Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 12.2s

Alternatives: 16

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(s * Float32(3.0))))) / Float32(s * Float32(r * 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))))) / (s * (r * (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. Taylor expanded in s around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*r* 99.4%

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

   3. associate-*r* 99.5%

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

   4. associate-*l* 99.5%

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

4. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-r}{s \cdot 3}}}{r} \cdot \left(0.75 \cdot \frac{0.16666666666666666}{s \cdot \pi}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ 0.25 (* s (* 2.0 PI))) (/ (exp (/ (- r) s)) r))
  (*
   (/ (exp (/ (- r) (* s 3.0))) r)
   (* 0.75 (/ 0.16666666666666666 (* s PI))))))

C

float code(float s, float r) {
	return ((0.25f / (s * (2.0f * ((float) M_PI)))) * (expf((-r / s)) / r)) + ((expf((-r / (s * 3.0f))) / r) * (0.75f * (0.16666666666666666f / (s * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(s * Float32(Float32(2.0) * Float32(pi)))) * Float32(exp(Float32(Float32(-r) / s)) / r)) + Float32(Float32(exp(Float32(Float32(-r) / Float32(s * Float32(3.0)))) / r) * Float32(Float32(0.75) * Float32(Float32(0.16666666666666666) / Float32(s * Float32(pi))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (s * (single(2.0) * single(pi)))) * (exp((-r / s)) / r)) + ((exp((-r / (s * single(3.0)))) / r) * (single(0.75) * (single(0.16666666666666666) / (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. 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. times-frac 99.5%

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

   4. *-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. distribute-frac-neg 99.5%

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

   7. *-commutative 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in s around 0 99.5%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \left(0.75 \cdot \color{blue}{\frac{0.16666666666666666}{s \cdot \pi}}\right) \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]

7. Final simplification 99.5%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-r}{s \cdot 3}}}{r} \cdot \left(0.75 \cdot \frac{0.16666666666666666}{s \cdot \pi}\right) \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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

   4. *-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. distribute-frac-neg 99.5%

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

   7. *-commutative 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-r}{s \cdot 3}}}{r} \cdot \frac{0.125}{s \cdot \pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ 0.25 (* s (* 2.0 PI))) (/ (exp (/ (- r) s)) r))
  (* (/ (exp (/ (- r) (* s 3.0))) r) (/ 0.125 (* s PI)))))

C

float code(float s, float r) {
	return ((0.25f / (s * (2.0f * ((float) M_PI)))) * (expf((-r / s)) / r)) + ((expf((-r / (s * 3.0f))) / r) * (0.125f / (s * ((float) M_PI))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(s * Float32(Float32(2.0) * Float32(pi)))) * Float32(exp(Float32(Float32(-r) / s)) / r)) + Float32(Float32(exp(Float32(Float32(-r) / Float32(s * Float32(3.0)))) / r) * Float32(Float32(0.125) / Float32(s * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (s * (single(2.0) * single(pi)))) * (exp((-r / s)) / r)) + ((exp((-r / (s * single(3.0)))) / r) * (single(0.125) / (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. 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. times-frac 99.5%

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

   4. *-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. distribute-frac-neg 99.5%

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

   7. *-commutative 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around 0 99.4%

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

5. Final simplification 99.4%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-r}{s \cdot 3}}}{r} \cdot \frac{\frac{0.125}{s}}{\pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ 0.25 (* s (* 2.0 PI))) (/ (exp (/ (- r) s)) r))
  (* (/ (exp (/ (- r) (* s 3.0))) r) (/ (/ 0.125 s) PI))))

C

float code(float s, float r) {
	return ((0.25f / (s * (2.0f * ((float) M_PI)))) * (expf((-r / s)) / r)) + ((expf((-r / (s * 3.0f))) / r) * ((0.125f / s) / ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(s * Float32(Float32(2.0) * Float32(pi)))) * Float32(exp(Float32(Float32(-r) / s)) / r)) + Float32(Float32(exp(Float32(Float32(-r) / Float32(s * Float32(3.0)))) / r) * Float32(Float32(Float32(0.125) / s) / Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (s * (single(2.0) * single(pi)))) * (exp((-r / s)) / r)) + ((exp((-r / (s * single(3.0)))) / r) * ((single(0.125) / 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. 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. times-frac 99.5%

