Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 15.8s

Alternatives: 20

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / (s * -3.0f))) / r) + (expf((-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 * Float32(-3.0)))) / r) + Float32(exp(Float32(Float32(-r) / s)) / r)))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. times-frac N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{1}{4}}{\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4}}{2}}{\mathsf{PI}\left(\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{1}{8}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{\frac{3}{4}}{6}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{3}{4}}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{3}{4}}{\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   10. distribute-lft-out N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Final simplification 99.8%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{r} \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{\frac{-r}{s}}\right)}{\left(s \cdot \pi\right) \cdot 8} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (* (/ 1.0 r) (+ (exp (/ r (* s -3.0))) (exp (/ (- r) s))))
  (* (* s PI) 8.0)))

C

float code(float s, float r) {
	return ((1.0f / r) * (expf((r / (s * -3.0f))) + expf((-r / s)))) / ((s * ((float) M_PI)) * 8.0f);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(1.0) / r) * Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) + exp(Float32(Float32(-r) / s)))) / Float32(Float32(s * Float32(pi)) * Float32(8.0)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(1.0) / r) * (exp((r / (s * single(-3.0)))) + exp((-r / s)))) / ((s * single(pi)) * single(8.0));
end

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. times-frac N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{1}{4}}{\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4}}{2}}{\mathsf{PI}\left(\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{1}{8}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{\frac{3}{4}}{6}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{3}{4}}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{3}{4}}{\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   10. distribute-lft-out N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}} \]

   2. clear-num N/A

      \[\leadsto \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \cdot \color{blue}{\frac{1}{\frac{s \cdot \mathsf{PI}\left(\right)}{\frac{1}{8}}}} \]

   3. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}}{\frac{s \cdot \mathsf{PI}\left(\right)}{\frac{1}{8}}}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}}{\frac{s \cdot \mathsf{PI}\left(\right)}{\frac{1}{8}}}} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\frac{1}{r} \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{0 - \frac{r}{s}}\right)}{\left(s \cdot \pi\right) \cdot 8}} \]

7. Final simplification 99.7%

   \[\leadsto \frac{\frac{1}{r} \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{\frac{-r}{s}}\right)}{\left(s \cdot \pi\right) \cdot 8} \]
8. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. times-frac N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{1}{4}}{\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4}}{2}}{\mathsf{PI}\left(\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{1}{8}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{\frac{3}{4}}{6}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{3}{4}}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{3}{4}}{\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   10. distribute-lft-out N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in r around inf

   \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{e^{\mathsf{neg}\left(\frac{r}{s}\right)} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r}} \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{e^{\mathsf{neg}\left(\frac{r}{s}\right)} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{\color{blue}{e^{\frac{-1}{3} \cdot \frac{r}{s}} + e^{\mathsf{neg}\left(\frac{r}{s}\right)}}}{r} \]

   3. neg-mul-1 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}} + e^{\color{blue}{-1 \cdot \frac{r}{s}}}}{r} \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{\color{blue}{e^{\frac{-1}{3} \cdot \frac{r}{s}} + e^{-1 \cdot \frac{r}{s}}}}{r} \]

   5. exp-lowering-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{\color{blue}{e^{\frac{-1}{3} \cdot \frac{r}{s}}} + e^{-1 \cdot \frac{r}{s}}}{r} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot \frac{-1}{3}}} + e^{-1 \cdot \frac{r}{s}}}{r} \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot \frac{-1}{3}}} + e^{-1 \cdot \frac{r}{s}}}{r} \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{e^{\color{blue}{\frac{r}{s}} \cdot \frac{-1}{3}} + e^{-1 \cdot \frac{r}{s}}}{r} \]

   9. exp-lowering-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{e^{\frac{r}{s} \cdot \frac{-1}{3}} + \color{blue}{e^{-1 \cdot \frac{r}{s}}}}{r} \]

   10. neg-mul-1 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{e^{\frac{r}{s} \cdot \frac{-1}{3}} + e^{\color{blue}{\mathsf{neg}\left(\frac{r}{s}\right)}}}{r} \]

   11. neg-sub0 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{e^{\frac{r}{s} \cdot \frac{-1}{3}} + e^{\color{blue}{0 - \frac{r}{s}}}}{r} \]

   12. --lowering--.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{e^{\frac{r}{s} \cdot \frac{-1}{3}} + e^{\color{blue}{0 - \frac{r}{s}}}}{r} \]

