Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 12.7s

Alternatives: 18

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around inf 99.8%

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

   1. neg-mul-1 99.8%

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

   2. rec-exp 99.8%

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

   3. associate-*r/ 99.8%

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

   4. metadata-eval 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in r around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ 0.125 (* (* PI (exp (/ r s))) (* r s)))
  (/ (* 0.75 (exp (* r (/ -0.3333333333333333 s)))) (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	return (0.125f / ((((float) M_PI) * expf((r / s))) * (r * s))) + ((0.75f * expf((r * (-0.3333333333333333f / s)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(Float32(Float32(pi) * exp(Float32(r / s))) * Float32(r * s))) + Float32(Float32(Float32(0.75) * exp(Float32(r * Float32(Float32(-0.3333333333333333) / s)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / ((single(pi) * exp((r / s))) * (r * s))) + ((single(0.75) * exp((r * (single(-0.3333333333333333) / s)))) / (r * (s * (single(pi) * single(6.0)))));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around inf 99.8%

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

   1. neg-mul-1 99.8%

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

   2. rec-exp 99.8%

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

   3. associate-*r/ 99.8%

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

   4. metadata-eval 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in r around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in r around 0 99.7%

   \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
10. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    2. *-rgt-identity 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\frac{\color{blue}{r \cdot 1}}{s} \cdot -0.3333333333333333}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    3. associate-*r/ 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{\left(r \cdot \frac{1}{s}\right)} \cdot -0.3333333333333333}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    4. associate-*l* 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \left(\frac{1}{s} \cdot -0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    5. metadata-eval 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(\frac{1}{s} \cdot \color{blue}{\left(-0.3333333333333333\right)}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    6. distribute-rgt-neg-in 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \color{blue}{\left(-\frac{1}{s} \cdot 0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    7. *-commutative 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\color{blue}{0.3333333333333333 \cdot \frac{1}{s}}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    8. associate-*r/ 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{s}}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    9. metadata-eval 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\frac{\color{blue}{0.3333333333333333}}{s}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    10. distribute-neg-frac 99.7%

        \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \color{blue}{\frac{-0.3333333333333333}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    11. metadata-eval 99.7%

        \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \frac{\color{blue}{-0.3333333333333333}}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

11. Simplified 99.7%

    \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \frac{-0.3333333333333333}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

12. Final simplification 99.7%

    \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
13. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around 0 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

   1. pow-exp 99.7%

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

   2. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. clear-num 99.6%

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

   3. un-div-inv 99.7%

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

8. Applied egg-rr 99.7%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

   1. pow-exp 99.7%

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

   2. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 7: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around inf 99.6%

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

6. Final simplification 99.6%

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

Alternative 8: 48.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 30:\\ \;\;\;\;\frac{0.75 \cdot e^{\frac{r}{3 \cdot \left(-s\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{r \cdot \left(s \cdot \pi + r \cdot \left(\pi + \frac{\pi}{s} \cdot \left(r \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 30.0)
   (+
    (/ (* 0.75 (exp (/ r (* 3.0 (- s))))) (* r (* s (* PI 6.0))))
    (/ 0.125 (* r (+ (* s PI) (* r (+ PI (* (/ PI s) (* r 0.5))))))))
   (/ 0.25 (* s (log1p (expm1 (* r PI)))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 30.0f) {
		tmp = ((0.75f * expf((r / (3.0f * -s)))) / (r * (s * (((float) M_PI) * 6.0f)))) + (0.125f / (r * ((s * ((float) M_PI)) + (r * (((float) M_PI) + ((((float) M_PI) / s) * (r * 0.5f)))))));
	} else {
		tmp = 0.25f / (s * log1pf(expm1f((r * ((float) M_PI)))));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(30.0))
		tmp = Float32(Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(Float32(3.0) * Float32(-s))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))) + Float32(Float32(0.125) / Float32(r * Float32(Float32(s * Float32(pi)) + Float32(r * Float32(Float32(pi) + Float32(Float32(Float32(pi) / s) * Float32(r * Float32(0.5)))))))));
	else
		tmp = Float32(Float32(0.25) / Float32(s * log1p(expm1(Float32(r * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 30

