Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.8%

Time: 22.2s

Alternatives: 20

Speedup: 0.5×

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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \langle \left( \frac{0.125}{s \cdot \pi} \cdot \langle \left( \langle \left( \left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} + \frac{e^{\frac{r}{-s}}}{r}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (*
    (/ 0.125 (* s PI))
    (cast
     (!
      :precision
      binary32
      (cast
       (!
        :precision
        binary64
        (+
         (/ (exp (* -0.3333333333333333 (/ r s))) r)
         (/ (exp (/ r (- s))) r))))))))))

C

float code(float s, float r) {
	double tmp_6 = (exp((-0.3333333333333333 * (((double) r) / ((double) s)))) / ((double) r)) + (exp((((double) r) / -((double) s))) / ((double) r));
	double tmp_5 = (float) tmp_6;
	double tmp_3 = (0.125 / (((double) s) * ((double) M_PI))) * ((double) tmp_5);
	return (float) tmp_3;
}

Julia

function code(s, r)
	tmp_6 = Float64(Float64(exp(Float64(-0.3333333333333333 * Float64(Float64(r) / Float64(s)))) / Float64(r)) + Float64(exp(Float64(Float64(r) / Float64(-Float64(s)))) / Float64(r)))
	tmp_5 = Float32(tmp_6)
	tmp_3 = Float64(Float64(0.125 / Float64(Float64(s) * pi)) * Float64(tmp_5))
	return Float32(tmp_3)
end

MATLAB

function tmp_8 = code(s, r)
	tmp_7 = (exp((-0.3333333333333333 * (double(r) / double(s)))) / double(r)) + (exp((double(r) / -s)) / double(r));
	tmp_6 = single(tmp_7);
	tmp_4 = (0.125 / (double(s) * pi)) * double(tmp_6);
	tmp_8 = single(tmp_4);
end

Derivation

1. Initial program 99.8%

   \[\langle \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \rangle_{\text{binary64}} \]
2. Step-by-step derivation

   1. rewrite-binary64/binary32-simplify 99.9%

      \[\leadsto \color{blue}{\langle \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \langle \left( \langle \left( \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}} \rangle_{\text{binary64}}} \]

3. Applied rewrite-once 99.9%

   \[\leadsto \langle \frac{0.125}{\color{blue}{s \cdot \pi}} \cdot \langle \left( \langle \left( \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \rangle_{\text{binary64}} \]
4. Step-by-step derivation

   1. +-commutative 99.9%

      \[\leadsto \langle \frac{0.125}{s \cdot \pi} \cdot \langle \left( \langle \left( \left(\frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r} + \frac{e^{\frac{r}{-s}}}{r}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \rangle_{\text{binary64}} \]

   2. exp-prod 99.9%

      \[\leadsto \langle \frac{0.125}{s \cdot \pi} \cdot \langle \left( \langle \left( \left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} + \frac{e^{\frac{r}{-s}}}{r}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \rangle_{\text{binary64}} \]

5. Simplified 99.9%

   \[\leadsto \langle \frac{0.125}{\color{blue}{s \cdot \pi}} \cdot \langle \left( \langle \left( \left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} + \frac{e^{\frac{r}{-s}}}{r}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \rangle_{\text{binary64}} \]

6. Final simplification 99.9%

   \[\leadsto \langle \frac{0.125}{s \cdot \pi} \cdot \langle \left( \langle \left( \left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} + \frac{e^{\frac{r}{-s}}}{r}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \rangle_{\text{binary64}} \]

Alternative 2: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \langle \left( \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (*
    (/ 0.125 (* s PI))
    (+
     (/ (exp (/ r (- s))) r)
     (/ (pow (exp -0.3333333333333333) (/ r s)) r))))))

C

float code(float s, float r) {
	double tmp = (0.125 / (((double) s) * ((double) M_PI))) * ((exp((((double) r) / -((double) s))) / ((double) r)) + (pow(exp(-0.3333333333333333), (((double) r) / ((double) s))) / ((double) r)));
	return (float) tmp;
}