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

   4. *-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. distribute-frac-neg 99.5%

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

   7. *-commutative 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around 0 99.4%

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

   1. associate-/r* 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 6: 97.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / (r * single(pi))) / s) * (exp((-r / s)) + exp(((r / s) * single(-0.3333333333333333))));
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. associate-*l/ 97.5%

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

   2. associate-*l/ 97.5%

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

   3. associate-*l* 97.5%

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

   4. associate-*l* 97.5%

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

   5. associate-/r* 97.5%

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

   6. metadata-eval 97.5%

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

   7. metadata-eval 97.5%

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

   8. associate-/r* 97.5%

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

   9. associate-*l* 97.5%

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

   10. associate-*l* 97.5%

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

   11. distribute-lft-out 97.5%

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

3. Simplified 97.0%

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

4. Taylor expanded in r around inf 97.4%

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

5. Final simplification 97.4%

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

Alternative 7: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((-r / s)) + exp(((r / s) * single(-0.3333333333333333)))) / (s * (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. Step-by-step derivation

   1. associate-*l/ 97.5%

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

   2. associate-*l/ 97.5%

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

   3. associate-*l* 97.5%

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

   4. associate-*l* 97.5%

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

   5. associate-/r* 97.5%

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

   6. metadata-eval 97.5%

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

   7. metadata-eval 97.5%

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

   8. associate-/r* 97.5%

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

   9. associate-*l* 97.5%

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

   10. associate-*l* 97.5%

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

   11. distribute-lft-out 97.5%

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

3. Simplified 97.0%

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

4. Taylor expanded in r around inf 99.4%

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

5. Final simplification 99.4%

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

Alternative 8: 87.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-r}{s}}\\ \mathbf{if}\;s \leq 0.009999999776482582:\\ \;\;\;\;0 \cdot \left(t_0 + \left(\frac{r}{s} \cdot -0.3333333333333333 + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{s \cdot \pi} \cdot \left(\frac{t_0}{r} + \left(\frac{1}{r} - \frac{0.3333333333333333}{s}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (exp (/ (- r) s))))
   (if (<= s 0.009999999776482582)
     (* 0.0 (+ t_0 (+ (* (/ r s) -0.3333333333333333) 1.0)))
     (*
      (/ 0.125 (* s PI))
      (+ (/ t_0 r) (- (/ 1.0 r) (/ 0.3333333333333333 s)))))))

C

float code(float s, float r) {
	float t_0 = expf((-r / s));
	float tmp;
	if (s <= 0.009999999776482582f) {
		tmp = 0.0f * (t_0 + (((r / s) * -0.3333333333333333f) + 1.0f));
	} else {
		tmp = (0.125f / (s * ((float) M_PI))) * ((t_0 / r) + ((1.0f / r) - (0.3333333333333333f / s)));
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = exp(Float32(Float32(-r) / s))
	tmp = Float32(0.0)
	if (s <= Float32(0.009999999776482582))
		tmp = Float32(Float32(0.0) * Float32(t_0 + Float32(Float32(Float32(r / s) * Float32(-0.3333333333333333)) + Float32(1.0))));
	else
		tmp = Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(t_0 / r) + Float32(Float32(Float32(1.0) / r) - Float32(Float32(0.3333333333333333) / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	t_0 = exp((-r / s));
	tmp = single(0.0);
	if (s <= single(0.009999999776482582))
		tmp = single(0.0) * (t_0 + (((r / s) * single(-0.3333333333333333)) + single(1.0)));
	else
		tmp = (single(0.125) / (s * single(pi))) * ((t_0 / r) + ((single(1.0) / r) - (single(0.3333333333333333) / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.00999999978

