Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 18.0s

Alternatives: 21

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (exp (/ (* r -0.3333333333333333) 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 * -0.3333333333333333f) / s))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(r * Float32(-0.3333333333333333)) / 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(-0.3333333333333333)) / s))) / (r * (s * (single(pi) * single(6.0)))));
end

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in r around 0 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

   1. associate-*r/ 99.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Final simplification 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (exp (* -0.3333333333333333 (/ r 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((-0.3333333333333333f * (r / s)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-0.3333333333333333) * Float32(r / 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((single(-0.3333333333333333) * (r / s)))) / (r * (s * (single(pi) * single(6.0)))));
end

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in r around 0 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Final simplification 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

Alternative 3: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

   1. associate-/l/ 99.4%

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

   2. metadata-eval 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-/r* 99.4%

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

   5. div-inv 99.4%

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

   6. associate-*r* 99.4%

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

   7. *-commutative 99.4%

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

   8. *-commutative 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in r around inf 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around inf 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 43.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 9.3%

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

5. Taylor expanded in s around inf 8.8%

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

   1. log1p-expm1-u 47.3%

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

   2. *-commutative 47.3%

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

7. Applied egg-rr 47.3%

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

8. Final simplification 47.3%

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

Alternative 6: 10.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \frac{r}{s} \cdot \left(-0.3333333333333333 + \frac{r}{s} \cdot 0.05555555555555555\right)}{r}\right) \cdot \frac{0.125}{s \cdot \pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (+
   (/ (exp (/ r (- s))) r)
   (/
    (+ 1.0 (* (/ r s) (+ -0.3333333333333333 (* (/ r s) 0.05555555555555555))))
    r))
  (/ 0.125 (* s PI))))

C

float code(float s, float r) {
	return ((expf((r / -s)) / r) + ((1.0f + ((r / s) * (-0.3333333333333333f + ((r / s) * 0.05555555555555555f)))) / r)) * (0.125f / (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(Float32(1.0) + Float32(Float32(r / s) * Float32(Float32(-0.3333333333333333) + Float32(Float32(r / s) * Float32(0.05555555555555555))))) / r)) * Float32(Float32(0.125) / Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((r / -s)) / r) + ((single(1.0) + ((r / s) * (single(-0.3333333333333333) + ((r / s) * single(0.05555555555555555))))) / r)) * (single(0.125) / (s * single(pi)));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \left(0.05555555555555555 \cdot \frac{{r}^{2}}{{s}^{2}} + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]
5. Step-by-step derivation

   1. fma-def 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\mathsf{fma}\left(0.05555555555555555, \frac{{r}^{2}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

   2. unpow2 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{\color{blue}{r \cdot r}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

   3. unpow2 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{\color{blue}{s \cdot s}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

6. Simplified 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{s \cdot s}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

7. Taylor expanded in s around 0 10.6%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{s \cdot s}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

8. Taylor expanded in r around 0 10.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\left(0.05555555555555555 \cdot \frac{{r}^{2}}{{s}^{2}} + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]
9. Step-by-step derivation

   1. +-commutative 10.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\left(-0.3333333333333333 \cdot \frac{r}{s} + 0.05555555555555555 \cdot \frac{{r}^{2}}{{s}^{2}}\right)}}{r}\right) \]

   2. unpow2 10.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \left(-0.3333333333333333 \cdot \frac{r}{s} + 0.05555555555555555 \cdot \frac{\color{blue}{r \cdot r}}{{s}^{2}}\right)}{r}\right) \]

   3. unpow2 10.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \left(-0.3333333333333333 \cdot \frac{r}{s} + 0.05555555555555555 \cdot \frac{r \cdot r}{\color{blue}{s \cdot s}}\right)}{r}\right) \]

   4. times-frac 10.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \left(-0.3333333333333333 \cdot \frac{r}{s} + 0.05555555555555555 \cdot \color{blue}{\left(\frac{r}{s} \cdot \frac{r}{s}\right)}\right)}{r}\right) \]