Julia

function code(s, r)
	tmp = Float64(Float64(0.125 / Float64(Float64(s) * pi)) * Float64(Float64(exp(Float64(Float64(r) / Float64(-Float64(s)))) / Float64(r)) + Float64((exp(-0.3333333333333333) ^ Float64(Float64(r) / Float64(s))) / Float64(r))))
	return Float32(tmp)
end

MATLAB

function tmp_2 = code(s, r)
	tmp = (0.125 / (double(s) * pi)) * ((exp((double(r) / -s)) / double(r)) + ((exp(-0.3333333333333333) ^ (double(r) / double(s))) / double(r)));
	tmp_2 = single(tmp);
end

Derivation

1. Initial program 99.8%

   \[\langle \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \rangle_{\text{binary64}} \]

2. Final simplification 99.8%

   \[\leadsto \langle \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \rangle_{\text{binary64}} \]

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\langle \left( e^{-0.3333333333333333 \cdot \frac{r}{s}} \cdot 0.75 \right)_{\text{binary64}} \rangle_{\text{binary32}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* PI 2.0))))
  (/
   (cast
    (! :precision binary64 (* (exp (* -0.3333333333333333 (/ r s))) 0.75)))
   (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	double tmp = exp((-0.3333333333333333 * (((double) r) / ((double) s)))) * 0.75;
	return ((0.25f * expf((-r / s))) / (r * (s * (((float) M_PI) * 2.0f)))) + (((float) tmp) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	tmp = Float64(exp(Float64(-0.3333333333333333 * Float64(Float64(r) / Float64(s)))) * 0.75)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(tmp) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp_2 = code(s, r)
	tmp = exp((-0.3333333333333333 * (double(r) / double(s)))) * 0.75;
	tmp_2 = ((single(0.25) * exp((-r / s))) / (r * (s * (single(pi) * single(2.0))))) + single((single(tmp) / double((r * (s * (single(pi) * single(6.0)))))));
end

Derivation

1. Initial program 99.4%

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

   1. rewrite-binary32/binary64 99.8%

      \[\leadsto \color{blue}{\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\langle 0.75 \cdot e^{\frac{-r}{3 \cdot s}} \rangle_{\text{binary64}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

3. Applied rewrite-once 99.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\langle \color{blue}{0.75 \cdot e^{\frac{-r}{3 \cdot s}}} \rangle_{\text{binary64}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. add-exp-log_binary64 99.8%

      \[\leadsto \color{blue}{\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\langle 0.75 \cdot e^{\log \left(e^{\frac{-r}{3 \cdot s}}\right)} \rangle_{\text{binary64}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

5. Applied rewrite-once 99.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\langle 0.75 \cdot e^{\log \left(e^{\frac{-r}{3 \cdot s}}\right)} \rangle_{\text{binary64}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

   1. rem-exp-log 99.8%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\langle 0.75 \cdot e^{\frac{-r}{3 \cdot s}} \rangle_{\text{binary64}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. neg-mul-1 99.8%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\langle 0.75 \cdot e^{\frac{-1 \cdot r}{3 \cdot s}} \rangle_{\text{binary64}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. times-frac 99.8%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\langle 0.75 \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}} \rangle_{\text{binary64}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. metadata-eval 99.8%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\langle 0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}} \rangle_{\text{binary64}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Simplified 99.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\langle 0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}} \rangle_{\text{binary64}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Final simplification 99.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\langle e^{-0.3333333333333333 \cdot \frac{r}{s}} \cdot 0.75 \rangle_{\text{binary64}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\langle \left( e^{-0.3333333333333333 \cdot \frac{r}{s}} \right)_{\text{binary64}} \rangle_{\text{binary32}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (exp (/ r (- s))) r)
   (/
    (cast (! :precision binary64 (exp (* -0.3333333333333333 (/ r s)))))
    r))))

C

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

Julia

function code(s, r)
	tmp = exp(Float64(-0.3333333333333333 * Float64(Float64(r) / Float64(s))))
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(tmp) / r)))
end

MATLAB

function tmp_2 = code(s, r)
	tmp = exp((-0.3333333333333333 * (double(r) / double(s))));
	tmp_2 = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + single((single(tmp) / double(r))));
end

