Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 13.3s

Alternatives: 18

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around inf 99.5%

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

   1. neg-mul-1 99.5%

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

   2. rec-exp 99.5%

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

   3. associate-*r/ 99.5%

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

   4. metadata-eval 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. associate-*r* 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((r / -s)) + (single(2.71828182845904523536) ^ (r * (single(-0.3333333333333333) / s)))) / (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 98.7%

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

5. Taylor expanded in r around inf 99.4%

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

   1. associate-*r/ 99.5%

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

   2. *-commutative 99.5%

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

   3. *-un-lft-identity 99.5%

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

   4. exp-prod 99.5%

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

   5. associate-/l* 99.5%

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

7. Applied egg-rr 99.5%

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

   1. exp-1-e 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((r / -s)) + exp(((r * single(-0.3333333333333333)) / s))) / (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 98.7%

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

5. Taylor expanded in r around inf 99.4%

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

   1. associate-*r/ 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 98.7%

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

5. Taylor expanded in r around inf 99.4%

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

   1. mul-1-neg 99.4%

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

   2. distribute-frac-neg2 99.4%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 9.0%

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

   5. sqr-neg 9.0%

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

   6. sqrt-unprod 9.0%

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

   7. add-sqr-sqrt 9.0%

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

   8. frac-2neg 9.0%

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

   9. add-sqr-sqrt -0.0%

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

   10. sqrt-unprod 99.4%

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

   11. sqr-neg 99.4%

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

   12. sqrt-unprod 99.4%

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

   13. add-sqr-sqrt 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 5: 27.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp((r / (s * -single(3.0))))) / (r * (s * (single(pi) * single(6.0))))) + (single(0.125) / (r * ((s * single(pi)) + (r * (single(pi) + (single(0.5) * ((r * single(pi)) / s)))))));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around inf 99.5%

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

   1. neg-mul-1 99.5%

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

   2. rec-exp 99.5%

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

   3. associate-*r/ 99.5%

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

   4. metadata-eval 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. associate-*r* 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in r around 0 30.5%

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

10. Final simplification 30.5%

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

Alternative 6: 16.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around inf 99.5%

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

   1. neg-mul-1 99.5%

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

   2. rec-exp 99.5%

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

   3. associate-*r/ 99.5%

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

   4. metadata-eval 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. associate-*r* 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in r around 0 14.8%

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

10. Taylor expanded in s around inf 19.5%

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

    1. associate-/l* 20.6%

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

12. Simplified 20.6%

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

13. Final simplification 20.6%

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

Alternative 7: 16.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}} + \frac{0.75}{1 + \frac{r}{s} \cdot 0.3333333333333333} \cdot \frac{\frac{0.16666666666666666}{\pi}}{r \cdot s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ 0.125 (* (* r (* s PI)) (exp (/ r s))))
  (*
   (/ 0.75 (+ 1.0 (* (/ r s) 0.3333333333333333)))
   (/ (/ 0.16666666666666666 PI) (* r s)))))

C

float code(float s, float r) {
	return (0.125f / ((r * (s * ((float) M_PI))) * expf((r / s)))) + ((0.75f / (1.0f + ((r / s) * 0.3333333333333333f))) * ((0.16666666666666666f / ((float) M_PI)) / (r * s)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(Float32(r * Float32(s * Float32(pi))) * exp(Float32(r / s)))) + Float32(Float32(Float32(0.75) / Float32(Float32(1.0) + Float32(Float32(r / s) * Float32(0.3333333333333333)))) * Float32(Float32(Float32(0.16666666666666666) / Float32(pi)) / Float32(r * s))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / ((r * (s * single(pi))) * exp((r / s)))) + ((single(0.75) / (single(1.0) + ((r / s) * single(0.3333333333333333)))) * ((single(0.16666666666666666) / single(pi)) / (r * s)));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around inf 99.5%