   13. /-lowering-/.f32 99.7

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333} + e^{0 - \color{blue}{\frac{r}{s}}}}{r} \]

7. Simplified 99.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\frac{e^{\frac{r}{s} \cdot -0.3333333333333333} + e^{0 - \frac{r}{s}}}{r}} \]

8. Final simplification 99.7%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. times-frac N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{1}{4}}{\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4}}{2}}{\mathsf{PI}\left(\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{1}{8}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{\frac{3}{4}}{6}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{3}{4}}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{3}{4}}{\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   10. distribute-lft-out N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{\mathsf{neg}\left(\frac{r}{s}\right)} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \left(e^{\mathsf{neg}\left(\frac{r}{s}\right)} + e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{8} \cdot -1\right)} \cdot \left(e^{\mathsf{neg}\left(\frac{r}{s}\right)} + e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot \left(-1 \cdot \left(e^{\mathsf{neg}\left(\frac{r}{s}\right)} + e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   4. neg-mul-1 N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(-1 \cdot \left(e^{\color{blue}{-1 \cdot \frac{r}{s}}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   5. distribute-lft-out N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \color{blue}{\left(-1 \cdot e^{-1 \cdot \frac{r}{s}} + -1 \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(-1 \cdot e^{-1 \cdot \frac{r}{s}} + -1 \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

7. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{\frac{r}{s} \cdot -0.3333333333333333} + e^{0 - \frac{r}{s}}\right)}{r \cdot \left(s \cdot \pi\right)}} \]

8. Final simplification 99.7%

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

Alternative 6: 10.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right), \frac{0.125}{\pi}\right)}{r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/
   (/
    (fma
     r
     (fma
      r
      (/ 0.006944444444444444 (* s (* s PI)))
      (/ -0.041666666666666664 (* s PI)))
     (/ 0.125 PI))
    r)
   s)))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (r * (s * (2.0f * ((float) M_PI))))) + ((fmaf(r, fmaf(r, (0.006944444444444444f / (s * (s * ((float) M_PI)))), (-0.041666666666666664f / (s * ((float) M_PI)))), (0.125f / ((float) M_PI))) / r) / s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(fma(r, fma(r, Float32(Float32(0.006944444444444444) / Float32(s * Float32(s * Float32(pi)))), Float32(Float32(-0.041666666666666664) / Float32(s * Float32(pi)))), Float32(Float32(0.125) / Float32(pi))) / r) / s))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{1296} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Simplified 12.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.0007716049382716049, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.006944444444444444}{\pi}\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \pi} + \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]

6. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}}{s} \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}}{s} \]

8. Simplified 12.9%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right), \frac{0.125}{\pi}\right)}{r}}}{s} \]

9. Final simplification 12.9%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right), \frac{0.125}{\pi}\right)}{r}}{s} \]
10. Add Preprocessing

Alternative 7: 10.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(\frac{1}{s \cdot \pi}, -0.041666666666666664, \frac{r \cdot 0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/
   (+
    (/ 0.125 (* r PI))
    (fma
     (/ 1.0 (* s PI))
     -0.041666666666666664
     (/ (* r 0.006944444444444444) (* s (* s PI)))))
   s)))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (r * (s * (2.0f * ((float) M_PI))))) + (((0.125f / (r * ((float) M_PI))) + fmaf((1.0f / (s * ((float) M_PI))), -0.041666666666666664f, ((r * 0.006944444444444444f) / (s * (s * ((float) M_PI)))))) / s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(Float32(0.125) / Float32(r * Float32(pi))) + fma(Float32(Float32(1.0) / Float32(s * Float32(pi))), Float32(-0.041666666666666664), Float32(Float32(r * Float32(0.006944444444444444)) / Float32(s * Float32(s * Float32(pi)))))) / s))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{24}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Simplified 12.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \color{blue}{\left(\frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)} + r \cdot \frac{\frac{1}{144}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)}}{s} \]

   2. clear-num N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \left(\color{blue}{\frac{1}{\frac{s \cdot \mathsf{PI}\left(\right)}{\frac{-1}{24}}}} + r \cdot \frac{\frac{1}{144}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)}{s} \]