   1. Initial program 99.7%

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

   3. Taylor expanded in r around inf 99.7%

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

      1. neg-mul-1 99.7%

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

      2. rec-exp 99.6%

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

      3. associate-*r/ 99.6%

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

      4. metadata-eval 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in r around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l* 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in r around 0 20.7%

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

       1. associate-*r/ 20.7%

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

   11. Applied egg-rr 20.7%

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

       1. associate-*r/ 20.7%

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

       2. *-commutative 20.7%

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

       3. associate-*r/ 21.9%

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

       4. *-commutative 21.9%

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

       5. associate-*l* 21.9%

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

   13. Simplified 21.9%

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

   if 30 < r

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around inf 5.3%

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

      1. *-commutative 5.3%

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

      2. associate-*l* 5.3%

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

      3. *-commutative 5.3%

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

   7. Simplified 5.3%

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

      1. log1p-expm1-u 98.3%

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

      2. *-commutative 98.3%

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

   9. Applied egg-rr 98.3%

      \[\leadsto \frac{0.25}{s \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 30:\\ \;\;\;\;\frac{0.75 \cdot e^{\frac{r}{3 \cdot \left(-s\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{r \cdot \left(s \cdot \pi + r \cdot \left(\pi + \frac{\pi}{s} \cdot \left(r \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.5% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.75 (exp (/ r (* 3.0 (- s))))) (* r (* s (* PI 6.0))))
  (/ 0.125 (* r (+ (* s PI) (* r (+ PI (* (/ PI s) (* r 0.5)))))))))

C

float code(float s, float r) {
	return ((0.75f * expf((r / (3.0f * -s)))) / (r * (s * (((float) M_PI) * 6.0f)))) + (0.125f / (r * ((s * ((float) M_PI)) + (r * (((float) M_PI) + ((((float) M_PI) / s) * (r * 0.5f)))))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(Float32(3.0) * Float32(-s))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))) + Float32(Float32(0.125) / Float32(r * Float32(Float32(s * Float32(pi)) + Float32(r * Float32(Float32(pi) + Float32(Float32(Float32(pi) / s) * Float32(r * Float32(0.5)))))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp((r / (single(3.0) * -s)))) / (r * (s * (single(pi) * single(6.0))))) + (single(0.125) / (r * ((s * single(pi)) + (r * (single(pi) + ((single(pi) / s) * (r * single(0.5))))))));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around inf 99.8%

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

   1. neg-mul-1 99.8%

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

   2. rec-exp 99.8%

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

   3. associate-*r/ 99.8%

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

   4. metadata-eval 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in r around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in r around 0 28.7%

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

    1. associate-*r/ 28.7%

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

11. Applied egg-rr 28.7%

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

    1. associate-*r/ 28.7%

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

    2. *-commutative 28.7%

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

    3. associate-*r/ 29.4%

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

    4. *-commutative 29.4%

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

    5. associate-*l* 29.4%

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

13. Simplified 29.4%

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

14. Final simplification 29.4%

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

Alternative 10: 26.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{r \cdot \left(s \cdot \pi + r \cdot \left(\pi + 0.5 \cdot \frac{r \cdot \pi}{s}\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.75 (exp (* r (/ -0.3333333333333333 s)))) (* r (* s (* PI 6.0))))
  (/ 0.125 (* r (+ (* s PI) (* r (+ PI (* 0.5 (/ (* r PI) s)))))))))

C

float code(float s, float r) {
	return ((0.75f * expf((r * (-0.3333333333333333f / s)))) / (r * (s * (((float) M_PI) * 6.0f)))) + (0.125f / (r * ((s * ((float) M_PI)) + (r * (((float) M_PI) + (0.5f * ((r * ((float) M_PI)) / s)))))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.75) * exp(Float32(r * Float32(Float32(-0.3333333333333333) / s)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))) + Float32(Float32(0.125) / Float32(r * Float32(Float32(s * Float32(pi)) + Float32(r * Float32(Float32(pi) + Float32(Float32(0.5) * Float32(Float32(r * Float32(pi)) / s))))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp((r * (single(-0.3333333333333333) / s)))) / (r * (s * (single(pi) * single(6.0))))) + (single(0.125) / (r * ((s * single(pi)) + (r * (single(pi) + (single(0.5) * ((r * single(pi)) / s)))))));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around inf 99.8%

