Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 14.3s

Alternatives: 20

Speedup: 1.0×

• Rival profile vector overflowed, profile may not be complete

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

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(r / Float32(-s)))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + 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(0.25) * exp((r / -s))) / (r * (s * (single(2.0) * single(pi))))) + ((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. Final simplification 99.5%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(s \cdot \pi\right)\\ \frac{0.125}{e^{\frac{r}{s}} \cdot t\_0} + \frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{6 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* r (* s PI))))
   (+
    (/ 0.125 (* (exp (/ r s)) t_0))
    (/ (* 0.75 (exp (/ r (* s (- 3.0))))) (* 6.0 t_0)))))

C

float code(float s, float r) {
	float t_0 = r * (s * ((float) M_PI));
	return (0.125f / (expf((r / s)) * t_0)) + ((0.75f * expf((r / (s * -3.0f)))) / (6.0f * t_0));
}

Julia

function code(s, r)
	t_0 = Float32(r * Float32(s * Float32(pi)))
	return Float32(Float32(Float32(0.125) / Float32(exp(Float32(r / s)) * t_0)) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-Float32(3.0)))))) / Float32(Float32(6.0) * t_0)))
end

MATLAB

function tmp = code(s, r)
	t_0 = r * (s * single(pi));
	tmp = (single(0.125) / (exp((r / s)) * t_0)) + ((single(0.75) * exp((r / (s * -single(3.0))))) / (single(6.0) * t_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 s around 0 99.5%

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

7. Taylor expanded in r around inf 99.5%

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

   1. associate-*r* 99.5%

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

   2. associate-*l* 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*r* 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (* 0.125 (/ (exp (* (/ r s) -0.3333333333333333)) (* r PI)))
   (* 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)) / (r * ((float) M_PI)))) + (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(r * Float32(pi)))) + 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))) / (r * single(pi)))) + (single(0.125) * (single(1.0) / (r * (single(pi) * exp((r / 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. 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 0 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.5%

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

   3. fma-define 99.5%

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

   4. associate-*l* 99.5%

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

   5. associate-/r* 99.5%

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

   6. metadata-eval 99.5%

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

   7. *-commutative 99.5%

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

   8. neg-mul-1 99.5%

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

   9. times-frac 99.5%

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

   10. metadata-eval 99.5%

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

   11. times-frac 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around 0 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 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 99.5%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around inf 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 25.9% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.75 (exp (/ r (* s (- 3.0))))) (* 6.0 (* r (* s PI))))
  (/ 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)))) / (6.0f * (r * (s * ((float) M_PI))))) + (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(Float32(6.0) * Float32(r * Float32(s * Float32(pi))))) + 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))))) / (single(6.0) * (r * (s * single(pi))))) + (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 s around 0 99.5%

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

7. Taylor expanded in r around inf 99.5%

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

   1. associate-*r* 99.5%

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

   2. associate-*l* 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*r* 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in r around 0 25.8%

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

11. Final simplification 25.8%

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

Alternative 7: 15.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} + \frac{\frac{0.25}{\frac{r}{s} + 1}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.75 (exp (/ r (* s (- 3.0))))) (* 6.0 (* r (* s PI))))
  (/ (/ 0.25 (+ (/ r s) 1.0)) (* r (* s (* 2.0 PI))))))

C

float code(float s, float r) {
	return ((0.75f * expf((r / (s * -3.0f)))) / (6.0f * (r * (s * ((float) M_PI))))) + ((0.25f / ((r / s) + 1.0f)) / (r * (s * (2.0f * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-Float32(3.0)))))) / Float32(Float32(6.0) * Float32(r * Float32(s * Float32(pi))))) + Float32(Float32(Float32(0.25) / Float32(Float32(r / s) + Float32(1.0))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp((r / (s * -single(3.0))))) / (single(6.0) * (r * (s * single(pi))))) + ((single(0.25) / ((r / s) + single(1.0))) / (r * (s * (single(2.0) * single(pi)))));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 0 99.5%

