Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 11.5s

Alternatives: 21

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ 0.25 (* s (* 2.0 PI))) (/ (exp (/ (- r) s)) r))
  (* (exp (log (/ 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)) + (expf(logf((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(exp(log(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)) + (exp(log((single(0.75) / (s * (single(pi) * single(6.0)))))) * (exp((-r / (s * single(3.0)))) / r));
end

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. distribute-frac-neg 99.7%

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

   7. *-commutative 99.7%

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

3. Simplified 99.7%

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

   1. add-exp-log 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.6% 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 99.7%

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

   1. times-frac 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. distribute-frac-neg 99.7%

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

   7. *-commutative 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(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(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

2. Final simplification 99.7%

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

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 99.7%

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

   1. times-frac 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. distribute-frac-neg 99.7%

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

   7. *-commutative 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in s around 0 99.7%

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

5. Final simplification 99.7%

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

Alternative 5: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. times-frac 99.7%

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

   3. associate-*l* 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. metadata-eval 99.7%

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

   7. associate-/r* 99.7%

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

   8. associate-*l* 99.7%

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

   9. distribute-lft-out 99.7%

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

3. Simplified 99.3%

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

   1. pow-exp 97.7%

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

   2. associate-*r/ 97.6%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 6: 97.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*l/ 97.8%

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

   3. associate-*l* 97.8%

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

   4. associate-*l* 97.8%

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

   5. associate-/r* 97.8%

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

   6. metadata-eval 97.8%

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

   7. metadata-eval 97.8%

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

   8. associate-/r* 97.8%

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

   9. associate-*l* 97.8%

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

   10. associate-*l* 97.8%

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

   11. distribute-lft-out 97.7%

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

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 inf 97.7%

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

5. Final simplification 97.7%

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

Alternative 7: 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 99.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*l/ 97.8%

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

   3. associate-*l* 97.8%

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

   4. associate-*l* 97.8%

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

   5. associate-/r* 97.8%

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

   6. metadata-eval 97.8%

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

   7. metadata-eval 97.8%

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

   8. associate-/r* 97.8%

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

   9. associate-*l* 97.8%

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

   10. associate-*l* 97.8%

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

   11. distribute-lft-out 97.7%

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

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. Step-by-step derivation

   1. pow-exp 97.7%

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

   2. associate-*r/ 97.6%

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

5. Applied egg-rr 97.6%

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

6. Taylor expanded in r around inf 99.6%

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

   1. associate-*r/ 99.6%

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

   2. mul-1-neg 99.6%

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

   3. associate-*r/ 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-*r/ 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-/r/ 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 8: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*l/ 97.8%

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

   3. associate-*l* 97.8%

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

   4. associate-*l* 97.8%

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

   5. associate-/r* 97.8%

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

   6. metadata-eval 97.8%

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

   7. metadata-eval 97.8%

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

   8. associate-/r* 97.8%

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

   9. associate-*l* 97.8%

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

   10. associate-*l* 97.8%

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

   11. distribute-lft-out 97.7%

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

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 inf 99.6%

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

5. Final simplification 99.6%

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

Alternative 9: 11.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*l/ 97.8%

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

   3. associate-*l* 97.8%

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

   4. associate-*l* 97.8%

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

   5. associate-/r* 97.8%

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

   6. metadata-eval 97.8%

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

   7. metadata-eval 97.8%

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

   8. associate-/r* 97.8%

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

   9. associate-*l* 97.8%

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

   10. associate-*l* 97.8%

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

   11. distribute-lft-out 97.7%

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

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 7.3%

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

   1. associate-*r* 7.3%

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

   2. *-commutative 7.3%

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

   3. associate-*l* 7.3%

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

6. Simplified 7.3%

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

   1. *-commutative 7.3%

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

   2. associate-*r* 7.3%

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

   3. log1p-expm1-u 10.4%

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

   4. associate-*r* 10.4%

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

   5. *-commutative 10.4%

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

   6. associate-*r* 10.4%

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

8. Applied egg-rr 10.4%

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

9. Final simplification 10.4%

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

Alternative 10: 43.2% 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 99.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*l/ 97.8%

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

   3. associate-*l* 97.8%

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

   4. associate-*l* 97.8%

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

   5. associate-/r* 97.8%

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

   6. metadata-eval 97.8%

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

   7. metadata-eval 97.8%

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

   8. associate-/r* 97.8%

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

   9. associate-*l* 97.8%

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

   10. associate-*l* 97.8%

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

   11. distribute-lft-out 97.7%

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

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 7.3%

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

   1. *-commutative 7.3%

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

6. Simplified 7.3%

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

   1. log1p-expm1-u 44.0%

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

8. Applied egg-rr 44.0%

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

9. Final simplification 44.0%

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

Alternative 11: 9.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. times-frac 99.7%