   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/ 97.5%

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

      2. associate-*l/ 97.5%

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

      3. associate-*l* 97.6%

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

      4. associate-*l* 97.5%

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

      5. associate-/r* 97.5%

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

      6. metadata-eval 97.5%

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

      7. metadata-eval 97.5%

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

      8. associate-/r* 97.5%

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

      9. associate-*l* 97.5%

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

      10. associate-*l* 97.5%

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

      11. distribute-lft-out 97.5%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in r around 0 5.3%

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

      1. expm1-log1p-u 5.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.125}{r \cdot \pi}}{s}\right)\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

      2. expm1-udef 5.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{0.125}{r \cdot \pi}}{s}\right)} - 1\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

      3. associate-/l/ 5.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{0.125}{s \cdot \left(r \cdot \pi\right)}}\right)} - 1\right) \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   6. Applied egg-rr 5.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.125}{s \cdot \left(r \cdot \pi\right)}\right)} - 1\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   7. Taylor expanded in s around inf 91.8%

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

   if 0.00999999978 < s

   1. Initial program 97.6%

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

      1. times-frac 97.3%

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

      2. times-frac 97.4%

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

      3. associate-*l* 97.0%

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

      4. associate-/r* 96.8%

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

      5. metadata-eval 96.8%

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

      6. metadata-eval 96.8%

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

      7. associate-/r* 96.8%

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

      8. associate-*l* 96.8%

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

      9. distribute-lft-out 96.8%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in r around 0 54.4%

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

   5. Taylor expanded in r around 0 54.3%

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

      1. associate-*r/ 54.3%

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

      2. metadata-eval 54.3%

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

   7. Simplified 54.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\left(\frac{1}{r} - \frac{0.3333333333333333}{s}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.009999999776482582:\\ \;\;\;\;0 \cdot \left(e^{\frac{-r}{s}} + \left(\frac{r}{s} \cdot -0.3333333333333333 + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \left(\frac{1}{r} - \frac{0.3333333333333333}{s}\right)\right)\\ \end{array} \]

Alternative 9: 87.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-r}{s}} + \left(\frac{r}{s} \cdot -0.3333333333333333 + 1\right)\\ \mathbf{if}\;s \leq 0.009999999776482582:\\ \;\;\;\;0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{0.125}{s \cdot \left(r \cdot \pi\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (+ (exp (/ (- r) s)) (+ (* (/ r s) -0.3333333333333333) 1.0))))
   (if (<= s 0.009999999776482582)
     (* 0.0 t_0)
     (* t_0 (/ 0.125 (* s (* r PI)))))))

C

float code(float s, float r) {
	float t_0 = expf((-r / s)) + (((r / s) * -0.3333333333333333f) + 1.0f);
	float tmp;
	if (s <= 0.009999999776482582f) {
		tmp = 0.0f * t_0;
	} else {
		tmp = t_0 * (0.125f / (s * (r * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(exp(Float32(Float32(-r) / s)) + Float32(Float32(Float32(r / s) * Float32(-0.3333333333333333)) + Float32(1.0)))
	tmp = Float32(0.0)
	if (s <= Float32(0.009999999776482582))
		tmp = Float32(Float32(0.0) * t_0);
	else
		tmp = Float32(t_0 * Float32(Float32(0.125) / Float32(s * Float32(r * Float32(pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	t_0 = exp((-r / s)) + (((r / s) * single(-0.3333333333333333)) + single(1.0));
	tmp = single(0.0);
	if (s <= single(0.009999999776482582))
		tmp = single(0.0) * t_0;
	else
		tmp = t_0 * (single(0.125) / (s * (r * single(pi))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.00999999978

   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/ 97.5%

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

      2. associate-*l/ 97.5%

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

      3. associate-*l* 97.6%

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

      4. associate-*l* 97.5%

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

      5. associate-/r* 97.5%

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

      6. metadata-eval 97.5%

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

      7. metadata-eval 97.5%

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

      8. associate-/r* 97.5%

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

      9. associate-*l* 97.5%

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

      10. associate-*l* 97.5%

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

      11. distribute-lft-out 97.5%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in r around 0 5.3%

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

      1. expm1-log1p-u 5.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.125}{r \cdot \pi}}{s}\right)\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

      2. expm1-udef 5.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{0.125}{r \cdot \pi}}{s}\right)} - 1\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

