Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 12.7s

Alternatives: 21

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf 99.7%

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

   1. neg-mul-1 99.7%

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

   2. rec-exp 99.7%

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

   3. associate-*r/ 99.7%

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

   4. metadata-eval 99.7%

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

5. Simplified 99.7%

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

   1. +-commutative 99.7%

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

   2. times-frac 99.7%

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

   3. fma-define 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. distribute-frac-neg 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-/l/ 99.8%

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

   9. associate-*l* 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf 99.7%

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

   1. neg-mul-1 99.7%

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

   2. rec-exp 99.7%

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

   3. associate-*r/ 99.7%

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

   4. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in s around 0 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf 99.7%

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

   1. neg-mul-1 99.7%

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

   2. rec-exp 99.7%

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

   3. associate-*r/ 99.7%

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

   4. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in s around 0 99.8%

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

7. Taylor expanded in r around inf 99.8%

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

   1. associate-*r* 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + ((single(2.71828182845904523536) ^ ((r * single(-0.3333333333333333)) / s)) / r));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

   1. pow-exp 99.7%

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

   2. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

8. Applied egg-rr 99.7%

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

   1. exp-1-e 99.7%

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

   2. associate-*l/ 99.7%

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

10. Simplified 99.7%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

   1. pow-exp 99.7%

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

   2. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in r around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l/ 99.7%

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

9. Simplified 99.7%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around inf 99.7%

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

6. Final simplification 99.7%

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

Alternative 7: 97.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

   1. pow-exp 99.7%

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

   2. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in r around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l/ 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in r around inf 99.7%

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

    1. associate-*r/ 99.7%

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

    2. associate-*r* 99.7%

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

    3. times-frac 98.6%

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

    4. mul-1-neg 98.6%

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

    5. distribute-frac-neg2 98.6%

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

    6. associate-*r/ 98.6%

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

12. Simplified 98.6%

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

13. Final simplification 98.6%

    \[\leadsto \frac{0.125}{s \cdot r} \cdot \frac{e^{\frac{r}{-s}} + e^{\frac{r \cdot -0.3333333333333333}{s}}}{\pi} \]
14. Add Preprocessing

Alternative 8: 44.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 9.6%

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

   1. pow1 9.6%

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

   2. *-commutative 9.6%

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

7. Applied egg-rr 9.6%

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

   1. unpow1 9.6%

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

   2. *-commutative 9.6%

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

   3. *-commutative 9.6%

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

   4. associate-*l* 9.6%

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

9. Simplified 9.6%

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

    1. log1p-expm1-u 44.6%

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

11. Applied egg-rr 44.6%

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

Alternative 9: 11.9% 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 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 9.6%

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

   1. log1p-expm1-u 12.2%

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

   2. *-commutative 12.2%

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

7. Applied egg-rr 12.2%

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

8. Final simplification 12.2%

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

Alternative 10: 10.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \frac{1}{r \cdot \left(\pi \cdot e^{\frac{r}{s}}\right)} + 0.125 \cdot \left(\frac{0.3333333333333333 \cdot \frac{-1}{\pi} - -0.05555555555555555 \cdot \frac{r}{s \cdot \pi}}{s} + \frac{1}{\pi \cdot r}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (* 0.125 (/ 1.0 (* r (* PI (exp (/ r s))))))
   (*
    0.125
    (+
     (/
      (-
       (* 0.3333333333333333 (/ -1.0 PI))
       (* -0.05555555555555555 (/ r (* s PI))))
      s)
     (/ 1.0 (* PI r)))))
  s))

C

float code(float s, float r) {
	return ((0.125f * (1.0f / (r * (((float) M_PI) * expf((r / s)))))) + (0.125f * ((((0.3333333333333333f * (-1.0f / ((float) M_PI))) - (-0.05555555555555555f * (r / (s * ((float) M_PI))))) / s) + (1.0f / (((float) M_PI) * r))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) * Float32(Float32(1.0) / Float32(r * Float32(Float32(pi) * exp(Float32(r / s)))))) + Float32(Float32(0.125) * Float32(Float32(Float32(Float32(Float32(0.3333333333333333) * Float32(Float32(-1.0) / Float32(pi))) - Float32(Float32(-0.05555555555555555) * Float32(r / Float32(s * Float32(pi))))) / s) + Float32(Float32(1.0) / Float32(Float32(pi) * r))))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) * (single(1.0) / (r * (single(pi) * exp((r / s)))))) + (single(0.125) * ((((single(0.3333333333333333) * (single(-1.0) / single(pi))) - (single(-0.05555555555555555) * (r / (s * single(pi))))) / s) + (single(1.0) / (single(pi) * r))))) / s;
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf 99.7%