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

   1. rewrite-binary32/binary64 99.7%

      \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\langle {\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)} \rangle_{\text{binary64}}}{r}\right)} \]

5. Applied rewrite-once 99.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\langle \color{blue}{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}} \rangle_{\text{binary64}}}}{r}\right) \]
6. Step-by-step derivation

   1. exp-prod 99.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\langle e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}} \rangle_{\text{binary64}}}{r}\right) \]

7. Simplified 99.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\langle \color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}} \rangle_{\text{binary64}}}}{r}\right) \]

8. Final simplification 99.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\langle e^{-0.3333333333333333 \cdot \frac{r}{s}} \rangle_{\text{binary64}}}{r}\right) \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (r * (s * (((float) M_PI) * 2.0f)))) + ((0.75f * expf((-r / (s * 3.0f)))) / (r * (((float) M_PI) * (s * 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(pi) * Float32(2.0))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(s * Float32(3.0))))) / Float32(r * Float32(Float32(pi) * Float32(s * Float32(6.0))))))
end

MATLAB

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in s around 0 99.4%

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

   1. *-commutative 99.4%

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

4. Simplified 99.4%

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

5. Taylor expanded in r around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*r* 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.5%

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

   2. times-frac 99.5%

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

   3. associate-*l* 99.5%

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

   4. *-commutative 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in r around 0 99.4%

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

   1. metadata-eval 99.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot \left(s \cdot r\right)} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\color{blue}{\frac{0.3333333333333333}{-1}} \cdot \frac{r}{s}}}{r} \]

   2. times-frac 99.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot \left(s \cdot r\right)} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\color{blue}{\frac{0.3333333333333333 \cdot r}{-1 \cdot s}}}}{r} \]

   3. *-commutative 99.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot \left(s \cdot r\right)} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{\color{blue}{r \cdot 0.3333333333333333}}{-1 \cdot s}}}{r} \]

   4. associate-/r* 99.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot \left(s \cdot r\right)} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\color{blue}{\frac{\frac{r \cdot 0.3333333333333333}{-1}}{s}}}}{r} \]

   5. associate-/l* 99.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(2 \cdot \pi\right) \cdot \left(s \cdot r\right)} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{\color{blue}{\frac{r}{\frac{-1}{0.3333333333333333}}}}{s}}}{r} \]

   6. metadata-eval 99.5%

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

   7. associate-/r* 99.5%

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

   8. *-commutative 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 7: 99.5% accurate, 1.3× speedup

Math

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

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 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

   1. pow-exp 99.3%

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

   2. clear-num 99.3%

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

   3. un-div-inv 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 8: 99.5% accurate, 1.3× speedup

Math

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

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(Float32(0.125) / 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 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

   1. div-inv 99.0%

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

   2. associate-/r* 99.0%

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

5. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.0%

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

   2. associate-*r/ 99.0%

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

   3. metadata-eval 99.0%

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

7. Simplified 99.0%

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

   1. pow-exp 99.3%

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

   2. clear-num 99.3%

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

   3. un-div-inv 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 9: 12.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around inf 9.5%

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

   1. log1p-expm1-u_binary32 11.9%

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

7. Applied rewrite-once 11.9%

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

8. Final simplification 11.9%

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

Alternative 10: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + ((1.0f + (-0.3333333333333333f * (r / s))) / 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(Float32(1.0) + Float32(Float32(-0.3333333333333333) * Float32(r / s))) / r)))
end

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

4. Taylor expanded in r around 0 10.3%

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

5. Final simplification 10.3%

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

Alternative 11: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. distribute-rgt-in 10.0%

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

   3. *-commutative 10.0%

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

   4. associate-*r/ 10.0%

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

   5. metadata-eval 10.0%

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

   6. mul-1-neg 10.0%

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

   7. distribute-frac-neg 10.0%

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

7. Simplified 10.0%

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

9. Applied egg-rr 10.0%

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

    1. *-commutative 10.0%

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

    2. associate-/r* 10.0%

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

    3. times-frac 10.0%

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

    4. associate-*r/ 10.0%

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

    5. neg-sub0 10.0%

       \[\leadsto \frac{\color{blue}{0 - \left(r + \frac{r}{e^{\frac{r}{s}}}\right)}}{r} \cdot \frac{0.125}{r \cdot \left(\pi \cdot \left(-s\right)\right)} \]

