Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.4%

Time: 12.5s

Alternatives: 15

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: 99.4% accurate, 1.0× speedup

Math

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

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(Float32(0.75) / 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. Step-by-step derivation

   1. expm1-log1p-u 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.75}{s \cdot \left(\pi \cdot 6\right)}\right)\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]

   2. expm1-udef 98.1%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.75}{s \cdot \left(\pi \cdot 6\right)}\right)} - 1\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]

5. Applied egg-rr 98.1%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.75}{s \cdot \left(\pi \cdot 6\right)}\right)} - 1\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]
6. Step-by-step derivation

   1. expm1-def 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.75}{s \cdot \left(\pi \cdot 6\right)}\right)\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]

   2. expm1-log1p 99.5%

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

   3. associate-/r* 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

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 / (s * (((float) M_PI) * 6.0f))));
}

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. times-frac 99.5%

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

   2. *-commutative 99.5%

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

   3. 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%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) / r) * ((single(0.125) / s) / single(pi))) + (((single(0.75) / s) / (single(pi) * single(6.0))) * ((single(1.0) / 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. Step-by-step derivation

   1. expm1-log1p-u 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.75}{s \cdot \left(\pi \cdot 6\right)}\right)\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]

   2. expm1-udef 98.1%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.75}{s \cdot \left(\pi \cdot 6\right)}\right)} - 1\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]

5. Applied egg-rr 98.1%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.75}{s \cdot \left(\pi \cdot 6\right)}\right)} - 1\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]
6. Step-by-step derivation

   1. expm1-def 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.75}{s \cdot \left(\pi \cdot 6\right)}\right)\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]

   2. expm1-log1p 99.5%

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

   3. associate-/r* 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in s around 0 99.5%

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

   1. associate-/r* 99.5%

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

10. Simplified 99.5%

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

    1. exp-neg 99.5%

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

12. Applied egg-rr 99.5%

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

13. Final simplification 99.5%

    \[\leadsto \frac{e^{\frac{-r}{s}}}{r} \cdot \frac{\frac{0.125}{s}}{\pi} + \frac{\frac{0.75}{s}}{\pi \cdot 6} \cdot \frac{\frac{1}{e^{\frac{r}{s \cdot 3}}}}{r} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / (s * single(3.0)))) / r) * (single(0.75) / (s * (single(pi) * single(6.0))))) + ((exp((-r / s)) / 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 99.5%

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

   1. associate-/r* 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{0.125}{s}}{\pi}\\ \frac{e^{\frac{-r}{s}}}{r} \cdot t_0 + t_0 \cdot \frac{\frac{1}{e^{\frac{r}{s \cdot 3}}}}{r} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	t_0 = (single(0.125) / s) / single(pi);
	tmp = ((exp((-r / s)) / r) * t_0) + (t_0 * ((single(1.0) / 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. Step-by-step derivation

   1. expm1-log1p-u 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.75}{s \cdot \left(\pi \cdot 6\right)}\right)\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]

   2. expm1-udef 98.1%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.75}{s \cdot \left(\pi \cdot 6\right)}\right)} - 1\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]

5. Applied egg-rr 98.1%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.75}{s \cdot \left(\pi \cdot 6\right)}\right)} - 1\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]
6. Step-by-step derivation

   1. expm1-def 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.75}{s \cdot \left(\pi \cdot 6\right)}\right)\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]

   2. expm1-log1p 99.5%

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

   3. associate-/r* 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in s around 0 99.5%

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

   1. associate-/r* 99.5%

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

10. Simplified 99.5%

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

    1. exp-neg 99.5%

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

12. Applied egg-rr 99.5%

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

13. Taylor expanded in s around 0 99.5%

    \[\leadsto \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{\frac{1}{e^{\frac{r}{s \cdot 3}}}}{r} \]
14. Step-by-step derivation

    1. associate-/r* 99.5%

       \[\leadsto \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{\frac{1}{e^{\frac{r}{s \cdot 3}}}}{r} \]

15. Simplified 99.5%

    \[\leadsto \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{\frac{1}{e^{\frac{r}{s \cdot 3}}}}{r} \]

16. Final simplification 99.5%

    \[\leadsto \frac{e^{\frac{-r}{s}}}{r} \cdot \frac{\frac{0.125}{s}}{\pi} + \frac{\frac{0.125}{s}}{\pi} \cdot \frac{\frac{1}{e^{\frac{r}{s \cdot 3}}}}{r} \]