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

   1. neg-mul-1 99.7%

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

   2. rec-exp 99.7%

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

   3. associate-*r/ 99.7%

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

   4. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in s around 0 99.8%

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

7. Taylor expanded in s around -inf 11.6%

   \[\leadsto \frac{0.125 \cdot \color{blue}{\left(-1 \cdot \frac{-0.05555555555555555 \cdot \frac{r}{s \cdot \pi} + 0.3333333333333333 \cdot \frac{1}{\pi}}{s} + \frac{1}{r \cdot \pi}\right)} + 0.125 \cdot \frac{1}{r \cdot \left(\pi \cdot e^{\frac{r}{s}}\right)}}{s} \]

8. Final simplification 11.6%

   \[\leadsto \frac{0.125 \cdot \frac{1}{r \cdot \left(\pi \cdot e^{\frac{r}{s}}\right)} + 0.125 \cdot \left(\frac{0.3333333333333333 \cdot \frac{-1}{\pi} - -0.05555555555555555 \cdot \frac{r}{s \cdot \pi}}{s} + \frac{1}{\pi \cdot r}\right)}{s} \]
9. Add Preprocessing

Alternative 11: 10.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi \cdot r} + 0.125 \cdot \left(\frac{\frac{1}{r}}{\pi} - \frac{\frac{1}{\pi} + \frac{r}{s \cdot \pi} \cdot -0.5}{s}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (* 0.125 (/ (exp (* (/ r s) -0.3333333333333333)) (* PI r)))
   (* 0.125 (- (/ (/ 1.0 r) PI) (/ (+ (/ 1.0 PI) (* (/ r (* s PI)) -0.5)) s))))
  s))

C

float code(float s, float r) {
	return ((0.125f * (expf(((r / s) * -0.3333333333333333f)) / (((float) M_PI) * r))) + (0.125f * (((1.0f / r) / ((float) M_PI)) - (((1.0f / ((float) M_PI)) + ((r / (s * ((float) M_PI))) * -0.5f)) / s)))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(Float32(pi) * r))) + Float32(Float32(0.125) * Float32(Float32(Float32(Float32(1.0) / r) / Float32(pi)) - Float32(Float32(Float32(Float32(1.0) / Float32(pi)) + Float32(Float32(r / Float32(s * Float32(pi))) * Float32(-0.5))) / s)))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) * (exp(((r / s) * single(-0.3333333333333333))) / (single(pi) * r))) + (single(0.125) * (((single(1.0) / r) / single(pi)) - (((single(1.0) / single(pi)) + ((r / (s * single(pi))) * single(-0.5))) / s)))) / s;
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf 99.7%

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

   1. neg-mul-1 99.7%

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

   2. rec-exp 99.7%

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

   3. associate-*r/ 99.7%

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

   4. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in s around 0 99.8%

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

7. Taylor expanded in s around -inf 10.8%

   \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{r}{s \cdot \pi} + \left(0.5 \cdot \frac{r}{s \cdot \pi} + \frac{1}{\pi}\right)}{s} + \frac{1}{r \cdot \pi}\right)}}{s} \]
8. Step-by-step derivation

   1. +-commutative 10.8%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \color{blue}{\left(\frac{1}{r \cdot \pi} + -1 \cdot \frac{-1 \cdot \frac{r}{s \cdot \pi} + \left(0.5 \cdot \frac{r}{s \cdot \pi} + \frac{1}{\pi}\right)}{s}\right)}}{s} \]

   2. mul-1-neg 10.8%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \left(\frac{1}{r \cdot \pi} + \color{blue}{\left(-\frac{-1 \cdot \frac{r}{s \cdot \pi} + \left(0.5 \cdot \frac{r}{s \cdot \pi} + \frac{1}{\pi}\right)}{s}\right)}\right)}{s} \]

   3. unsub-neg 10.8%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \color{blue}{\left(\frac{1}{r \cdot \pi} - \frac{-1 \cdot \frac{r}{s \cdot \pi} + \left(0.5 \cdot \frac{r}{s \cdot \pi} + \frac{1}{\pi}\right)}{s}\right)}}{s} \]

   4. associate-/r* 10.9%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \left(\color{blue}{\frac{\frac{1}{r}}{\pi}} - \frac{-1 \cdot \frac{r}{s \cdot \pi} + \left(0.5 \cdot \frac{r}{s \cdot \pi} + \frac{1}{\pi}\right)}{s}\right)}{s} \]