      3. associate-/l/ 5.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{0.125}{s \cdot \left(r \cdot \pi\right)}}\right)} - 1\right) \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   6. Applied egg-rr 5.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.125}{s \cdot \left(r \cdot \pi\right)}\right)} - 1\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   7. Taylor expanded in s around inf 91.8%

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

   if 0.00999999978 < s

   1. Initial program 97.6%

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

      1. associate-*l/ 97.7%

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

      2. associate-*l/ 97.3%

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

      3. associate-*l* 97.4%

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

      4. associate-*l* 97.1%

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

      5. associate-/r* 97.2%

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

      6. metadata-eval 97.2%

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

      7. metadata-eval 97.2%

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

      8. associate-/r* 97.2%

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

      9. associate-*l* 97.2%

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

      10. associate-*l* 97.3%

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

      11. distribute-lft-out 97.4%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in r around 0 54.4%

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

   5. Taylor expanded in r around 0 54.7%

      \[\leadsto \color{blue}{\frac{0.125}{s \cdot \left(r \cdot \pi\right)}} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.009999999776482582:\\ \;\;\;\;0 \cdot \left(e^{\frac{-r}{s}} + \left(\frac{r}{s} \cdot -0.3333333333333333 + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\frac{-r}{s}} + \left(\frac{r}{s} \cdot -0.3333333333333333 + 1\right)\right) \cdot \frac{0.125}{s \cdot \left(r \cdot \pi\right)}\\ \end{array} \]

Alternative 10: 87.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-r}{s}} + \left(\frac{r}{s} \cdot -0.3333333333333333 + 1\right)\\ \mathbf{if}\;s \leq 0.009999999776482582:\\ \;\;\;\;0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\frac{\frac{0.125}{r}}{\pi}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (+ (exp (/ (- r) s)) (+ (* (/ r s) -0.3333333333333333) 1.0))))
   (if (<= s 0.009999999776482582)
     (* 0.0 t_0)
     (* t_0 (/ (/ (/ 0.125 r) PI) s)))))

C

float code(float s, float r) {
	float t_0 = expf((-r / s)) + (((r / s) * -0.3333333333333333f) + 1.0f);
	float tmp;
	if (s <= 0.009999999776482582f) {
		tmp = 0.0f * t_0;
	} else {
		tmp = t_0 * (((0.125f / r) / ((float) M_PI)) / s);
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(exp(Float32(Float32(-r) / s)) + Float32(Float32(Float32(r / s) * Float32(-0.3333333333333333)) + Float32(1.0)))
	tmp = Float32(0.0)
	if (s <= Float32(0.009999999776482582))
		tmp = Float32(Float32(0.0) * t_0);
	else
		tmp = Float32(t_0 * Float32(Float32(Float32(Float32(0.125) / r) / Float32(pi)) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	t_0 = exp((-r / s)) + (((r / s) * single(-0.3333333333333333)) + single(1.0));
	tmp = single(0.0);
	if (s <= single(0.009999999776482582))
		tmp = single(0.0) * t_0;
	else
		tmp = t_0 * (((single(0.125) / r) / single(pi)) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.00999999978

   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/ 97.5%

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

      2. associate-*l/ 97.5%

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

      3. associate-*l* 97.6%

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

      4. associate-*l* 97.5%

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

      5. associate-/r* 97.5%

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

      6. metadata-eval 97.5%

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

      7. metadata-eval 97.5%

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

      8. associate-/r* 97.5%

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

      9. associate-*l* 97.5%

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

      10. associate-*l* 97.5%

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

      11. distribute-lft-out 97.5%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in r around 0 5.3%

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

      1. expm1-log1p-u 5.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.125}{r \cdot \pi}}{s}\right)\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

      2. expm1-udef 5.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{0.125}{r \cdot \pi}}{s}\right)} - 1\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

      3. associate-/l/ 5.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{0.125}{s \cdot \left(r \cdot \pi\right)}}\right)} - 1\right) \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   6. Applied egg-rr 5.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.125}{s \cdot \left(r \cdot \pi\right)}\right)} - 1\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   7. Taylor expanded in s around inf 91.8%

