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

   1. associate-*r* 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*r* 99.7%

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

5. Simplified 99.7%

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

   1. neg-mul-1 99.7%

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

   2. times-frac 99.7%

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

   3. metadata-eval 99.7%

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

   4. *-commutative 99.7%

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

7. Applied egg-rr 99.7%

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

   1. expm1-log1p-u 99.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(6 \cdot \left(s \cdot r\right)\right)\right)} \cdot \pi} \]

   2. expm1-undefine 21.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\color{blue}{\left(e^{\mathsf{log1p}\left(6 \cdot \left(s \cdot r\right)\right)} - 1\right)} \cdot \pi} \]

   3. *-commutative 21.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\left(e^{\mathsf{log1p}\left(\color{blue}{\left(s \cdot r\right) \cdot 6}\right)} - 1\right) \cdot \pi} \]

   4. *-commutative 21.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\left(e^{\mathsf{log1p}\left(\color{blue}{\left(r \cdot s\right)} \cdot 6\right)} - 1\right) \cdot \pi} \]

   5. associate-*l* 21.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\left(e^{\mathsf{log1p}\left(\color{blue}{r \cdot \left(s \cdot 6\right)}\right)} - 1\right) \cdot \pi} \]

9. Applied egg-rr 21.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\color{blue}{\left(e^{\mathsf{log1p}\left(r \cdot \left(s \cdot 6\right)\right)} - 1\right)} \cdot \pi} \]
10. Step-by-step derivation

    1. expm1-define 99.7%

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(r \cdot \left(s \cdot 6\right)\right)\right)} \cdot \pi} \]

11. Simplified 99.7%

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(r \cdot \left(s \cdot 6\right)\right)\right)} \cdot \pi} \]

12. Final simplification 99.7%

    \[\leadsto \frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(r \cdot \left(s \cdot 6\right)\right)\right)} \]
13. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. *-commutative 99.6%

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

   3. distribute-frac-neg 99.6%

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

   4. associate-/l* 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l* 99.6%

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

3. Simplified 99.6%

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

   1. neg-mul-1 99.6%

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

   2. *-commutative 99.6%

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

   3. times-frac 99.6%

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

   4. metadata-eval 99.6%

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

   5. *-commutative 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0 99.6%

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

   1. associate-*r* 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*r* 99.7%

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

5. Simplified 99.7%

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

   1. neg-mul-1 99.7%

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

   2. times-frac 99.7%

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

   3. metadata-eval 99.7%

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

   4. *-commutative 99.7%

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in s around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 5: 11.7% 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%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.6%

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

   3. fma-define 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 8.7%

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

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 10.5%

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

8. Applied egg-rr 10.5%

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

9. Final simplification 10.5%

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

Alternative 6: 44.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.6%

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

   3. fma-define 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 8.7%

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

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. *-commutative 8.3%

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

   2. associate-*r* 8.3%

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

8. Simplified 8.3%

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

   1. log1p-expm1-u 41.7%

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

10. Applied egg-rr 41.7%

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

11. Final simplification 41.7%

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

Alternative 7: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.6%

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

   3. fma-define 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 8.7%

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

   1. associate-/r* 8.7%

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

   2. div-inv 8.7%

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

7. Applied egg-rr 8.7%

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

8. Taylor expanded in s around 0 8.7%

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

   1. associate-/r* 8.7%

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

10. Simplified 8.7%

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

11. Taylor expanded in r around inf 8.7%

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

    1. +-commutative 8.7%

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

    2. associate-*r/ 8.7%

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

    3. metadata-eval 8.7%

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

    4. *-commutative 8.7%

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

    5. associate-*r/ 8.7%

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

    6. *-commutative 8.7%

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

    7. mul-1-neg 8.7%

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

    8. rec-exp 8.7%

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

    9. associate-*r/ 8.7%

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

    10. metadata-eval 8.7%

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

13. Simplified 8.7%

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

14. Final simplification 8.7%

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

Alternative 8: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.6%

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

   3. fma-define 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in r around inf 8.7%