   5. associate-*r* 10.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \left(-0.3333333333333333 \cdot \frac{r}{s} + \color{blue}{\left(0.05555555555555555 \cdot \frac{r}{s}\right) \cdot \frac{r}{s}}\right)}{r}\right) \]

   6. distribute-rgt-out 10.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\frac{r}{s} \cdot \left(-0.3333333333333333 + 0.05555555555555555 \cdot \frac{r}{s}\right)}}{r}\right) \]

   7. *-commutative 10.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \frac{r}{s} \cdot \left(-0.3333333333333333 + \color{blue}{\frac{r}{s} \cdot 0.05555555555555555}\right)}{r}\right) \]

10. Simplified 10.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\frac{r}{s} \cdot \left(-0.3333333333333333 + \frac{r}{s} \cdot 0.05555555555555555\right)}}{r}\right) \]

11. Final simplification 10.7%

    \[\leadsto \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \frac{r}{s} \cdot \left(-0.3333333333333333 + \frac{r}{s} \cdot 0.05555555555555555\right)}{r}\right) \cdot \frac{0.125}{s \cdot \pi} \]

Alternative 7: 10.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} + \left(\frac{r}{s} \cdot \frac{0.05555555555555555}{s} + \frac{-0.3333333333333333}{s}\right)\right)\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ (/ 0.125 PI) s)
  (+
   (/ (exp (/ r (- s))) r)
   (+
    (/ 1.0 r)
    (+ (* (/ r s) (/ 0.05555555555555555 s)) (/ -0.3333333333333333 s))))))

C

float code(float s, float r) {
	return ((0.125f / ((float) M_PI)) / s) * ((expf((r / -s)) / r) + ((1.0f / r) + (((r / s) * (0.05555555555555555f / s)) + (-0.3333333333333333f / s))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(pi)) / s) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(Float32(1.0) / r) + Float32(Float32(Float32(r / s) * Float32(Float32(0.05555555555555555) / s)) + Float32(Float32(-0.3333333333333333) / s)))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / single(pi)) / s) * ((exp((r / -s)) / r) + ((single(1.0) / r) + (((r / s) * (single(0.05555555555555555) / s)) + (single(-0.3333333333333333) / s))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \left(0.05555555555555555 \cdot \frac{{r}^{2}}{{s}^{2}} + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]
5. Step-by-step derivation

   1. fma-def 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\mathsf{fma}\left(0.05555555555555555, \frac{{r}^{2}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

   2. unpow2 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{\color{blue}{r \cdot r}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

   3. unpow2 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{\color{blue}{s \cdot s}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

6. Simplified 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{s \cdot s}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

7. Taylor expanded in r around 0 10.7%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \color{blue}{\left(\left(\frac{1}{r} + 0.05555555555555555 \cdot \frac{r}{{s}^{2}}\right) - 0.3333333333333333 \cdot \frac{1}{s}\right)}\right) \]
8. Step-by-step derivation

   1. associate--l+ 10.7%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \color{blue}{\left(\frac{1}{r} + \left(0.05555555555555555 \cdot \frac{r}{{s}^{2}} - 0.3333333333333333 \cdot \frac{1}{s}\right)\right)}\right) \]

   2. associate-*r/ 10.7%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} + \left(\color{blue}{\frac{0.05555555555555555 \cdot r}{{s}^{2}}} - 0.3333333333333333 \cdot \frac{1}{s}\right)\right)\right) \]

   3. unpow2 10.7%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} + \left(\frac{0.05555555555555555 \cdot r}{\color{blue}{s \cdot s}} - 0.3333333333333333 \cdot \frac{1}{s}\right)\right)\right) \]

   4. times-frac 10.7%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} + \left(\color{blue}{\frac{0.05555555555555555}{s} \cdot \frac{r}{s}} - 0.3333333333333333 \cdot \frac{1}{s}\right)\right)\right) \]

   5. fma-neg 10.7%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} + \color{blue}{\mathsf{fma}\left(\frac{0.05555555555555555}{s}, \frac{r}{s}, -0.3333333333333333 \cdot \frac{1}{s}\right)}\right)\right) \]