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

   1. neg-mul-1 99.5%

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

   2. rec-exp 99.5%

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

   3. associate-*r/ 99.5%

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

   4. metadata-eval 99.5%

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

5. Simplified 99.5%

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

   1. div-inv 96.8%

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

   2. fma-define 96.8%

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

   3. associate-*l* 96.9%

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

   4. associate-/l* 96.8%

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

   5. *-commutative 96.8%

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

   6. associate-*l* 96.8%

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

   7. *-commutative 96.8%

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

7. Applied egg-rr 96.8%

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

9. Simplified 95.6%

   \[\leadsto \color{blue}{\frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}} + \frac{0.75}{\sqrt[3]{e^{\frac{r}{s}}}} \cdot \frac{\frac{0.16666666666666666}{\pi}}{r \cdot s}} \]

10. Taylor expanded in r around 0 19.2%

    \[\leadsto \frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}} + \frac{0.75}{\color{blue}{1 + 0.3333333333333333 \cdot \frac{r}{s}}} \cdot \frac{\frac{0.16666666666666666}{\pi}}{r \cdot s} \]
11. Step-by-step derivation

    1. *-commutative 19.2%

       \[\leadsto \frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}} + \frac{0.75}{1 + \color{blue}{\frac{r}{s} \cdot 0.3333333333333333}} \cdot \frac{\frac{0.16666666666666666}{\pi}}{r \cdot s} \]

12. Simplified 19.2%

    \[\leadsto \frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}} + \frac{0.75}{\color{blue}{1 + \frac{r}{s} \cdot 0.3333333333333333}} \cdot \frac{\frac{0.16666666666666666}{\pi}}{r \cdot s} \]
13. Add Preprocessing

Alternative 8: 12.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{\frac{1}{r}}{\frac{\pi \cdot \left(r + s\right)}{0.125}} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.75 (exp (/ r (* s (- 3.0))))) (* r (* s (* PI 6.0))))
  (/ (/ 1.0 r) (/ (* PI (+ r s)) 0.125))))

C

float code(float s, float r) {
	return ((0.75f * expf((r / (s * -3.0f)))) / (r * (s * (((float) M_PI) * 6.0f)))) + ((1.0f / r) / ((((float) M_PI) * (r + s)) / 0.125f));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-Float32(3.0)))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))) + Float32(Float32(Float32(1.0) / r) / Float32(Float32(Float32(pi) * Float32(r + s)) / Float32(0.125))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp((r / (s * -single(3.0))))) / (r * (s * (single(pi) * single(6.0))))) + ((single(1.0) / r) / ((single(pi) * (r + s)) / single(0.125)));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around inf 99.5%

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

   1. neg-mul-1 99.5%

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

   2. rec-exp 99.5%

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

   3. associate-*r/ 99.5%

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

   4. metadata-eval 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. associate-*r* 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in r around 0 14.8%

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

    1. clear-num 14.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{r \cdot \left(r \cdot \pi + s \cdot \pi\right)}{0.125}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    2. inv-pow 14.8%

       \[\leadsto \color{blue}{{\left(\frac{r \cdot \left(r \cdot \pi + s \cdot \pi\right)}{0.125}\right)}^{-1}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    3. distribute-rgt-out 14.8%

       \[\leadsto {\left(\frac{r \cdot \color{blue}{\left(\pi \cdot \left(r + s\right)\right)}}{0.125}\right)}^{-1} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

11. Applied egg-rr 14.8%

    \[\leadsto \color{blue}{{\left(\frac{r \cdot \left(\pi \cdot \left(r + s\right)\right)}{0.125}\right)}^{-1}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
12. Step-by-step derivation

    1. unpow-1 14.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{r \cdot \left(\pi \cdot \left(r + s\right)\right)}{0.125}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    2. associate-/l* 14.8%

       \[\leadsto \frac{1}{\color{blue}{r \cdot \frac{\pi \cdot \left(r + s\right)}{0.125}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    3. associate-/r* 14.9%

       \[\leadsto \color{blue}{\frac{\frac{1}{r}}{\frac{\pi \cdot \left(r + s\right)}{0.125}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

13. Simplified 14.9%

    \[\leadsto \color{blue}{\frac{\frac{1}{r}}{\frac{\pi \cdot \left(r + s\right)}{0.125}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

14. Final simplification 14.9%

    \[\leadsto \frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{\frac{1}{r}}{\frac{\pi \cdot \left(r + s\right)}{0.125}} \]
15. Add Preprocessing