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

   if 0.00999999978 < s

   1. Initial program 97.6%

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

      1. associate-*l/ 97.7%

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

      2. associate-*l/ 97.3%

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

      3. associate-*l* 97.4%

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

      4. associate-*l* 97.1%

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

      5. associate-/r* 97.2%

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

      6. metadata-eval 97.2%

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

      7. metadata-eval 97.2%

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

      8. associate-/r* 97.2%

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

      9. associate-*l* 97.2%

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

      10. associate-*l* 97.3%

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

      11. distribute-lft-out 97.4%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in r around 0 54.4%

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

      1. expm1-log1p-u 53.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{r \cdot \pi}\right)\right)}}{s} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

      2. expm1-udef 53.8%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{0.125}{r \cdot \pi}\right)} - 1}}{s} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

      3. *-commutative 53.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{0.125}{\color{blue}{\pi \cdot r}}\right)} - 1}{s} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   6. Applied egg-rr 53.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{0.125}{\pi \cdot r}\right)} - 1}}{s} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-def 53.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{\pi \cdot r}\right)\right)}}{s} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

      2. expm1-log1p 54.4%

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

      3. *-commutative 54.4%

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

      4. associate-/r* 54.7%

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

   8. Simplified 54.7%

      \[\leadsto \frac{\color{blue}{\frac{\frac{0.125}{r}}{\pi}}}{s} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.009999999776482582:\\ \;\;\;\;0 \cdot \left(e^{\frac{-r}{s}} + \left(\frac{r}{s} \cdot -0.3333333333333333 + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\frac{-r}{s}} + \left(\frac{r}{s} \cdot -0.3333333333333333 + 1\right)\right) \cdot \frac{\frac{\frac{0.125}{r}}{\pi}}{s}\\ \end{array} \]

Alternative 11: 87.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 0.009999999776482582:\\ \;\;\;\;0 \cdot \left(e^{\frac{-r}{s}} + \left(\frac{r}{s} \cdot -0.3333333333333333 + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{r \cdot \left(s \cdot \pi\right)} + \frac{-0.16666666666666666}{s \cdot \left(s \cdot \pi\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= s 0.009999999776482582)
   (* 0.0 (+ (exp (/ (- r) s)) (+ (* (/ r s) -0.3333333333333333) 1.0)))
   (+ (/ 0.25 (* r (* s PI))) (/ -0.16666666666666666 (* s (* s PI))))))

C

float code(float s, float r) {
	float tmp;
	if (s <= 0.009999999776482582f) {
		tmp = 0.0f * (expf((-r / s)) + (((r / s) * -0.3333333333333333f) + 1.0f));
	} else {
		tmp = (0.25f / (r * (s * ((float) M_PI)))) + (-0.16666666666666666f / (s * (s * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (s <= Float32(0.009999999776482582))
		tmp = Float32(Float32(0.0) * Float32(exp(Float32(Float32(-r) / s)) + Float32(Float32(Float32(r / s) * Float32(-0.3333333333333333)) + Float32(1.0))));
	else
		tmp = Float32(Float32(Float32(0.25) / Float32(r * Float32(s * Float32(pi)))) + Float32(Float32(-0.16666666666666666) / Float32(s * Float32(s * Float32(pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if (s <= single(0.009999999776482582))
		tmp = single(0.0) * (exp((-r / s)) + (((r / s) * single(-0.3333333333333333)) + single(1.0)));
	else
		tmp = (single(0.25) / (r * (s * single(pi)))) + (single(-0.16666666666666666) / (s * (s * single(pi))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.00999999978

   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/ 97.5%

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

      2. associate-*l/ 97.5%

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

      3. associate-*l* 97.6%

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

      4. associate-*l* 97.5%

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

      5. associate-/r* 97.5%

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

      6. metadata-eval 97.5%

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

      7. metadata-eval 97.5%

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

      8. associate-/r* 97.5%

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

      9. associate-*l* 97.5%

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

      10. associate-*l* 97.5%

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

      11. distribute-lft-out 97.5%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in r around 0 5.3%