   6. associate-*r/ 10.7%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} + \mathsf{fma}\left(\frac{0.05555555555555555}{s}, \frac{r}{s}, -\color{blue}{\frac{0.3333333333333333 \cdot 1}{s}}\right)\right)\right) \]

   7. metadata-eval 10.7%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} + \mathsf{fma}\left(\frac{0.05555555555555555}{s}, \frac{r}{s}, -\frac{\color{blue}{0.3333333333333333}}{s}\right)\right)\right) \]

   8. distribute-neg-frac 10.7%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} + \mathsf{fma}\left(\frac{0.05555555555555555}{s}, \frac{r}{s}, \color{blue}{\frac{-0.3333333333333333}{s}}\right)\right)\right) \]

   9. metadata-eval 10.7%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} + \mathsf{fma}\left(\frac{0.05555555555555555}{s}, \frac{r}{s}, \frac{\color{blue}{-0.3333333333333333}}{s}\right)\right)\right) \]

9. Simplified 10.7%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \color{blue}{\left(\frac{1}{r} + \mathsf{fma}\left(\frac{0.05555555555555555}{s}, \frac{r}{s}, \frac{-0.3333333333333333}{s}\right)\right)}\right) \]
10. Step-by-step derivation

    1. fma-udef 10.7%

       \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} + \color{blue}{\left(\frac{0.05555555555555555}{s} \cdot \frac{r}{s} + \frac{-0.3333333333333333}{s}\right)}\right)\right) \]

11. Applied egg-rr 10.7%

    \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} + \color{blue}{\left(\frac{0.05555555555555555}{s} \cdot \frac{r}{s} + \frac{-0.3333333333333333}{s}\right)}\right)\right) \]

12. Final simplification 10.7%

    \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} + \left(\frac{r}{s} \cdot \frac{0.05555555555555555}{s} + \frac{-0.3333333333333333}{s}\right)\right)\right) \]

Alternative 8: 10.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \frac{r}{s} \cdot \left(-0.3333333333333333 + \frac{r}{s} \cdot 0.05555555555555555\right)}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ (/ 0.125 PI) s)
  (+
   (/ (exp (/ r (- s))) r)
   (/
    (+ 1.0 (* (/ r s) (+ -0.3333333333333333 (* (/ r s) 0.05555555555555555))))
    r))))

C

float code(float s, float r) {
	return ((0.125f / ((float) M_PI)) / s) * ((expf((r / -s)) / r) + ((1.0f + ((r / s) * (-0.3333333333333333f + ((r / s) * 0.05555555555555555f)))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(pi)) / s) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(Float32(1.0) + Float32(Float32(r / s) * Float32(Float32(-0.3333333333333333) + Float32(Float32(r / s) * Float32(0.05555555555555555))))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / single(pi)) / s) * ((exp((r / -s)) / r) + ((single(1.0) + ((r / s) * (single(-0.3333333333333333) + ((r / s) * single(0.05555555555555555))))) / r));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \left(0.05555555555555555 \cdot \frac{{r}^{2}}{{s}^{2}} + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]
5. Step-by-step derivation

   1. fma-def 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\mathsf{fma}\left(0.05555555555555555, \frac{{r}^{2}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

   2. unpow2 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{\color{blue}{r \cdot r}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

   3. unpow2 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{\color{blue}{s \cdot s}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

6. Simplified 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{s \cdot s}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

7. Taylor expanded in r around 0 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\left(0.05555555555555555 \cdot \frac{{r}^{2}}{{s}^{2}} + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]
8. Step-by-step derivation

   1. +-commutative 10.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\left(-0.3333333333333333 \cdot \frac{r}{s} + 0.05555555555555555 \cdot \frac{{r}^{2}}{{s}^{2}}\right)}}{r}\right) \]

   2. unpow2 10.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \left(-0.3333333333333333 \cdot \frac{r}{s} + 0.05555555555555555 \cdot \frac{\color{blue}{r \cdot r}}{{s}^{2}}\right)}{r}\right) \]