   5. associate-+r+ 10.9%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \left(\frac{\frac{1}{r}}{\pi} - \frac{\color{blue}{\left(-1 \cdot \frac{r}{s \cdot \pi} + 0.5 \cdot \frac{r}{s \cdot \pi}\right) + \frac{1}{\pi}}}{s}\right)}{s} \]

   6. distribute-rgt-out 11.0%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \left(\frac{\frac{1}{r}}{\pi} - \frac{\color{blue}{\frac{r}{s \cdot \pi} \cdot \left(-1 + 0.5\right)} + \frac{1}{\pi}}{s}\right)}{s} \]

   7. metadata-eval 11.0%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \left(\frac{\frac{1}{r}}{\pi} - \frac{\frac{r}{s \cdot \pi} \cdot \color{blue}{-0.5} + \frac{1}{\pi}}{s}\right)}{s} \]

9. Simplified 11.0%

   \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \color{blue}{\left(\frac{\frac{1}{r}}{\pi} - \frac{\frac{r}{s \cdot \pi} \cdot -0.5 + \frac{1}{\pi}}{s}\right)}}{s} \]

10. Final simplification 11.0%

    \[\leadsto \frac{0.125 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi \cdot r} + 0.125 \cdot \left(\frac{\frac{1}{r}}{\pi} - \frac{\frac{1}{\pi} + \frac{r}{s \cdot \pi} \cdot -0.5}{s}\right)}{s} \]
11. Add Preprocessing

Alternative 12: 10.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{r \cdot \left(0.06944444444444445 \cdot \frac{r}{\pi \cdot {s}^{2}} + 0.16666666666666666 \cdot \frac{-1}{s \cdot \pi}\right) + 0.25 \cdot \frac{1}{\pi}}{r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (/
   (+
    (*
     r
     (+
      (* 0.06944444444444445 (/ r (* PI (pow s 2.0))))
      (* 0.16666666666666666 (/ -1.0 (* s PI)))))
    (* 0.25 (/ 1.0 PI)))
   r)
  s))

C

float code(float s, float r) {
	return (((r * ((0.06944444444444445f * (r / (((float) M_PI) * powf(s, 2.0f)))) + (0.16666666666666666f * (-1.0f / (s * ((float) M_PI)))))) + (0.25f * (1.0f / ((float) M_PI)))) / r) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(r * Float32(Float32(Float32(0.06944444444444445) * Float32(r / Float32(Float32(pi) * (s ^ Float32(2.0))))) + Float32(Float32(0.16666666666666666) * Float32(Float32(-1.0) / Float32(s * Float32(pi)))))) + Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(pi)))) / r) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (((r * ((single(0.06944444444444445) * (r / (single(pi) * (s ^ single(2.0))))) + (single(0.16666666666666666) * (single(-1.0) / (s * single(pi)))))) + (single(0.25) * (single(1.0) / single(pi)))) / r) / s;
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf 99.7%

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

   1. neg-mul-1 99.7%

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

   2. rec-exp 99.7%

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

   3. associate-*r/ 99.7%

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

   4. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in s around 0 99.8%

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

7. Taylor expanded in r around 0 11.0%

   \[\leadsto \frac{\color{blue}{\frac{r \cdot \left(0.06944444444444445 \cdot \frac{r}{{s}^{2} \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}\right) + 0.25 \cdot \frac{1}{\pi}}{r}}}{s} \]

8. Final simplification 11.0%

   \[\leadsto \frac{\frac{r \cdot \left(0.06944444444444445 \cdot \frac{r}{\pi \cdot {s}^{2}} + 0.16666666666666666 \cdot \frac{-1}{s \cdot \pi}\right) + 0.25 \cdot \frac{1}{\pi}}{r}}{s} \]
9. Add Preprocessing

Alternative 13: 10.1% accurate, 11.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.25f / (((float) M_PI) * r)) - (((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(Float32(pi) * r)) - 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) / (single(pi) * r)) - (((single(0.16666666666666666) / single(pi)) + (((r / single(pi)) * single(-0.06944444444444445)) / s)) / s)) / s;
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf 99.7%

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

   1. neg-mul-1 99.7%

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

   2. rec-exp 99.7%

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

   3. associate-*r/ 99.7%

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

   4. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in s around 0 99.8%

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

7. Taylor expanded in s around -inf 11.0%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-0.125 \cdot \left(-1 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}\right) + 0.006944444444444444 \cdot \frac{r}{\pi}}{s} + 0.16666666666666666 \cdot \frac{1}{\pi}}{s} + 0.25 \cdot \frac{1}{r \cdot \pi}}}{s} \]