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

7. Taylor expanded in r around 0 15.8%

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

   1. +-commutative 15.8%

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

9. Simplified 15.8%

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

10. Final simplification 15.8%

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

Alternative 8: 15.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(s \cdot \pi\right)\\ \frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{6 \cdot t\_0} + \frac{0.125}{t\_0 \cdot \left(\frac{r}{s} + 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* r (* s PI))))
   (+
    (/ (* 0.75 (exp (/ r (* s (- 3.0))))) (* 6.0 t_0))
    (/ 0.125 (* t_0 (+ (/ r s) 1.0))))))

C

float code(float s, float r) {
	float t_0 = r * (s * ((float) M_PI));
	return ((0.75f * expf((r / (s * -3.0f)))) / (6.0f * t_0)) + (0.125f / (t_0 * ((r / s) + 1.0f)));
}

Julia

function code(s, r)
	t_0 = Float32(r * Float32(s * Float32(pi)))
	return Float32(Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-Float32(3.0)))))) / Float32(Float32(6.0) * t_0)) + Float32(Float32(0.125) / Float32(t_0 * Float32(Float32(r / s) + Float32(1.0)))))
end

MATLAB

function tmp = code(s, r)
	t_0 = r * (s * single(pi));
	tmp = ((single(0.75) * exp((r / (s * -single(3.0))))) / (single(6.0) * t_0)) + (single(0.125) / (t_0 * ((r / s) + single(1.0))));
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 s around 0 99.5%

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

7. Taylor expanded in r around inf 99.5%

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

   1. associate-*r* 99.5%

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

   2. associate-*l* 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*r* 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in r around 0 15.8%

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

    1. +-commutative 15.8%

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

12. Simplified 15.8%

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

13. Final simplification 15.8%

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

Alternative 9: 12.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 0 99.5%

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

7. Taylor expanded in r around inf 99.5%

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

   1. associate-*r* 99.5%

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

   2. associate-*l* 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*r* 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in r around 0 11.1%

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

11. Final simplification 11.1%

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

Alternative 10: 12.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.75f * expf((r / (s * -3.0f)))) / (6.0f * (r * (s * ((float) M_PI))))) + (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(Float32(6.0) * Float32(r * Float32(s * Float32(pi))))) + 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))))) / (single(6.0) * (r * (s * single(pi))))) + (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 s around 0 99.5%

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

7. Taylor expanded in r around inf 99.5%

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

   1. associate-*r* 99.5%

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

   2. associate-*l* 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*r* 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in r around 0 11.1%

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

    1. +-commutative 11.1%

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

    2. distribute-rgt-out 11.1%

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

12. Simplified 11.1%

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

13. Final simplification 11.1%

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

Alternative 11: 9.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. times-frac 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-frac-neg 99.5%

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

   4. associate-/l* 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

   1. add-cube-cbrt 99.4%

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

   2. pow3 99.4%

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

   3. associate-*r* 99.4%

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

   4. *-commutative 99.4%

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

   5. *-commutative 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in r around 0 8.6%

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

   1. rem-cube-cbrt 8.6%

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

9. Simplified 8.6%

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

10. Final simplification 8.6%

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

Alternative 12: 9.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. *-commutative 99.5%

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

   2. associate-/r* 99.5%

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

   3. distribute-neg-frac 99.5%

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

   4. exp-cbrt 98.6%

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

   5. pow1/3 98.6%

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

   6. distribute-neg-frac2 98.6%

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

7. Applied egg-rr 98.6%

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

8. Taylor expanded in r around 0 8.5%

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

9. Final simplification 8.5%

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

Alternative 13: 9.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{e^{\frac{r}{s}}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (/ 0.25 (exp (/ r s))) (* r (* s (* 2.0 PI))))
  (/ 0.125 (* r (* s PI)))))

C

float code(float s, float r) {
	return ((0.25f / expf((r / s))) / (r * (s * (2.0f * ((float) M_PI))))) + (0.125f / (r * (s * ((float) M_PI))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / exp(Float32(r / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(0.125) / Float32(r * Float32(s * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / exp((r / s))) / (r * (s * (single(2.0) * single(pi))))) + (single(0.125) / (r * (s * single(pi))));
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. *-commutative 99.5%

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

   2. associate-/r* 99.5%

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

   3. distribute-neg-frac 99.5%

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

   4. exp-cbrt 98.6%

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

   5. pow1/3 98.6%

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

   6. distribute-neg-frac2 98.6%

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

7. Applied egg-rr 98.6%

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

8. Taylor expanded in r around 0 8.5%

   \[\leadsto \frac{\frac{0.25}{e^{\frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{0.125}{r \cdot \left(s \cdot \pi\right)}} \]