Alternative 6: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 97.2%

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

   2. associate-*l/ 97.2%

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

   3. associate-*l* 97.2%

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

   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* 97.1%

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

   6. metadata-eval 97.1%

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

   7. metadata-eval 97.1%

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

   8. associate-/r* 97.1%

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

   9. associate-*l* 97.1%

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

   10. associate-*l* 97.2%

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

   11. distribute-lft-out 97.1%

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

3. Simplified 96.8%

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

4. Taylor expanded in r around inf 97.1%

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

   1. associate-*r/ 97.1%

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

   2. *-commutative 97.1%

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

6. Simplified 97.1%

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

   1. add-exp-log 96.8%

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

   2. *-commutative 96.8%

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

8. Applied egg-rr 96.8%

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

9. Taylor expanded in r around 0 97.1%

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

    1. *-commutative 97.1%

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

    2. associate-*l* 97.1%

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

    3. *-commutative 97.1%

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

11. Simplified 97.1%

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

12. Final simplification 97.1%

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

Alternative 7: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 97.2%

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

   2. associate-*l/ 97.2%

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

   3. associate-*l* 97.2%

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

   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* 97.1%

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

   6. metadata-eval 97.1%

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

   7. metadata-eval 97.1%

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

   8. associate-/r* 97.1%

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

   9. associate-*l* 97.1%

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

   10. associate-*l* 97.2%

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

   11. distribute-lft-out 97.1%

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

3. Simplified 96.8%

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

4. Taylor expanded in r around inf 97.1%

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

   1. associate-*r/ 97.1%

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

   2. *-commutative 97.1%

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

6. Simplified 97.1%

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

7. Taylor expanded in r around 0 97.1%

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

   1. associate-*r/ 97.1%

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

   2. associate-/l* 97.2%

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

9. Simplified 97.2%

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

10. Final simplification 97.2%

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

Alternative 8: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 97.2%

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

   2. associate-*l/ 97.2%

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

   3. associate-*l* 97.2%

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

   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* 97.1%

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

   6. metadata-eval 97.1%

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

   7. metadata-eval 97.1%

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

   8. associate-/r* 97.1%

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

   9. associate-*l* 97.1%

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

   10. associate-*l* 97.2%

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

   11. distribute-lft-out 97.1%

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

3. Simplified 96.8%

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

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. Step-by-step derivation

   1. associate-*r/ 99.4%

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

   2. mul-1-neg 99.4%

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

   3. distribute-frac-neg 99.4%

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

   4. associate-*r/ 99.4%

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

   5. *-commutative 99.4%

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

   6. *-commutative 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in r around 0 99.4%

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

   1. associate-*r/ 97.1%

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

   2. associate-/l* 97.2%

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

9. Simplified 99.4%

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

10. Final simplification 99.4%

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

Alternative 9: 43.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 97.2%

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

   2. associate-*l/ 97.2%

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

   3. associate-*l* 97.2%

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

   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* 97.1%

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

   6. metadata-eval 97.1%

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

   7. metadata-eval 97.1%

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

   8. associate-/r* 97.1%

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

   9. associate-*l* 97.1%

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

   10. associate-*l* 97.2%

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

   11. distribute-lft-out 97.1%

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

3. Simplified 96.8%

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

4. Taylor expanded in r around 0 10.1%

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

   1. *-commutative 10.1%

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

6. Simplified 10.1%

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

   1. log1p-expm1-u 42.0%

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

8. Applied egg-rr 42.0%

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

9. Final simplification 42.0%

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

Alternative 10: 10.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / s) * (((single(1.0) / r) + ((exp((-r / s)) / r) + (single(-0.3333333333333333) / 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. times-frac 99.5%

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

   3. associate-*l* 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. metadata-eval 99.5%

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

   7. associate-/r* 99.5%

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

   8. associate-*l* 99.5%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 11.5%

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

   1. associate-*r/ 11.5%

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

   2. *-commutative 11.5%

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

6. Simplified 11.5%

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

7. Taylor expanded in r around 0 11.5%

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

   1. associate-*r/ 11.5%

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

   2. metadata-eval 11.5%

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

9. Simplified 11.5%

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

    1. distribute-lft-in 11.5%

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

    2. associate-/r* 11.5%

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

    3. associate-/r* 11.5%

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

11. Applied egg-rr 11.5%

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

    1. distribute-lft-in 11.5%

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

    2. associate-*l/ 11.5%

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

    3. associate-*l/ 11.5%

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

    4. associate-/r* 11.5%

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

    5. times-frac 11.5%

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

    6. +-commutative 11.5%

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

    7. sub-neg 11.5%

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

    8. associate-+l+ 11.5%

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

    9. distribute-neg-frac 11.5%

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

    10. metadata-eval 11.5%

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

13. Simplified 11.5%

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

14. Final simplification 11.5%

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

Alternative 11: 9.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.125}{r}}{\pi} \cdot \frac{-1 - e^{\frac{-r}{s}}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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