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

      1. expm1-log1p-u 5.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.125}{r \cdot \pi}}{s}\right)\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

      2. expm1-udef 5.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{0.125}{r \cdot \pi}}{s}\right)} - 1\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

      3. associate-/l/ 5.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{0.125}{s \cdot \left(r \cdot \pi\right)}}\right)} - 1\right) \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   6. Applied egg-rr 5.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.125}{s \cdot \left(r \cdot \pi\right)}\right)} - 1\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   7. Taylor expanded in s around inf 91.8%

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

   if 0.00999999978 < s

   1. Initial program 97.6%

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

      1. associate-*l/ 97.7%

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

      2. associate-*l/ 97.3%

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

      3. associate-*l* 97.4%

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

      4. associate-*l* 97.1%

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

      5. associate-/r* 97.2%

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

      6. metadata-eval 97.2%

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

      7. metadata-eval 97.2%

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

      8. associate-/r* 97.2%

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

      9. associate-*l* 97.2%

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

      10. associate-*l* 97.3%

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

      11. distribute-lft-out 97.4%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in r around 0 51.2%

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

      1. sub-neg 51.2%

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

      2. associate-*r/ 51.2%

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

      3. metadata-eval 51.2%

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

      4. associate-*r* 51.5%

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

      5. *-commutative 51.5%

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

      6. associate-*l* 51.5%

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

      7. associate-*r/ 51.5%

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

      8. metadata-eval 51.5%

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

      9. distribute-neg-frac 51.5%

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

      10. metadata-eval 51.5%

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

      11. unpow2 51.5%

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

      12. associate-*r* 51.5%

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

   6. Simplified 51.5%

      \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)} + \frac{-0.16666666666666666}{s \cdot \left(s \cdot \pi\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.009999999776482582:\\ \;\;\;\;0 \cdot \left(e^{\frac{-r}{s}} + \left(\frac{r}{s} \cdot -0.3333333333333333 + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{r \cdot \left(s \cdot \pi\right)} + \frac{-0.16666666666666666}{s \cdot \left(s \cdot \pi\right)}\\ \end{array} \]

Alternative 12: 85.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 0 \cdot \left(e^{\frac{-r}{s}} + \left(\frac{r}{s} \cdot -0.3333333333333333 + 1\right)\right) \end{array} \]

FPCore

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

C

float code(float s, float r) {
	return 0.0f * (expf((-r / s)) + (((r / s) * -0.3333333333333333f) + 1.0f));
}

Fortran

real(4) function code(s, r)
    real(4), intent (in) :: s
    real(4), intent (in) :: r
    code = 0.0e0 * (exp((-r / s)) + (((r / s) * (-0.3333333333333333e0)) + 1.0e0))
end function

Julia

function code(s, r)
	return Float32(Float32(0.0) * Float32(exp(Float32(Float32(-r) / s)) + Float32(Float32(Float32(r / s) * Float32(-0.3333333333333333)) + Float32(1.0))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.0) * (exp((-r / s)) + (((r / s) * single(-0.3333333333333333)) + 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. Step-by-step derivation

   1. associate-*l/ 97.5%

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

   2. associate-*l/ 97.5%

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

   3. associate-*l* 97.5%

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

   4. associate-*l* 97.5%

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

   5. associate-/r* 97.5%

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

   6. metadata-eval 97.5%

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

   7. metadata-eval 97.5%

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

   8. associate-/r* 97.5%

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

   9. associate-*l* 97.5%

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

   10. associate-*l* 97.5%

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

   11. distribute-lft-out 97.5%

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

3. Simplified 97.0%

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

4. Taylor expanded in r around 0 9.9%

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

   1. expm1-log1p-u 9.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.125}{r \cdot \pi}}{s}\right)\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   2. expm1-udef 9.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{0.125}{r \cdot \pi}}{s}\right)} - 1\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   3. associate-/l/ 9.9%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{0.125}{s \cdot \left(r \cdot \pi\right)}}\right)} - 1\right) \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

6. Applied egg-rr 9.9%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.125}{s \cdot \left(r \cdot \pi\right)}\right)} - 1\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