Alternative 9: 12.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around inf 99.5%

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

   1. neg-mul-1 99.5%

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

   2. rec-exp 99.5%

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

   3. associate-*r/ 99.5%

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

   4. metadata-eval 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. associate-*r* 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in r around 0 14.8%

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

    1. div-inv 14.8%

       \[\leadsto \color{blue}{0.125 \cdot \frac{1}{r \cdot \left(r \cdot \pi + s \cdot \pi\right)}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    2. distribute-rgt-out 14.8%

       \[\leadsto 0.125 \cdot \frac{1}{r \cdot \color{blue}{\left(\pi \cdot \left(r + s\right)\right)}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

11. Applied egg-rr 14.8%

    \[\leadsto \color{blue}{0.125 \cdot \frac{1}{r \cdot \left(\pi \cdot \left(r + s\right)\right)}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
12. Step-by-step derivation

    1. associate-*r/ 14.8%

       \[\leadsto \color{blue}{\frac{0.125 \cdot 1}{r \cdot \left(\pi \cdot \left(r + s\right)\right)}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    2. metadata-eval 14.8%

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

    3. associate-*r* 14.8%

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

    4. associate-/r* 14.8%

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

13. Simplified 14.8%

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

14. Final simplification 14.8%

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

Alternative 10: 12.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around inf 99.5%

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

   1. neg-mul-1 99.5%

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

   2. rec-exp 99.5%

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

   3. associate-*r/ 99.5%

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

   4. metadata-eval 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. associate-*r* 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in r around 0 14.8%

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

    1. distribute-rgt-out 14.8%

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

11. Simplified 14.8%

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

12. Final simplification 14.8%

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

Alternative 11: 10.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * (single(1.0) / (r * single(pi)))) - (((single(0.16666666666666666) * (single(1.0) / single(pi))) - (single(0.125) * (((single(0.05555555555555555) * (r / single(pi))) + (single(0.5) * (r / single(pi)))) / s))) / s)) / 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 98.7%

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

5. Taylor expanded in s around -inf 12.8%

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

6. Final simplification 12.8%

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

Alternative 12: 10.4% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. times-frac 99.2%

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

   3. fma-define 99.2%

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

   4. associate-*l* 99.2%

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

   5. associate-/r* 99.2%

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

   6. metadata-eval 99.2%

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

   7. *-commutative 99.2%

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

   8. neg-mul-1 99.2%

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

   9. times-frac 99.1%

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

   10. metadata-eval 99.1%

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

   11. times-frac 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 12.8%

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

7. Simplified 12.8%

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

   1. unpow2 12.8%

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

9. Applied egg-rr 12.8%

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

Alternative 13: 10.4% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. times-frac 99.2%

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

   3. fma-define 99.2%

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

   4. associate-*l* 99.2%

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

   5. associate-/r* 99.2%

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

   6. metadata-eval 99.2%

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

   7. *-commutative 99.2%

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

   8. neg-mul-1 99.2%

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

   9. times-frac 99.1%

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

   10. metadata-eval 99.1%

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

   11. times-frac 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 12.8%

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

   1. mul-1-neg 12.8%

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

7. Simplified 12.8%

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

8. Final simplification 12.8%

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

Alternative 14: 9.4% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * single(pi))) - (single(0.16666666666666666) / (s * 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 98.7%

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

5. Taylor expanded in s around inf 11.6%

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

   1. associate-*r/ 11.6%

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

   2. metadata-eval 11.6%

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

   3. associate-*r/ 11.6%

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

   4. metadata-eval 11.6%

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

7. Simplified 11.6%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]
8. Add Preprocessing

Alternative 15: 9.3% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around inf 99.5%

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

   1. neg-mul-1 99.5%

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

   2. rec-exp 99.5%

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

   3. associate-*r/ 99.5%

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

   4. metadata-eval 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in s around inf 11.6%