   3. associate-*l* 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. metadata-eval 99.5%

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

   7. associate-/r* 99.5%

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

   8. associate-*l* 99.5%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 10.7%

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

5. Taylor expanded in r around -inf 10.7%

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

   1. associate-*r/ 10.7%

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

   2. *-commutative 10.7%

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

   3. times-frac 10.7%

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

   4. associate-/r* 10.7%

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

   5. sub-neg 10.7%

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

   6. metadata-eval 10.7%

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

   7. +-commutative 10.7%

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

   8. mul-1-neg 10.7%

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

   9. mul-1-neg 10.7%

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

   10. distribute-frac-neg 10.7%

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

   11. unsub-neg 10.7%

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

7. Simplified 10.7%

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

8. Final simplification 10.7%

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

Alternative 12: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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

   3. associate-*l* 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. metadata-eval 99.5%

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

   7. associate-/r* 99.5%

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

   8. associate-*l* 99.5%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 10.7%

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

5. Taylor expanded in r around inf 10.7%

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

   1. associate-*r/ 10.7%

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

   2. +-commutative 10.7%

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

   3. mul-1-neg 10.7%

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

   4. distribute-frac-neg 10.7%

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

   5. distribute-lft-in 10.7%

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

   6. metadata-eval 10.7%

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

   7. distribute-frac-neg 10.7%

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

   8. rec-exp 10.7%

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

   9. rem-cube-cbrt 10.7%

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

   10. associate-*r/ 10.7%

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

   11. metadata-eval 10.7%

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

   12. rem-cube-cbrt 10.7%

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

   13. *-commutative 10.7%

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

   14. associate-*l* 10.7%

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

   15. *-commutative 10.7%

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

7. Simplified 10.7%

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

8. Final simplification 10.7%

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

Alternative 13: 9.3% 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/ 97.2%

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

   2. associate-*l/ 97.2%

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

   3. associate-*l* 97.2%

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

   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* 97.1%

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

   6. metadata-eval 97.1%

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

   7. metadata-eval 97.1%

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

   8. associate-/r* 97.1%

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

   9. associate-*l* 97.1%

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

   10. associate-*l* 97.2%

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

   11. distribute-lft-out 97.1%

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

3. Simplified 96.8%

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

4. Taylor expanded in r around 0 10.1%

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

   1. *-commutative 10.1%

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

6. Simplified 10.1%

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

7. Taylor expanded in s around 0 10.1%

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

   1. *-commutative 10.1%

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

   2. associate-*l* 10.1%

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

   3. associate-/r* 10.2%

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

9. Simplified 10.2%

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

10. Taylor expanded in s around 0 10.1%

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

    1. associate-/r* 10.1%

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

    2. *-commutative 10.1%

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

12. Simplified 10.1%

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

13. Taylor expanded in s around 0 10.1%

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

    1. *-commutative 10.1%

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

    2. associate-*l* 10.1%

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

    3. *-commutative 10.1%

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

15. Simplified 10.1%

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

16. Final simplification 10.1%

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

Alternative 14: 9.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 97.2%

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

   2. associate-*l/ 97.2%

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

   3. associate-*l* 97.2%

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

   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* 97.1%

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

   6. metadata-eval 97.1%

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

   7. metadata-eval 97.1%

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

   8. associate-/r* 97.1%

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

   9. associate-*l* 97.1%

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

   10. associate-*l* 97.2%

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

   11. distribute-lft-out 97.1%

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

3. Simplified 96.8%

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

4. Taylor expanded in r around 0 10.1%

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

   1. *-commutative 10.1%

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

6. Simplified 10.1%

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

7. Final simplification 10.1%

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

Alternative 15: 9.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 97.2%

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

   2. associate-*l/ 97.2%

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

   3. associate-*l* 97.2%

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

   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* 97.1%

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

   6. metadata-eval 97.1%

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

   7. metadata-eval 97.1%

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

   8. associate-/r* 97.1%

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

   9. associate-*l* 97.1%

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

   10. associate-*l* 97.2%

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

   11. distribute-lft-out 97.1%

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

3. Simplified 96.8%

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

4. Taylor expanded in r around 0 10.1%

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

   1. *-commutative 10.1%

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

6. Simplified 10.1%

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

7. Taylor expanded in s around 0 10.1%

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

   1. *-commutative 10.1%

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

   2. associate-*l* 10.1%

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

   3. associate-/r* 10.2%

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

9. Simplified 10.2%

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

10. Final simplification 10.2%

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

Reproduce

herbie shell --seed 2023222 
(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))))