   3. associate-/r/ N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \left(\color{blue}{\frac{1}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{-1}{24}} + r \cdot \frac{\frac{1}{144}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)}{s} \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \color{blue}{\mathsf{fma}\left(\frac{1}{s \cdot \mathsf{PI}\left(\right)}, \frac{-1}{24}, r \cdot \frac{\frac{1}{144}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)}}{s} \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \mathsf{fma}\left(\color{blue}{\frac{1}{s \cdot \mathsf{PI}\left(\right)}}, \frac{-1}{24}, r \cdot \frac{\frac{1}{144}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)}{s} \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \mathsf{fma}\left(\frac{1}{\color{blue}{s \cdot \mathsf{PI}\left(\right)}}, \frac{-1}{24}, r \cdot \frac{\frac{1}{144}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)}{s} \]

   7. PI-lowering-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \mathsf{fma}\left(\frac{1}{s \cdot \color{blue}{\mathsf{PI}\left(\right)}}, \frac{-1}{24}, r \cdot \frac{\frac{1}{144}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)}{s} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \mathsf{fma}\left(\frac{1}{s \cdot \mathsf{PI}\left(\right)}, \frac{-1}{24}, \color{blue}{\frac{r \cdot \frac{1}{144}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)}{s} \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \mathsf{fma}\left(\frac{1}{s \cdot \mathsf{PI}\left(\right)}, \frac{-1}{24}, \color{blue}{\frac{r \cdot \frac{1}{144}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)}{s} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \mathsf{fma}\left(\frac{1}{s \cdot \mathsf{PI}\left(\right)}, \frac{-1}{24}, \frac{\color{blue}{r \cdot \frac{1}{144}}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)}{s} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \mathsf{fma}\left(\frac{1}{s \cdot \mathsf{PI}\left(\right)}, \frac{-1}{24}, \frac{r \cdot \frac{1}{144}}{\color{blue}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)}{s} \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \mathsf{fma}\left(\frac{1}{s \cdot \mathsf{PI}\left(\right)}, \frac{-1}{24}, \frac{r \cdot \frac{1}{144}}{s \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)}{s} \]

   13. PI-lowering-PI.f32 12.8

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(\frac{1}{s \cdot \pi}, -0.041666666666666664, \frac{r \cdot 0.006944444444444444}{s \cdot \left(s \cdot \color{blue}{\pi}\right)}\right)}{s} \]

7. Applied egg-rr 12.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{0.125}{r \cdot \pi} + \color{blue}{\mathsf{fma}\left(\frac{1}{s \cdot \pi}, -0.041666666666666664, \frac{r \cdot 0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}\right)}}{s} \]

8. Final simplification 12.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(\frac{1}{s \cdot \pi}, -0.041666666666666664, \frac{r \cdot 0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}\right)}{s} \]
9. Add Preprocessing

Alternative 8: 10.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right) + \frac{0.125}{r \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/
   (+
    (fma
     r
     (/ 0.006944444444444444 (* s (* s PI)))
     (/ -0.041666666666666664 (* s PI)))
    (/ 0.125 (* r PI)))
   s)))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (r * (s * (2.0f * ((float) M_PI))))) + ((fmaf(r, (0.006944444444444444f / (s * (s * ((float) M_PI)))), (-0.041666666666666664f / (s * ((float) M_PI)))) + (0.125f / (r * ((float) M_PI)))) / s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(fma(r, Float32(Float32(0.006944444444444444) / Float32(s * Float32(s * Float32(pi)))), Float32(Float32(-0.041666666666666664) / Float32(s * Float32(pi)))) + Float32(Float32(0.125) / Float32(r * Float32(pi)))) / s))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{24}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Simplified 12.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]

6. Final simplification 12.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right) + \frac{0.125}{r \cdot \pi}}{s} \]
7. Add Preprocessing

Alternative 9: 10.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. times-frac N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{1}{4}}{\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4}}{2}}{\mathsf{PI}\left(\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{1}{8}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{\frac{3}{4}}{6}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{3}{4}}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{3}{4}}{\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   10. distribute-lft-out N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}} \]

   2. frac-add N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{r}{s \cdot -3}} \cdot r + r \cdot e^{0 - \frac{r}{s}}}{r \cdot r}} \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left(e^{\frac{r}{s \cdot -3}} \cdot r + r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\left(e^{\frac{r}{s \cdot -3}} \cdot r + r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r}} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(r, e^{\frac{r}{s \cdot -3}}, r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{0.125}{s \cdot \pi}}{r \cdot r}} \]

