Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 17.9s

Alternatives: 12

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ r (- s)))) (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (exp (/ r (* s (- 3.0))))) (* (* PI (* r s)) 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)))) / ((((float) M_PI) * (r * s)) * 6.0f));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(r / Float32(-s)))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-Float32(3.0)))))) / Float32(Float32(Float32(pi) * Float32(r * s)) * 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))))) / ((single(pi) * (r * s)) * single(6.0)));
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 s around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*r* 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-/l* 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. times-frac 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-/l* 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. times-frac 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*l* 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \frac{e^{\frac{r}{-s}}}{\pi \cdot \left(2 \cdot \left(r \cdot s\right)\right)} + 0.125 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

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

C

float code(float s, float r) {
	return (0.25f * (expf((r / -s)) / (((float) M_PI) * (2.0f * (r * s))))) + (0.125f * (expf(((r / s) * -0.3333333333333333f)) / (r * (s * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) * Float32(exp(Float32(r / Float32(-s))) / Float32(Float32(pi) * Float32(Float32(2.0) * Float32(r * s))))) + Float32(Float32(0.125) * Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(r * Float32(s * Float32(pi))))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) * (exp((r / -s)) / (single(pi) * (single(2.0) * (r * s))))) + (single(0.125) * (exp(((r / s) * single(-0.3333333333333333))) / (r * (s * single(pi)))));
end

Derivation

1. Initial program 99.7%

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

   1. associate-/l* 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. times-frac 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around 0 99.7%

   \[\leadsto 0.25 \cdot \frac{e^{-\frac{r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
6. Step-by-step derivation

   1. associate-/r* 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in r around 0 99.7%

   \[\leadsto 0.25 \cdot \frac{e^{-\frac{r}{s}}}{\color{blue}{2 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)}} + \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
9. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*l* 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in s around 0 99.7%

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

12. Final simplification 99.7%

    \[\leadsto 0.25 \cdot \frac{e^{\frac{r}{-s}}}{\pi \cdot \left(2 \cdot \left(r \cdot s\right)\right)} + 0.125 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(s \cdot \pi\right)} \]
13. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.75) * Float32(Float32(0.16666666666666666) / 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(0.75) * (single(0.16666666666666666) / 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 99.7%

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

   1. metadata-eval 99.7%

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

   2. associate-/r* 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

   7. div-inv 99.7%

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

   8. *-commutative 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-*l* 99.6%

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

   11. *-commutative 99.6%

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

   12. associate-/r* 99.7%

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

   13. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/r* 99.7%

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

   3. associate-*l/ 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

   \[\leadsto \frac{0.75 \cdot \frac{0.16666666666666666}{s}}{\pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}\right) \]
10. Add Preprocessing

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

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(r * Float32(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 99.7%

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

5. Taylor expanded in r around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 7: 9.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.8%

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

   2. fma-define 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. *-commutative 99.8%

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

   7. distribute-frac-neg 99.8%

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

   8. distribute-frac-neg2 99.8%

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

   9. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 9.2%

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

   1. add0 9.2%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{r}{-s}}}{r}, \color{blue}{\frac{0.125}{r \cdot \left(s \cdot \pi\right)} + 0}\right) \]

   2. *-commutative 9.2%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{r}{-s}}}{r}, \frac{0.125}{\color{blue}{\left(s \cdot \pi\right) \cdot r}} + 0\right) \]

   3. associate-/r* 9.2%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{r}{-s}}}{r}, \color{blue}{\frac{\frac{0.125}{s \cdot \pi}}{r}} + 0\right) \]

   4. associate-/r* 9.2%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{r}{-s}}}{r}, \frac{\color{blue}{\frac{\frac{0.125}{s}}{\pi}}}{r} + 0\right) \]

7. Applied egg-rr 9.2%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{r}{-s}}}{r}, \color{blue}{\frac{\frac{\frac{0.125}{s}}{\pi}}{r} + 0}\right) \]
8. Step-by-step derivation

   1. associate-/l/ 9.2%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{r}{-s}}}{r}, \color{blue}{\frac{\frac{0.125}{s}}{r \cdot \pi}} + 0\right) \]

   2. add0 9.2%

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

9. Simplified 9.2%

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

10. Taylor expanded in s around 0 9.2%

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

    1. associate-/r* 9.2%

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

12. Simplified 9.2%

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

13. Final simplification 9.2%

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

Alternative 8: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-/l* 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. times-frac 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around 0 99.7%

   \[\leadsto 0.25 \cdot \frac{e^{-\frac{r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
6. Step-by-step derivation

   1. associate-/r* 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in r around 0 99.7%

   \[\leadsto 0.25 \cdot \frac{e^{-\frac{r}{s}}}{\color{blue}{2 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)}} + \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
9. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*l* 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in s around inf 9.2%