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

   1. neg-mul-1 99.8%

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

   2. rec-exp 99.8%

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

   3. associate-*r/ 99.8%

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

   4. metadata-eval 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in r around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in r around 0 28.7%

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

10. Taylor expanded in r around 0 28.7%

    \[\leadsto \frac{0.125}{r \cdot \left(r \cdot \left(\pi + 0.5 \cdot \frac{r \cdot \pi}{s}\right) + s \cdot \pi\right)} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
11. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    2. *-rgt-identity 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\frac{\color{blue}{r \cdot 1}}{s} \cdot -0.3333333333333333}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    3. associate-*r/ 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{\left(r \cdot \frac{1}{s}\right)} \cdot -0.3333333333333333}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    4. associate-*l* 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \left(\frac{1}{s} \cdot -0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    5. metadata-eval 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(\frac{1}{s} \cdot \color{blue}{\left(-0.3333333333333333\right)}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    6. distribute-rgt-neg-in 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \color{blue}{\left(-\frac{1}{s} \cdot 0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    7. *-commutative 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\color{blue}{0.3333333333333333 \cdot \frac{1}{s}}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    8. associate-*r/ 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{s}}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    9. metadata-eval 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\frac{\color{blue}{0.3333333333333333}}{s}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    10. distribute-neg-frac 99.7%

        \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \color{blue}{\frac{-0.3333333333333333}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    11. metadata-eval 99.7%

        \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \frac{\color{blue}{-0.3333333333333333}}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

12. Simplified 28.7%

    \[\leadsto \frac{0.125}{r \cdot \left(r \cdot \left(\pi + 0.5 \cdot \frac{r \cdot \pi}{s}\right) + s \cdot \pi\right)} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \frac{-0.3333333333333333}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

13. Final simplification 28.7%

    \[\leadsto \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{r \cdot \left(s \cdot \pi + r \cdot \left(\pi + 0.5 \cdot \frac{r \cdot \pi}{s}\right)\right)} \]
14. Add Preprocessing

Alternative 11: 16.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{\left(r \cdot s\right) \cdot \left(\pi + r \cdot \frac{\pi}{s}\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.75 (exp (* r (/ -0.3333333333333333 s)))) (* r (* s (* PI 6.0))))
  (/ 0.125 (* (* r s) (+ PI (* r (/ PI s)))))))

C

float code(float s, float r) {
	return ((0.75f * expf((r * (-0.3333333333333333f / s)))) / (r * (s * (((float) M_PI) * 6.0f)))) + (0.125f / ((r * s) * (((float) M_PI) + (r * (((float) M_PI) / s)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.75) * exp(Float32(r * Float32(Float32(-0.3333333333333333) / s)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))) + Float32(Float32(0.125) / Float32(Float32(r * s) * Float32(Float32(pi) + Float32(r * Float32(Float32(pi) / s))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp((r * (single(-0.3333333333333333) / s)))) / (r * (s * (single(pi) * single(6.0))))) + (single(0.125) / ((r * s) * (single(pi) + (r * (single(pi) / s)))));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around inf 99.8%

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

   1. neg-mul-1 99.8%

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

   2. rec-exp 99.8%

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

   3. associate-*r/ 99.8%

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

   4. metadata-eval 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in r around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in r around 0 99.7%

   \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
10. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    2. *-rgt-identity 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\frac{\color{blue}{r \cdot 1}}{s} \cdot -0.3333333333333333}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    3. associate-*r/ 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{\left(r \cdot \frac{1}{s}\right)} \cdot -0.3333333333333333}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    4. associate-*l* 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \left(\frac{1}{s} \cdot -0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    5. metadata-eval 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(\frac{1}{s} \cdot \color{blue}{\left(-0.3333333333333333\right)}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    6. distribute-rgt-neg-in 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \color{blue}{\left(-\frac{1}{s} \cdot 0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    7. *-commutative 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\color{blue}{0.3333333333333333 \cdot \frac{1}{s}}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    8. associate-*r/ 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{s}}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    9. metadata-eval 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\frac{\color{blue}{0.3333333333333333}}{s}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    10. distribute-neg-frac 99.7%