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

   3. associate-*l* 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. metadata-eval 99.7%

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

   7. associate-/r* 99.7%

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

   8. associate-*l* 99.7%

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

   9. distribute-lft-out 99.7%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 7.7%

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

5. Taylor expanded in r around 0 7.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{0.5 \cdot \frac{{r}^{2}}{{s}^{2}} + \left(1 + -1 \cdot \frac{r}{s}\right)}}{r} + \frac{1 + -0.3333333333333333 \cdot \frac{r}{s}}{r}\right) \]
6. Step-by-step derivation

   1. associate-+r+ 7.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{\left(0.5 \cdot \frac{{r}^{2}}{{s}^{2}} + 1\right) + -1 \cdot \frac{r}{s}}}{r} + \frac{1 + -0.3333333333333333 \cdot \frac{r}{s}}{r}\right) \]

   2. mul-1-neg 7.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\left(0.5 \cdot \frac{{r}^{2}}{{s}^{2}} + 1\right) + \color{blue}{\left(-\frac{r}{s}\right)}}{r} + \frac{1 + -0.3333333333333333 \cdot \frac{r}{s}}{r}\right) \]

   3. unsub-neg 7.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{\left(0.5 \cdot \frac{{r}^{2}}{{s}^{2}} + 1\right) - \frac{r}{s}}}{r} + \frac{1 + -0.3333333333333333 \cdot \frac{r}{s}}{r}\right) \]

   4. fma-def 7.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{{r}^{2}}{{s}^{2}}, 1\right)} - \frac{r}{s}}{r} + \frac{1 + -0.3333333333333333 \cdot \frac{r}{s}}{r}\right) \]

   5. unpow2 7.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\mathsf{fma}\left(0.5, \frac{\color{blue}{r \cdot r}}{{s}^{2}}, 1\right) - \frac{r}{s}}{r} + \frac{1 + -0.3333333333333333 \cdot \frac{r}{s}}{r}\right) \]

   6. associate-/l* 7.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\mathsf{fma}\left(0.5, \color{blue}{\frac{r}{\frac{{s}^{2}}{r}}}, 1\right) - \frac{r}{s}}{r} + \frac{1 + -0.3333333333333333 \cdot \frac{r}{s}}{r}\right) \]

   7. unpow2 7.7%

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

7. Simplified 7.7%

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

8. Final simplification 7.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\mathsf{fma}\left(0.5, \frac{r}{\frac{s \cdot s}{r}}, 1\right) - \frac{r}{s}}{r} + \frac{-0.3333333333333333 \cdot \frac{r}{s} + 1}{r}\right) \]

Alternative 12: 9.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. times-frac 99.7%

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

   3. associate-*l* 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. metadata-eval 99.7%

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

   7. associate-/r* 99.7%

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

   8. associate-*l* 99.7%

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

   9. distribute-lft-out 99.7%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 7.7%

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

5. Taylor expanded in r around 0 7.7%

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

   1. cancel-sign-sub-inv 7.7%

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

   2. metadata-eval 7.7%

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

   3. associate-*r/ 7.7%

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

   4. metadata-eval 7.7%

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

7. Simplified 7.7%

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

8. Final simplification 7.7%

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

Alternative 13: 9.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. times-frac 99.7%

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

   3. associate-*l* 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. metadata-eval 99.7%

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

   7. associate-/r* 99.7%

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

   8. associate-*l* 99.7%

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

   9. distribute-lft-out 99.7%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 7.7%

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

5. Taylor expanded in r around inf 7.7%

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

   1. mul-1-neg 7.7%

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

7. Simplified 7.7%

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

8. Final simplification 7.7%

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

Alternative 14: 9.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. times-frac 99.7%

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

   3. associate-*l* 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. metadata-eval 99.7%

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

   7. associate-/r* 99.7%

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

   8. associate-*l* 99.7%

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

   9. distribute-lft-out 99.7%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 7.7%