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

12. Final simplification 9.2%

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

Alternative 9: 9.7% accurate, 1.9× speedup

Math

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

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(0.25) * Float32(exp(Float32(r / Float32(-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 99.7%

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

   1. associate-/l* 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. times-frac 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around 0 99.7%

   \[\leadsto 0.25 \cdot \frac{e^{-\frac{r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
6. Step-by-step derivation

   1. associate-/r* 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in r around 0 99.7%

   \[\leadsto 0.25 \cdot \frac{e^{-\frac{r}{s}}}{\color{blue}{2 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)}} + \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
9. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*l* 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in s around inf 9.2%

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

12. Taylor expanded in r around 0 9.2%

    \[\leadsto 0.25 \cdot \frac{e^{-\frac{r}{s}}}{\color{blue}{2 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)}} + \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \]
13. Step-by-step derivation

    1. associate-*r* 9.2%

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

    2. *-commutative 9.2%

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

    3. associate-*r* 9.2%

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

    4. *-commutative 9.2%

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

    5. *-commutative 9.2%

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

    6. associate-*r* 9.2%

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

    7. *-commutative 9.2%

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

14. Simplified 9.2%

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

15. Final simplification 9.2%

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

Alternative 10: 7.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.8%

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

   2. fma-define 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. *-commutative 99.8%

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

   7. distribute-frac-neg 99.8%

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

   8. distribute-frac-neg2 99.8%

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

   9. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 9.2%

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

   1. frac-2neg 9.2%

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

   2. distribute-frac-neg2 9.2%

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

   3. add-sqr-sqrt 9.2%

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

   4. sqrt-unprod 9.2%

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

   5. sqr-neg 9.2%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 4.4%

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

   8. distribute-lft-neg-in 4.4%

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

   9. frac-2neg 4.4%

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

   10. *-commutative 4.4%

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

   11. associate-/r* 4.4%

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

   12. associate-/r* 4.4%

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

7. Applied egg-rr 4.4%

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

   1. fma-undefine 4.4%

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

   2. *-commutative 4.4%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 4.6%

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

   5. sqr-neg 4.6%

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

   6. sqrt-unprod 4.6%

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

   7. add-sqr-sqrt 4.6%

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

   8. add-sqr-sqrt -0.0%

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

   9. sqrt-unprod 7.1%

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

   10. sqr-neg 7.1%

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

   11. sqrt-unprod 7.1%

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

   12. add-sqr-sqrt 7.1%

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

   13. associate-/l/ 7.1%

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

9. Applied egg-rr 7.1%

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

10. Final simplification 7.1%

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

Alternative 11: 4.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.8%

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

   2. fma-define 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. *-commutative 99.8%

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

   7. distribute-frac-neg 99.8%

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

   8. distribute-frac-neg2 99.8%

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

   9. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 9.2%

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

   1. frac-2neg 9.2%

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

   2. distribute-frac-neg2 9.2%

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

   3. add-sqr-sqrt 9.2%

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

   4. sqrt-unprod 9.2%

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

   5. sqr-neg 9.2%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 4.4%

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

   8. distribute-lft-neg-in 4.4%

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

   9. frac-2neg 4.4%

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

   10. *-commutative 4.4%

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

   11. associate-/r* 4.4%

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

   12. associate-/r* 4.4%

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

7. Applied egg-rr 4.4%

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

8. Taylor expanded in s around 0 4.4%

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

   1. distribute-lft-out-- 4.4%

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

   2. mul-1-neg 4.4%

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

10. Simplified 4.4%

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

11. Final simplification 4.4%

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

Alternative 12: 3.7% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.125}{\pi \cdot \left(s \cdot s\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.8%

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

   2. fma-define 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. *-commutative 99.8%

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

   7. distribute-frac-neg 99.8%

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

   8. distribute-frac-neg2 99.8%

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

   9. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 9.2%

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

   1. frac-2neg 9.2%

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

   2. distribute-frac-neg2 9.2%

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

   3. add-sqr-sqrt 9.2%

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

   4. sqrt-unprod 9.2%

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

   5. sqr-neg 9.2%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 4.4%

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

   8. distribute-lft-neg-in 4.4%

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

   9. frac-2neg 4.4%

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

   10. *-commutative 4.4%

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

   11. associate-/r* 4.4%

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

   12. associate-/r* 4.4%

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

7. Applied egg-rr 4.4%

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

8. Taylor expanded in s around inf 3.7%

   \[\leadsto \color{blue}{\frac{-0.125}{{s}^{2} \cdot \pi}} \]
9. Step-by-step derivation

   1. unpow2 3.7%

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

10. Applied egg-rr 3.7%

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

11. Final simplification 3.7%

    \[\leadsto \frac{-0.125}{\pi \cdot \left(s \cdot s\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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))))