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

   1. cancel-sign-sub-inv 11.6%

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

   2. associate-*r/ 11.6%

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

   3. metadata-eval 11.6%

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

   4. *-commutative 11.6%

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

   5. associate-/r* 11.6%

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

   6. metadata-eval 11.6%

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

   7. associate-*r/ 11.6%

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

   8. metadata-eval 11.6%

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

8. Simplified 11.6%

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

9. Final simplification 11.6%

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

Alternative 16: 9.3% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{r} \cdot \frac{1}{s \cdot \left(\pi \cdot 4\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (* (/ 1.0 r) (/ 1.0 (* s (* PI 4.0)))))

C

float code(float s, float r) {
	return (1.0f / r) * (1.0f / (s * (((float) M_PI) * 4.0f)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(1.0) / r) * Float32(Float32(1.0) / Float32(s * Float32(Float32(pi) * Float32(4.0)))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / r) * (single(1.0) / (s * (single(pi) * single(4.0))));
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 98.7%

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

5. Taylor expanded in s around inf 10.6%

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

   1. clear-num 10.6%

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

   2. inv-pow 10.6%

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

   3. associate-*r* 10.6%

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

7. Applied egg-rr 10.6%

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

   1. unpow-1 10.6%

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

   2. associate-/l* 10.6%

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

9. Simplified 10.6%

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

    1. inv-pow 10.6%

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

    2. associate-*l* 10.6%

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

    3. unpow-prod-down 10.6%

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

    4. inv-pow 10.6%

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

    5. div-inv 10.6%

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

    6. metadata-eval 10.6%

       \[\leadsto \frac{1}{r} \cdot {\left(s \cdot \left(\pi \cdot \color{blue}{4}\right)\right)}^{-1} \]

11. Applied egg-rr 10.6%

    \[\leadsto \color{blue}{\frac{1}{r} \cdot {\left(s \cdot \left(\pi \cdot 4\right)\right)}^{-1}} \]
12. Step-by-step derivation

    1. unpow-1 10.6%

       \[\leadsto \frac{1}{r} \cdot \color{blue}{\frac{1}{s \cdot \left(\pi \cdot 4\right)}} \]

13. Simplified 10.6%

    \[\leadsto \color{blue}{\frac{1}{r} \cdot \frac{1}{s \cdot \left(\pi \cdot 4\right)}} \]
14. Add Preprocessing

Alternative 17: 9.3% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 98.7%

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

5. Taylor expanded in s around inf 10.6%

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

   1. clear-num 10.6%

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

   2. inv-pow 10.6%

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

   3. associate-*r* 10.6%

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

7. Applied egg-rr 10.6%

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

   1. unpow-1 10.6%

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

   2. associate-/l* 10.6%

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

9. Simplified 10.6%

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

    1. inv-pow 10.6%

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

    2. associate-*l* 10.6%

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

    3. unpow-prod-down 10.6%

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

    4. inv-pow 10.6%

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

    5. div-inv 10.6%

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

    6. metadata-eval 10.6%

       \[\leadsto \frac{1}{r} \cdot {\left(s \cdot \left(\pi \cdot \color{blue}{4}\right)\right)}^{-1} \]

11. Applied egg-rr 10.6%

    \[\leadsto \color{blue}{\frac{1}{r} \cdot {\left(s \cdot \left(\pi \cdot 4\right)\right)}^{-1}} \]
12. Step-by-step derivation

    1. associate-*l/ 10.6%

       \[\leadsto \color{blue}{\frac{1 \cdot {\left(s \cdot \left(\pi \cdot 4\right)\right)}^{-1}}{r}} \]

    2. *-lft-identity 10.6%

       \[\leadsto \frac{\color{blue}{{\left(s \cdot \left(\pi \cdot 4\right)\right)}^{-1}}}{r} \]

    3. unpow-1 10.6%

       \[\leadsto \frac{\color{blue}{\frac{1}{s \cdot \left(\pi \cdot 4\right)}}}{r} \]

13. Simplified 10.6%

    \[\leadsto \color{blue}{\frac{\frac{1}{s \cdot \left(\pi \cdot 4\right)}}{r}} \]

14. Taylor expanded in s around 0 10.6%

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

Alternative 18: 9.3% 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 99.5%

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

3. Simplified 98.7%

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

5. Taylor expanded in s around inf 10.6%

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

Reproduce

herbie shell --seed 2024172 
(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))))