   3. unpow2 10.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \left(-0.3333333333333333 \cdot \frac{r}{s} + 0.05555555555555555 \cdot \frac{r \cdot r}{\color{blue}{s \cdot s}}\right)}{r}\right) \]

   4. times-frac 10.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \left(-0.3333333333333333 \cdot \frac{r}{s} + 0.05555555555555555 \cdot \color{blue}{\left(\frac{r}{s} \cdot \frac{r}{s}\right)}\right)}{r}\right) \]

   5. associate-*r* 10.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \left(-0.3333333333333333 \cdot \frac{r}{s} + \color{blue}{\left(0.05555555555555555 \cdot \frac{r}{s}\right) \cdot \frac{r}{s}}\right)}{r}\right) \]

   6. distribute-rgt-out 10.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\frac{r}{s} \cdot \left(-0.3333333333333333 + 0.05555555555555555 \cdot \frac{r}{s}\right)}}{r}\right) \]

   7. *-commutative 10.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \frac{r}{s} \cdot \left(-0.3333333333333333 + \color{blue}{\frac{r}{s} \cdot 0.05555555555555555}\right)}{r}\right) \]

9. Simplified 10.7%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\frac{r}{s} \cdot \left(-0.3333333333333333 + \frac{r}{s} \cdot 0.05555555555555555\right)}}{r}\right) \]

10. Final simplification 10.7%

    \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \frac{r}{s} \cdot \left(-0.3333333333333333 + \frac{r}{s} \cdot 0.05555555555555555\right)}{r}\right) \]

Alternative 9: 9.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

   1. associate-/l/ 99.4%

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

   2. metadata-eval 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-/r* 99.4%

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

   5. div-inv 99.4%

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

   6. associate-*r* 99.4%

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

   7. *-commutative 99.4%

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

   8. *-commutative 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in r around 0 9.5%

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

7. Final simplification 9.5%

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

Alternative 10: 9.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 9.5%

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

5. Final simplification 9.5%

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

Alternative 11: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \left(0.05555555555555555 \cdot \frac{{r}^{2}}{{s}^{2}} + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]
5. Step-by-step derivation

   1. fma-def 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\mathsf{fma}\left(0.05555555555555555, \frac{{r}^{2}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

   2. unpow2 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{\color{blue}{r \cdot r}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

   3. unpow2 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{\color{blue}{s \cdot s}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

6. Simplified 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{s \cdot s}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

7. Taylor expanded in s around 0 10.6%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{s \cdot s}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

8. Taylor expanded in r around 0 9.5%

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

   1. associate-*r/ 9.5%

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

   2. metadata-eval 9.5%

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

10. Simplified 9.5%

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

11. Final simplification 9.5%

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

Alternative 12: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \left(0.05555555555555555 \cdot \frac{{r}^{2}}{{s}^{2}} + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]
5. Step-by-step derivation

   1. fma-def 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\mathsf{fma}\left(0.05555555555555555, \frac{{r}^{2}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

   2. unpow2 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{\color{blue}{r \cdot r}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

   3. unpow2 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{\color{blue}{s \cdot s}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

6. Simplified 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{s \cdot s}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

7. Taylor expanded in r around 0 9.5%

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

   1. associate-*r/ 9.5%

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

   2. metadata-eval 9.5%

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

9. Simplified 9.5%

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

10. Final simplification 9.5%

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

Alternative 13: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 9.3%

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

   1. div-inv 9.3%

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

6. Applied egg-rr 9.3%

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

7. Final simplification 9.3%

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

Alternative 14: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 9.3%

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

5. Taylor expanded in s around 0 9.3%

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

   1. associate-*r/ 9.3%

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

   2. mul-1-neg 9.3%

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

7. Simplified 9.3%

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

8. Final simplification 9.3%

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

Alternative 15: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + (1.0f / 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(Float32(1.0) / r)))
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 9.3%