7. Taylor expanded in s around inf 84.5%

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

8. Final simplification 84.5%

   \[\leadsto 0 \cdot \left(e^{\frac{-r}{s}} + \left(\frac{r}{s} \cdot -0.3333333333333333 + 1\right)\right) \]

Alternative 13: 9.2% accurate, 4.0× 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%

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

   1. associate-*l/ 97.5%

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

   2. associate-*l/ 97.5%

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

   3. associate-*l* 97.5%

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

   4. associate-*l* 97.5%

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

   5. associate-/r* 97.5%

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

   6. metadata-eval 97.5%

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

   7. metadata-eval 97.5%

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

   8. associate-/r* 97.5%

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

   9. associate-*l* 97.5%

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

   10. associate-*l* 97.5%

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

   11. distribute-lft-out 97.5%

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

3. Simplified 97.0%

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

4. Taylor expanded in r around 0 9.2%

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

   1. associate-*r* 9.2%

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

   2. *-commutative 9.2%

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

   3. associate-*l* 9.2%

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

6. Simplified 9.2%

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

7. Final simplification 9.2%

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

Alternative 14: 9.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / (s * (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. Step-by-step derivation

   1. associate-*l/ 97.5%

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

   2. associate-*l/ 97.5%

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

   3. associate-*l* 97.5%

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

   4. associate-*l* 97.5%

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

   5. associate-/r* 97.5%

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

   6. metadata-eval 97.5%

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

   7. metadata-eval 97.5%

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

   8. associate-/r* 97.5%

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

   9. associate-*l* 97.5%

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

   10. associate-*l* 97.5%

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

   11. distribute-lft-out 97.5%

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

3. Simplified 97.0%

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

4. Taylor expanded in r around 0 9.2%

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

   1. *-commutative 9.2%

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

6. Simplified 9.2%

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

7. Final simplification 9.2%

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

Alternative 15: 9.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / s) / (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. Step-by-step derivation

   1. associate-*l/ 97.5%

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

   2. associate-*l/ 97.5%

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

   3. associate-*l* 97.5%

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

   4. associate-*l* 97.5%

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

   5. associate-/r* 97.5%

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

   6. metadata-eval 97.5%

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

   7. metadata-eval 97.5%

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

   8. associate-/r* 97.5%

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

   9. associate-*l* 97.5%

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

   10. associate-*l* 97.5%

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

   11. distribute-lft-out 97.5%

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

3. Simplified 97.0%

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

4. Taylor expanded in r around 0 9.2%

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

   1. associate-*r* 9.2%

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

   2. *-commutative 9.2%

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

   3. associate-*l* 9.2%

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

6. Simplified 9.2%

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

7. Taylor expanded in r around 0 9.2%

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

   1. associate-/r* 9.2%

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

9. Simplified 9.2%

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

10. Final simplification 9.2%

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

Alternative 16: 9.2% accurate, 4.0× speedup

Math

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

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(Float32(0.25) / 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%

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

   1. associate-*l/ 97.5%

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

   2. associate-*l/ 97.5%

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

   3. associate-*l* 97.5%

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

   4. associate-*l* 97.5%

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

   5. associate-/r* 97.5%

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

   6. metadata-eval 97.5%

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

   7. metadata-eval 97.5%

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

   8. associate-/r* 97.5%

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

   9. associate-*l* 97.5%

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

   10. associate-*l* 97.5%

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

   11. distribute-lft-out 97.5%

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

3. Simplified 97.0%

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

4. Taylor expanded in r around 0 9.2%

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

   1. associate-*r* 9.2%

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

   2. *-commutative 9.2%

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

   3. associate-*l* 9.2%

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

6. Simplified 9.2%

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

7. Taylor expanded in r around 0 9.2%

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

   1. *-commutative 9.2%

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

   2. associate-*r* 9.2%

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

   3. associate-/l/ 9.2%

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

   4. associate-/r* 9.2%

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

9. Simplified 9.2%

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

10. Final simplification 9.2%

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

Reproduce

herbie shell --seed 2023255 
(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))))