9. Simplified 11.0%

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

10. Final simplification 11.0%

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

Alternative 14: 9.2% accurate, 13.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (((single(0.25) * (single(1.0) / single(pi))) + ((r / (s * single(pi))) * single(-0.16666666666666666))) / r) / s;
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf 99.7%

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

   1. neg-mul-1 99.7%

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

   2. rec-exp 99.7%

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

   3. associate-*r/ 99.7%

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

   4. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in s around 0 99.8%

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

7. Taylor expanded in r around 0 10.2%

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

8. Final simplification 10.2%

   \[\leadsto \frac{\frac{0.25 \cdot \frac{1}{\pi} + \frac{r}{s \cdot \pi} \cdot -0.16666666666666666}{r}}{s} \]
9. Add Preprocessing

Alternative 15: 9.2% accurate, 15.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 10.2%

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

6. Taylor expanded in r around inf 10.2%

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

   1. associate-*r/ 10.2%

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

   2. metadata-eval 10.2%

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

   3. associate-*r/ 10.2%

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

   4. metadata-eval 10.2%

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

   5. *-commutative 10.2%

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

8. Simplified 10.2%

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

9. Taylor expanded in s around inf 10.2%

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

    1. associate-*r/ 10.2%

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

    2. metadata-eval 10.2%

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

    3. *-commutative 10.2%

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

    4. associate-/r* 10.2%

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

    5. associate-*r/ 10.2%

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

    6. metadata-eval 10.2%

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

11. Simplified 10.2%

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

12. Taylor expanded in s around 0 10.2%

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

    1. associate-*r/ 10.2%

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

    2. associate-*r/ 10.2%

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

    3. metadata-eval 10.2%

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

14. Simplified 10.2%

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

15. Final simplification 10.2%

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

Alternative 16: 9.2% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 10.2%

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

6. Taylor expanded in r around inf 10.2%

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

   1. associate-*r/ 10.2%

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

   2. metadata-eval 10.2%

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

   3. associate-*r/ 10.2%

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

   4. metadata-eval 10.2%

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

   5. *-commutative 10.2%

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

8. Simplified 10.2%

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

9. Taylor expanded in s around inf 10.2%

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

    1. associate-*r/ 10.2%

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

    2. metadata-eval 10.2%

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

    3. *-commutative 10.2%

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

    4. associate-/r* 10.2%

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

    5. associate-*r/ 10.2%

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

    6. metadata-eval 10.2%

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

11. Simplified 10.2%

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

Alternative 17: 9.2% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 10.2%

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

   1. associate-*r/ 10.2%

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

   2. metadata-eval 10.2%

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

   3. associate-*r/ 10.2%

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

   4. metadata-eval 10.2%

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

7. Simplified 10.2%

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

8. Final simplification 10.2%

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

Alternative 18: 9.1% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{r}{\frac{0.25}{\pi}}}}{s} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ (/ 1.0 (/ r (/ 0.25 PI))) s))

C

float code(float s, float r) {
	return (1.0f / (r / (0.25f / ((float) M_PI)))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(1.0) / Float32(r / Float32(Float32(0.25) / Float32(pi)))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / (r / (single(0.25) / single(pi)))) / s;
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 9.6%

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

   1. *-commutative 9.6%

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

   2. associate-*l* 9.6%

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

   3. *-commutative 9.6%

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

   4. associate-/l/ 9.6%

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

7. Simplified 9.6%

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

   1. associate-/l/ 9.6%

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

   2. clear-num 9.6%

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

   3. inv-pow 9.6%

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

9. Applied egg-rr 9.6%

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

    1. unpow-1 9.6%

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

11. Simplified 9.6%

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{r}{\frac{0.25}{\pi}}}}}{s} \]
12. Add Preprocessing

Alternative 19: 9.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 10.2%

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

6. Taylor expanded in r around 0 9.6%

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

   1. associate-*r* 9.6%

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

   2. *-commutative 9.6%

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

   3. associate-/r* 9.6%

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

8. Simplified 9.6%

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

Alternative 20: 9.1% accurate, 33.0× speedup

Math

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

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(0.25) / Float32(s * Float32(Float32(pi) * r)))
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 9.6%

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

   1. pow1 9.6%

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

   2. *-commutative 9.6%

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

7. Applied egg-rr 9.6%

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

   1. unpow1 9.6%

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

   2. *-commutative 9.6%

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

   3. *-commutative 9.6%

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

   4. associate-*l* 9.6%

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

9. Simplified 9.6%

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

Alternative 21: 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 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 9.6%

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

Reproduce

herbie shell --seed 2024137 
(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))))