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

5. Taylor expanded in s around 0 9.3%

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

6. Final simplification 9.3%

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

Alternative 16: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 9.3%

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

5. Taylor expanded in r around inf 9.3%

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

   1. mul-1-neg 9.3%

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

7. Simplified 9.3%

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

8. Final simplification 9.3%

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

Alternative 17: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 9.3%

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

5. Taylor expanded in r around inf 9.3%

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

   1. associate-*r/ 9.3%

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

   2. mul-1-neg 9.3%

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

   3. *-commutative 9.3%

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

   4. associate-*l* 9.3%

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

   5. *-commutative 9.3%

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

7. Simplified 9.3%

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

8. Final simplification 9.3%

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

Alternative 18: 9.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 9.3%

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

5. Taylor expanded in r around 0 8.8%

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

6. Final simplification 8.8%

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

Alternative 19: 7.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \left(0.05555555555555555 \cdot \frac{{r}^{2}}{{s}^{2}} + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]
5. Step-by-step derivation

   1. fma-def 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\mathsf{fma}\left(0.05555555555555555, \frac{{r}^{2}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

   2. unpow2 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{\color{blue}{r \cdot r}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

   3. unpow2 10.6%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{\color{blue}{s \cdot s}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

6. Simplified 10.6%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{s \cdot s}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

7. Taylor expanded in r around inf 6.0%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \color{blue}{0.05555555555555555 \cdot \frac{r}{{s}^{2}}}\right) \]
8. Step-by-step derivation

   1. *-commutative 6.0%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \color{blue}{\frac{r}{{s}^{2}} \cdot 0.05555555555555555}\right) \]

   2. associate-*l/ 6.0%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \color{blue}{\frac{r \cdot 0.05555555555555555}{{s}^{2}}}\right) \]

   3. unpow2 6.0%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{r \cdot 0.05555555555555555}{\color{blue}{s \cdot s}}\right) \]

9. Simplified 6.0%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \color{blue}{\frac{r \cdot 0.05555555555555555}{s \cdot s}}\right) \]

10. Taylor expanded in s around inf 7.1%

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

11. Final simplification 7.1%

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

Alternative 20: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 9.3%

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

5. Taylor expanded in s around inf 8.8%

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

6. Final simplification 8.8%

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

Alternative 21: 9.1% accurate, 4.0× speedup

Math

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

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(Float32(Float32(0.25) / Float32(pi)) / r) / s)
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 9.3%

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

5. Taylor expanded in s around inf 8.0%

   \[\leadsto \color{blue}{0.25 \cdot \frac{1}{s \cdot \left(r \cdot \pi\right)} - 0.125 \cdot \frac{1}{{s}^{2} \cdot \pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 8.0%

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

   2. metadata-eval 8.0%

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

   3. associate-/r* 8.0%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{s}}{r \cdot \pi}} - 0.125 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   4. associate-*r/ 8.0%

      \[\leadsto \frac{\frac{0.25}{s}}{r \cdot \pi} - \color{blue}{\frac{0.125 \cdot 1}{{s}^{2} \cdot \pi}} \]

   5. metadata-eval 8.0%

      \[\leadsto \frac{\frac{0.25}{s}}{r \cdot \pi} - \frac{\color{blue}{0.125}}{{s}^{2} \cdot \pi} \]

   6. *-commutative 8.0%

      \[\leadsto \frac{\frac{0.25}{s}}{r \cdot \pi} - \frac{0.125}{\color{blue}{\pi \cdot {s}^{2}}} \]

   7. unpow2 8.0%

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

7. Simplified 8.0%

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

8. Taylor expanded in s around inf 8.8%

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

   1. associate-/r* 8.8%

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

   2. *-commutative 8.8%

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

10. Simplified 8.8%

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

11. Taylor expanded in s around 0 8.8%

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

    1. *-commutative 8.8%

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

    2. associate-/l/ 8.8%

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

    3. associate-/r* 8.8%

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

13. Simplified 8.8%

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

14. Final simplification 8.8%

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

Reproduce

herbie shell --seed 2023275 
(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))))