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

   1. associate-*r/ 8.7%

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

   2. metadata-eval 8.7%

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

   3. +-commutative 8.7%

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

   4. associate-/r* 8.7%

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

   5. associate-*r/ 8.7%

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

   6. mul-1-neg 8.7%

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

   7. rec-exp 8.7%

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

   8. associate-*r/ 8.7%

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

   9. metadata-eval 8.7%

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

8. Simplified 8.7%

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

9. Final simplification 8.7%

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

Alternative 9: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.6%

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

   3. fma-define 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in s around 0 8.7%

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

7. Taylor expanded in r around inf 8.7%

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

   1. associate-*r/ 8.7%

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

   2. mul-1-neg 8.7%

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

   3. rec-exp 8.7%

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

   4. associate-*r/ 8.7%

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

   5. metadata-eval 8.7%

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

   6. associate-*r/ 8.7%

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

   7. metadata-eval 8.7%

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

   8. *-commutative 8.7%

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

9. Simplified 8.7%

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

10. Final simplification 8.7%

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

Alternative 10: 9.6% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.6%

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

   3. fma-define 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 8.7%

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

   1. associate-/r* 8.7%

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

   2. div-inv 8.7%

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

7. Applied egg-rr 8.7%

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

8. Taylor expanded in s around 0 8.7%

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

   1. associate-*r/ 8.7%

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

   2. *-commutative 8.7%

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

   3. associate-/r* 8.7%

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

   4. mul-1-neg 8.7%

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

   5. rec-exp 8.7%

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

   6. associate-*r/ 8.7%

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

   7. metadata-eval 8.7%

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

   8. associate-*r/ 8.7%

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

   9. metadata-eval 8.7%

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

   10. associate-/l/ 8.7%

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

10. Simplified 8.7%

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

11. Taylor expanded in r around 0 8.7%

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

12. Final simplification 8.7%

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

Alternative 11: 9.1% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{0.125}{\pi}}{r}\\ \frac{t\_0 + t\_0}{s} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ (/ 0.125 PI) r))) (/ (+ t_0 t_0) s)))

C

float code(float s, float r) {
	float t_0 = (0.125f / ((float) M_PI)) / r;
	return (t_0 + t_0) / s;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(Float32(0.125) / Float32(pi)) / r)
	return Float32(Float32(t_0 + t_0) / s)
end

MATLAB

function tmp = code(s, r)
	t_0 = (single(0.125) / single(pi)) / r;
	tmp = (t_0 + t_0) / s;
end

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.6%

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

   3. fma-define 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 8.7%

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

   1. associate-/r* 8.7%

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

   2. div-inv 8.7%

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

7. Applied egg-rr 8.7%

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

8. Taylor expanded in s around 0 8.7%

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

   1. associate-*r/ 8.7%

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

   2. *-commutative 8.7%

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

   3. associate-/r* 8.7%

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

   4. mul-1-neg 8.7%

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

   5. rec-exp 8.7%

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

   6. associate-*r/ 8.7%

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

   7. metadata-eval 8.7%

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

   8. associate-*r/ 8.7%

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

   9. metadata-eval 8.7%

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

   10. associate-/l/ 8.7%

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

10. Simplified 8.7%

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

11. Taylor expanded in r around 0 8.3%

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

12. Final simplification 8.3%

    \[\leadsto \frac{\frac{\frac{0.125}{\pi}}{r} + \frac{\frac{0.125}{\pi}}{r}}{s} \]
13. Add Preprocessing

Alternative 12: 9.1% 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%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.6%

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

   3. fma-define 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 8.7%

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

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 13: 9.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.6%

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

   3. fma-define 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in s around 0 8.7%

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

7. Taylor expanded in r around 0 8.3%

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

   1. associate-/r* 8.3%

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

9. Simplified 8.3%

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

10. Final simplification 8.3%

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

Reproduce

herbie shell --seed 2024112 
(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))))