9. Final simplification 8.5%

   \[\leadsto \frac{\frac{0.25}{e^{\frac{r}{s}}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \]
10. Add Preprocessing

Alternative 14: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around 0 7.9%

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

   1. *-commutative 7.9%

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

7. Simplified 7.9%

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

8. Taylor expanded in s around 0 7.9%

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

9. Taylor expanded in s around inf 8.5%

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

Alternative 15: 9.8% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around -inf 8.5%

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

6. Final simplification 8.5%

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

Alternative 16: 9.8% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * single(pi))) - (((((r / single(pi)) * single(-0.06944444444444445)) / s) + (single(0.16666666666666666) / single(pi))) / 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 99.5%

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

   3. fma-define 99.5%

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

   4. associate-*l* 99.5%

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

   5. associate-/r* 99.5%

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

   6. metadata-eval 99.5%

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

   7. *-commutative 99.5%

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

   8. neg-mul-1 99.5%

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

   9. times-frac 99.5%

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

   10. metadata-eval 99.5%

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

   11. times-frac 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around -inf 8.5%

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

   1. mul-1-neg 8.5%

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

7. Simplified 8.5%

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

8. Final simplification 8.5%

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

Alternative 17: 9.7% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r \cdot \pi} + \frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/ 0.25 (* r PI))
   (/ (- (* 0.0625 (/ r (* s PI))) (/ 0.16666666666666666 PI)) s))
  s))

C

float code(float s, float r) {
	return ((0.25f / (r * ((float) M_PI))) + (((0.0625f * (r / (s * ((float) M_PI)))) - (0.16666666666666666f / ((float) M_PI))) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(r * Float32(pi))) + Float32(Float32(Float32(Float32(0.0625) * Float32(r / Float32(s * Float32(pi)))) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * single(pi))) + (((single(0.0625) * (r / (s * single(pi)))) - (single(0.16666666666666666) / single(pi))) / 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 99.0%

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

5. Taylor expanded in r around 0 7.9%

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

   1. *-commutative 7.9%

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

7. Simplified 7.9%

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

8. Taylor expanded in s around 0 7.9%

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

9. Taylor expanded in s around -inf 8.3%

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

    1. mul-1-neg 8.3%

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

    2. mul-1-neg 8.3%

       \[\leadsto -\frac{\color{blue}{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}\right)} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

    3. associate-*r/ 8.3%

       \[\leadsto -\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{\pi}}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

    4. metadata-eval 8.3%

       \[\leadsto -\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{\color{blue}{0.16666666666666666}}{\pi}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

    5. associate-*r/ 8.3%

       \[\leadsto -\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \color{blue}{\frac{0.25 \cdot 1}{r \cdot \pi}}}{s} \]

    6. metadata-eval 8.3%

       \[\leadsto -\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\color{blue}{0.25}}{r \cdot \pi}}{s} \]

11. Simplified 8.3%

    \[\leadsto \color{blue}{-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{r \cdot \pi}}{s}} \]

12. Final simplification 8.3%

    \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}}{s} \]
13. Add Preprocessing

Alternative 18: 8.9% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / Float32(r * 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 99.0%

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

5. Taylor expanded in s around inf 8.2%

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

   1. *-un-lft-identity 8.2%

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

   2. associate-*r* 8.2%

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

7. Applied egg-rr 8.2%

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

   1. *-un-lft-identity 8.2%

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

   2. associate-/r* 8.2%

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

9. Applied egg-rr 8.2%

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

Alternative 19: 8.9% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 8.2%

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

   1. *-un-lft-identity 8.2%

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

   2. associate-*r* 8.2%

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

7. Applied egg-rr 8.2%

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

   1. *-un-lft-identity 8.2%

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

   2. associate-/r* 8.2%

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

9. Applied egg-rr 8.2%

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

    1. associate-/l/ 8.2%

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

11. Simplified 8.2%

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

Alternative 20: 8.9% 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 99.0%

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

5. Taylor expanded in s around inf 8.2%

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

Reproduce

herbie shell --seed 2024155 
(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))))