    6. +-commutative 10.0%

       \[\leadsto \frac{0 - \color{blue}{\left(\frac{r}{e^{\frac{r}{s}}} + r\right)}}{r} \cdot \frac{0.125}{r \cdot \left(\pi \cdot \left(-s\right)\right)} \]

    7. associate--r+ 10.0%

       \[\leadsto \frac{\color{blue}{\left(0 - \frac{r}{e^{\frac{r}{s}}}\right) - r}}{r} \cdot \frac{0.125}{r \cdot \left(\pi \cdot \left(-s\right)\right)} \]

    8. neg-sub0 10.0%

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

    9. distribute-neg-frac 10.0%

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

    10. *-commutative 10.0%

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

    11. *-commutative 10.0%

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

    12. associate-*l* 10.0%

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

    13. *-commutative 10.0%

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

11. Simplified 10.0%

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

12. Final simplification 10.0%

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

Alternative 12: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. distribute-rgt-in 10.0%

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

   3. *-commutative 10.0%

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

   4. associate-*r/ 10.0%

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

   5. metadata-eval 10.0%

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

   6. mul-1-neg 10.0%

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

   7. distribute-frac-neg 10.0%

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

7. Simplified 10.0%

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

9. Applied egg-rr 10.0%

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

    1. associate-*l/ 10.0%

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

    2. mul-1-neg 10.0%

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

    3. neg-mul-1 10.0%

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

    4. times-frac 10.0%

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

    5. metadata-eval 10.0%

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

    6. *-commutative 10.0%

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

    7. associate-*l* 10.0%

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

11. Simplified 10.0%

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

12. Final simplification 10.0%

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

Alternative 13: 9.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-r\right) - \frac{r}{e^{\frac{r}{s}}}}{r \cdot \left(r \cdot \left(\left(s \cdot \pi\right) \cdot -8\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (- (- r) (/ r (exp (/ r s)))) (* r (* r (* (* s PI) -8.0)))))

C

float code(float s, float r) {
	return (-r - (r / expf((r / s)))) / (r * (r * ((s * ((float) M_PI)) * -8.0f)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(-r) - Float32(r / exp(Float32(r / s)))) / Float32(r * Float32(r * Float32(Float32(s * Float32(pi)) * Float32(-8.0)))))
end

MATLAB

function tmp = code(s, r)
	tmp = (-r - (r / exp((r / s)))) / (r * (r * ((s * single(pi)) * single(-8.0))));
end

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. distribute-rgt-in 10.0%

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

   3. *-commutative 10.0%

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

   4. associate-*r/ 10.0%

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

   5. metadata-eval 10.0%

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

   6. mul-1-neg 10.0%

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

   7. distribute-frac-neg 10.0%

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

7. Simplified 10.0%

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

9. Applied egg-rr 10.0%

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

11. Applied egg-rr 10.0%

    \[\leadsto \color{blue}{\frac{\left(-r\right) - \frac{r}{e^{\frac{r}{s}}}}{r \cdot \left(r \cdot \left(\left(\pi \cdot s\right) \cdot -8\right)\right)}} \]

12. Final simplification 10.0%

    \[\leadsto \frac{\left(-r\right) - \frac{r}{e^{\frac{r}{s}}}}{r \cdot \left(r \cdot \left(\left(s \cdot \pi\right) \cdot -8\right)\right)} \]

Alternative 14: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. distribute-rgt-in 10.0%