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

5. Taylor expanded in r around inf 7.7%

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

   1. associate-*r/ 7.7%

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

   2. associate-*r* 7.7%

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

   3. times-frac 7.7%

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

   4. mul-1-neg 7.7%

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

   5. distribute-neg-frac 7.7%

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

7. Simplified 7.7%

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

8. Final simplification 7.7%

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

Alternative 15: 9.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*l/ 97.8%

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

   3. associate-*l* 97.8%

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

   4. associate-*l* 97.8%

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

   5. associate-/r* 97.8%

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

   6. metadata-eval 97.8%

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

   7. metadata-eval 97.8%

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

   8. associate-/r* 97.8%

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

   9. associate-*l* 97.8%

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

   10. associate-*l* 97.8%

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

   11. distribute-lft-out 97.7%

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

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 7.7%

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

5. Taylor expanded in r around 0 7.7%

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

6. Final simplification 7.7%

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

Alternative 16: 9.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*l/ 97.8%

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

   3. associate-*l* 97.8%

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

   4. associate-*l* 97.8%

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

   5. associate-/r* 97.8%

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

   6. metadata-eval 97.8%

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

   7. metadata-eval 97.8%

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

   8. associate-/r* 97.8%

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

   9. associate-*l* 97.8%

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

   10. associate-*l* 97.8%

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

   11. distribute-lft-out 97.7%

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

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 7.7%

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

5. Taylor expanded in r around 0 7.7%

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

   1. associate-/r* 7.7%

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

   2. *-commutative 7.7%

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

7. Simplified 7.7%

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

8. Final simplification 7.7%

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

Alternative 17: 9.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*l/ 97.8%

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

   3. associate-*l* 97.8%

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

   4. associate-*l* 97.8%

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

   5. associate-/r* 97.8%

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

   6. metadata-eval 97.8%

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

   7. metadata-eval 97.8%

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

   8. associate-/r* 97.8%

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

   9. associate-*l* 97.8%

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

   10. associate-*l* 97.8%

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

   11. distribute-lft-out 97.7%

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

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 7.7%

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

5. Final simplification 7.7%

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

Alternative 18: 9.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*l/ 97.8%

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

   3. associate-*l* 97.8%

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

   4. associate-*l* 97.8%

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

   5. associate-/r* 97.8%

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

   6. metadata-eval 97.8%

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

   7. metadata-eval 97.8%

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

   8. associate-/r* 97.8%

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

   9. associate-*l* 97.8%

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

   10. associate-*l* 97.8%

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

   11. distribute-lft-out 97.7%

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

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 7.7%

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

   1. *-un-lft-identity 7.7%

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

   2. *-commutative 7.7%

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

   3. *-commutative 7.7%

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

   4. associate-/r* 7.7%

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

6. Applied egg-rr 7.7%

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

7. Final simplification 7.7%

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

Alternative 19: 8.6% 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 99.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*l/ 97.8%

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

   3. associate-*l* 97.8%

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

   4. associate-*l* 97.8%

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

   5. associate-/r* 97.8%

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

   6. metadata-eval 97.8%

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

   7. metadata-eval 97.8%

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

   8. associate-/r* 97.8%

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

   9. associate-*l* 97.8%

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

   10. associate-*l* 97.8%

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

   11. distribute-lft-out 97.7%

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

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 7.3%

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

   1. associate-*r* 7.3%

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

   2. *-commutative 7.3%

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

   3. associate-*l* 7.3%

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

6. Simplified 7.3%

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

7. Final simplification 7.3%

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

Alternative 20: 8.6% 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 99.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*l/ 97.8%

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

   3. associate-*l* 97.8%

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

   4. associate-*l* 97.8%

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

   5. associate-/r* 97.8%

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

   6. metadata-eval 97.8%

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

   7. metadata-eval 97.8%

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

   8. associate-/r* 97.8%

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

   9. associate-*l* 97.8%

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

   10. associate-*l* 97.8%

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

   11. distribute-lft-out 97.7%

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

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 7.3%

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

   1. *-commutative 7.3%

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

6. Simplified 7.3%

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

7. Final simplification 7.3%

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

Alternative 21: 8.6% accurate, 4.0× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*l/ 97.8%

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

   3. associate-*l* 97.8%

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

   4. associate-*l* 97.8%

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

   5. associate-/r* 97.8%

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

   6. metadata-eval 97.8%

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

   7. metadata-eval 97.8%

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

   8. associate-/r* 97.8%

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

   9. associate-*l* 97.8%

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

   10. associate-*l* 97.8%

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

   11. distribute-lft-out 97.7%

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

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 7.3%

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

   1. associate-*r* 7.3%

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

   2. *-commutative 7.3%

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

   3. associate-*l* 7.3%

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

6. Simplified 7.3%

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

7. Taylor expanded in r around 0 7.3%

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

   1. associate-/r* 7.3%

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

   2. *-commutative 7.3%

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

9. Simplified 7.3%

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

10. Final simplification 7.3%

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

Reproduce

herbie shell --seed 2023188 
(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))))