        \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \color{blue}{\frac{-0.3333333333333333}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    11. metadata-eval 99.7%

        \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \frac{\color{blue}{-0.3333333333333333}}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

11. Simplified 99.7%

    \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \frac{-0.3333333333333333}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

12. Taylor expanded in r around 0 17.1%

    \[\leadsto \frac{0.125}{\color{blue}{\left(\pi + \frac{r \cdot \pi}{s}\right)} \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
13. Step-by-step derivation

    1. associate-/l* 17.9%

       \[\leadsto \frac{0.125}{\left(\pi + \color{blue}{r \cdot \frac{\pi}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

14. Simplified 17.9%

    \[\leadsto \frac{0.125}{\color{blue}{\left(\pi + r \cdot \frac{\pi}{s}\right)} \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

15. Final simplification 17.9%

    \[\leadsto \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{\left(r \cdot s\right) \cdot \left(\pi + r \cdot \frac{\pi}{s}\right)} \]
16. Add Preprocessing

Alternative 12: 15.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{\left(r \cdot s\right) \cdot \left(\pi \cdot \left(\frac{r}{s} + 1\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.75 (exp (* r (/ -0.3333333333333333 s)))) (* r (* s (* PI 6.0))))
  (/ 0.125 (* (* r s) (* PI (+ (/ r s) 1.0))))))

C

float code(float s, float r) {
	return ((0.75f * expf((r * (-0.3333333333333333f / s)))) / (r * (s * (((float) M_PI) * 6.0f)))) + (0.125f / ((r * s) * (((float) M_PI) * ((r / s) + 1.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.75) * exp(Float32(r * Float32(Float32(-0.3333333333333333) / s)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))) + Float32(Float32(0.125) / Float32(Float32(r * s) * Float32(Float32(pi) * Float32(Float32(r / s) + Float32(1.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp((r * (single(-0.3333333333333333) / s)))) / (r * (s * (single(pi) * single(6.0))))) + (single(0.125) / ((r * s) * (single(pi) * ((r / s) + single(1.0)))));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around inf 99.8%

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

   1. neg-mul-1 99.8%

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

   2. rec-exp 99.8%

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

   3. associate-*r/ 99.8%

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

   4. metadata-eval 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in r around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in r around 0 99.7%

   \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
10. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    2. *-rgt-identity 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\frac{\color{blue}{r \cdot 1}}{s} \cdot -0.3333333333333333}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    3. associate-*r/ 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{\left(r \cdot \frac{1}{s}\right)} \cdot -0.3333333333333333}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    4. associate-*l* 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \left(\frac{1}{s} \cdot -0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    5. metadata-eval 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(\frac{1}{s} \cdot \color{blue}{\left(-0.3333333333333333\right)}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    6. distribute-rgt-neg-in 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \color{blue}{\left(-\frac{1}{s} \cdot 0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    7. *-commutative 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\color{blue}{0.3333333333333333 \cdot \frac{1}{s}}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    8. associate-*r/ 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{s}}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    9. metadata-eval 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\frac{\color{blue}{0.3333333333333333}}{s}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    10. distribute-neg-frac 99.7%

        \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \color{blue}{\frac{-0.3333333333333333}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    11. metadata-eval 99.7%

        \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \frac{\color{blue}{-0.3333333333333333}}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

11. Simplified 99.7%

    \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \frac{-0.3333333333333333}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

12. Taylor expanded in r around 0 17.1%

    \[\leadsto \frac{0.125}{\left(\pi \cdot \color{blue}{\left(1 + \frac{r}{s}\right)}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

13. Final simplification 17.1%

    \[\leadsto \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{\left(r \cdot s\right) \cdot \left(\pi \cdot \left(\frac{r}{s} + 1\right)\right)} \]
14. Add Preprocessing

Alternative 13: 12.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around inf 99.8%

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

   1. neg-mul-1 99.8%

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

   2. rec-exp 99.8%

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

   3. associate-*r/ 99.8%

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

   4. metadata-eval 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in r around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in r around 0 12.1%