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

   3. *-commutative 10.0%

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

   4. associate-*r/ 10.0%

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

   5. metadata-eval 10.0%

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

   6. mul-1-neg 10.0%

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

   7. distribute-frac-neg 10.0%

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

7. Simplified 10.0%

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

   1. associate-/r* 10.0%

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

   2. div-inv 10.0%

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

   3. *-lft-identity 10.0%

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

   4. div-inv 10.0%

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

   5. associate-*l* 10.0%

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

   6. *-commutative 10.0%

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

   7. div-inv 10.0%

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

   8. distribute-rgt-out 10.0%

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

   9. +-commutative 10.0%

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

   10. remove-double-neg 10.0%

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

   11. frac-2neg 10.0%

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

9. Applied egg-rr 10.0%

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

10. Final simplification 10.0%

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

Alternative 15: 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 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in r around inf 10.0%

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

   1. mul-1-neg 10.0%

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

7. Simplified 10.0%

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

8. Final simplification 10.0%

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

Alternative 16: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. distribute-rgt-in 10.0%

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

   3. *-commutative 10.0%

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

   4. associate-*r/ 10.0%

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

   5. metadata-eval 10.0%

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

   6. mul-1-neg 10.0%

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

   7. distribute-frac-neg 10.0%

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

7. Simplified 10.0%

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

9. Applied egg-rr 10.0%

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

    1. associate-*r/ 10.0%

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

    2. metadata-eval 10.0%

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

    3. *-commutative 10.0%

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

    4. +-commutative 10.0%

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

11. Simplified 10.0%

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

    1. associate-*r/ 10.0%

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

    2. associate-*r* 10.0%

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

    3. *-commutative 10.0%

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

    4. associate-/r/ 10.0%

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

    5. frac-2neg 10.0%

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

    6. distribute-frac-neg 10.0%

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

    7. remove-double-neg 10.0%

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

13. Applied egg-rr 10.0%

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

14. Final simplification 10.0%

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

Alternative 17: 9.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. distribute-rgt-in 10.0%

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

   3. *-commutative 10.0%

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

   4. associate-*r/ 10.0%

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

   5. metadata-eval 10.0%

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

   6. mul-1-neg 10.0%

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

   7. distribute-frac-neg 10.0%

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

7. Simplified 10.0%

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

9. Applied egg-rr 10.0%

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

10. Taylor expanded in r around 0 10.0%

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

11. Final simplification 10.0%

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

Alternative 18: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around inf 9.5%

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

6. Final simplification 9.5%

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

Alternative 19: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. distribute-rgt-in 10.0%

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

   3. *-commutative 10.0%

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

   4. associate-*r/ 10.0%

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

   5. metadata-eval 10.0%

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

   6. mul-1-neg 10.0%

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

   7. distribute-frac-neg 10.0%

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

7. Simplified 10.0%

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

8. Taylor expanded in r around 0 9.5%

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

   1. associate-*r* 9.5%

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

   2. *-commutative 9.5%

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

   3. *-commutative 9.5%

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

   4. *-commutative 9.5%

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

10. Simplified 9.5%

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

11. Taylor expanded in r around 0 9.5%

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

    1. associate-*r* 9.5%

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

    2. *-commutative 9.5%

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

    3. associate-/r* 9.5%

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

13. Simplified 9.5%

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

14. Final simplification 9.5%

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

Alternative 20: 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 99.4%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.0%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. distribute-rgt-in 10.0%

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

   3. *-commutative 10.0%

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

   4. associate-*r/ 10.0%

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

   5. metadata-eval 10.0%

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

   6. mul-1-neg 10.0%

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

   7. distribute-frac-neg 10.0%

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

7. Simplified 10.0%

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

8. Taylor expanded in r around 0 9.5%

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

   1. associate-*r* 9.5%

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

   2. *-commutative 9.5%

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

   3. *-commutative 9.5%

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

   4. *-commutative 9.5%

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

10. Simplified 9.5%

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

    1. associate-*r* 9.5%

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

    2. associate-/r* 9.5%

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

    3. div-inv 9.5%

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

12. Applied egg-rr 9.5%

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

    1. associate-*r/ 9.5%

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

    2. *-rgt-identity 9.5%

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

    3. associate-/r* 9.5%

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

14. Simplified 9.5%

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

15. Final simplification 9.5%

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

Reproduce

herbie shell --seed 2023297 
(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))))