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

    1. distribute-rgt-out 12.1%

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

11. Simplified 12.1%

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

12. Taylor expanded in r around 0 12.1%

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

    1. +-commutative 12.1%

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

    2. distribute-rgt-in 12.1%

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

    3. *-commutative 12.1%

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

    4. associate-*l* 12.1%

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

    5. +-commutative 12.1%

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

14. Simplified 12.1%

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

15. Final simplification 12.1%

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

Alternative 14: 12.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{r \cdot \left(\pi \cdot \left(r + s\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.75 (exp (* r (/ -0.3333333333333333 s)))) (* r (* s (* PI 6.0))))
  (/ 0.125 (* r (* PI (+ r s))))))

C

float code(float s, float r) {
	return ((0.75f * expf((r * (-0.3333333333333333f / s)))) / (r * (s * (((float) M_PI) * 6.0f)))) + (0.125f / (r * (((float) M_PI) * (r + s))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.75) * exp(Float32(r * Float32(Float32(-0.3333333333333333) / s)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))) + Float32(Float32(0.125) / Float32(r * Float32(Float32(pi) * Float32(r + s)))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp((r * (single(-0.3333333333333333) / s)))) / (r * (s * (single(pi) * single(6.0))))) + (single(0.125) / (r * (single(pi) * (r + s))));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around inf 99.8%

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

   1. neg-mul-1 99.8%

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

   2. rec-exp 99.8%

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

   3. associate-*r/ 99.8%

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

   4. metadata-eval 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in r around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in r around 0 12.1%

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

    1. distribute-rgt-out 12.1%

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

11. Simplified 12.1%

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

12. Taylor expanded in r around 0 12.1%

    \[\leadsto \frac{0.125}{r \cdot \left(\pi \cdot \left(r + s\right)\right)} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
13. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    2. *-rgt-identity 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\frac{\color{blue}{r \cdot 1}}{s} \cdot -0.3333333333333333}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    3. associate-*r/ 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{\left(r \cdot \frac{1}{s}\right)} \cdot -0.3333333333333333}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    4. associate-*l* 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \left(\frac{1}{s} \cdot -0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    5. metadata-eval 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(\frac{1}{s} \cdot \color{blue}{\left(-0.3333333333333333\right)}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    6. distribute-rgt-neg-in 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \color{blue}{\left(-\frac{1}{s} \cdot 0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    7. *-commutative 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\color{blue}{0.3333333333333333 \cdot \frac{1}{s}}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    8. associate-*r/ 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{s}}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    9. metadata-eval 99.7%

       \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \left(-\frac{\color{blue}{0.3333333333333333}}{s}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    10. distribute-neg-frac 99.7%

        \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \color{blue}{\frac{-0.3333333333333333}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    11. metadata-eval 99.7%

        \[\leadsto \frac{0.125}{\left(\pi \cdot e^{\frac{r}{s}}\right) \cdot \left(s \cdot r\right)} + \frac{0.75 \cdot e^{r \cdot \frac{\color{blue}{-0.3333333333333333}}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

14. Simplified 12.1%

    \[\leadsto \frac{0.125}{r \cdot \left(\pi \cdot \left(r + s\right)\right)} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \frac{-0.3333333333333333}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

15. Final simplification 12.1%

    \[\leadsto \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{r \cdot \left(\pi \cdot \left(r + s\right)\right)} \]
16. Add Preprocessing

Alternative 15: 10.5% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. times-frac 99.8%

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

   3. fma-define 99.8%

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

   4. associate-*l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around -inf 9.6%

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

   1. mul-1-neg 9.6%

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

7. Simplified 9.6%

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

8. Final simplification 9.6%

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

Alternative 16: 9.5% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * single(pi))) - (single(0.16666666666666666) / (s * single(pi)))) / s;
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around inf 9.1%

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

   1. associate-*r/ 9.1%

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

   2. metadata-eval 9.1%

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

   3. associate-*r/ 9.1%

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

   4. metadata-eval 9.1%

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

7. Simplified 9.1%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]
8. Add Preprocessing

Alternative 17: 9.3% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 9.1%

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

6. Taylor expanded in r around 0 8.8%

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

   1. associate-*r* 8.8%

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

   2. *-commutative 8.8%

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

8. Simplified 8.8%

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

9. Final simplification 8.8%

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

Alternative 18: 9.3% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around inf 8.8%

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

Reproduce

herbie shell --seed 2024139 
(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))))