7. Taylor expanded in r around 0

   \[\leadsto \frac{\mathsf{fma}\left(r, \color{blue}{1 + r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right)}, r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \color{blue}{r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) + 1}, r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   2. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(r, \frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, 1\right)}, r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   3. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{1}{18} \cdot \frac{r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)}, 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{r}{{s}^{2}} \cdot \frac{1}{18}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   5. associate-*l/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{r \cdot \frac{1}{18}}{{s}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{r \cdot \frac{\frac{1}{18}}{{s}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, r \cdot \frac{\color{blue}{\frac{1}{18} \cdot 1}}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, r \cdot \color{blue}{\left(\frac{1}{18} \cdot \frac{1}{{s}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   9. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(r, \frac{1}{18} \cdot \frac{1}{{s}^{2}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)}, 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{1}{18} \cdot 1}{{s}^{2}}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\color{blue}{\frac{1}{18}}}{{s}^{2}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{1}{18}}{{s}^{2}}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{\color{blue}{s \cdot s}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{\color{blue}{s \cdot s}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   15. associate-*r/ N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{s}}\right)\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{s}\right)\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   17. distribute-neg-frac N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{s}}\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   18. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \frac{\color{blue}{\frac{-1}{3}}}{s}\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}}{r \cdot r} \]

   19. /-lowering-/.f32 12.8

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \color{blue}{\frac{-0.3333333333333333}{s}}\right), 1\right), r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{0.125}{s \cdot \pi}}{r \cdot r} \]

9. Simplified 12.8%

   \[\leadsto \frac{\mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \frac{-0.3333333333333333}{s}\right), 1\right)}, r \cdot e^{0 - \frac{r}{s}}\right) \cdot \frac{0.125}{s \cdot \pi}}{r \cdot r} \]

10. Final simplification 12.8%

    \[\leadsto \frac{\frac{0.125}{s \cdot \pi} \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \frac{-0.3333333333333333}{s}\right), 1\right), r \cdot e^{\frac{-r}{s}}\right)}{r \cdot r} \]
11. Add Preprocessing

Alternative 10: 9.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.06944444444444445, \frac{r}{s \cdot \pi}, \frac{-0.16666666666666666}{\pi}\right)}{s \cdot s} + \frac{\frac{0.25}{s \cdot \pi}}{r} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right) - \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Simplified 12.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.06944444444444445, \frac{r}{s \cdot \pi}, \frac{-0.16666666666666666}{\pi}\right)}{s \cdot s} + \frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{72}, \frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)}\right)}{s \cdot s} + \frac{\frac{1}{4}}{\color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{72}, \frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)}\right)}{s \cdot s} + \color{blue}{\frac{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{72}, \frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)}\right)}{s \cdot s} + \color{blue}{\frac{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{72}, \frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)}\right)}{s \cdot s} + \frac{\color{blue}{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{72}, \frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)}\right)}{s \cdot s} + \frac{\frac{\frac{1}{4}}{\color{blue}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   6. PI-lowering-PI.f32 12.3

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

7. Applied egg-rr 12.3%

   \[\leadsto \frac{\mathsf{fma}\left(0.06944444444444445, \frac{r}{s \cdot \pi}, \frac{-0.16666666666666666}{\pi}\right)}{s \cdot s} + \color{blue}{\frac{\frac{0.25}{s \cdot \pi}}{r}} \]
8. Add Preprocessing

Alternative 11: 9.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\mathsf{fma}\left(r, \frac{0.5555555555555556}{s \cdot s}, \frac{2}{r}\right) + \frac{-1.3333333333333333}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (fma r (/ 0.5555555555555556 (* s s)) (/ 2.0 r))
   (/ -1.3333333333333333 s))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * (fmaf(r, (0.5555555555555556f / (s * s)), (2.0f / r)) + (-1.3333333333333333f / s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(fma(r, Float32(Float32(0.5555555555555556) / Float32(s * s)), Float32(Float32(2.0) / r)) + Float32(Float32(-1.3333333333333333) / s)))
end

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. times-frac N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{1}{4}}{\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4}}{2}}{\mathsf{PI}\left(\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{1}{8}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{\frac{3}{4}}{6}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{3}{4}}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{3}{4}}{\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   10. distribute-lft-out N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} + \left(\frac{1}{2} \cdot \frac{r}{{s}^{2}} + 2 \cdot \frac{1}{r}\right)\right) - \frac{4}{3} \cdot \frac{1}{s}\right)} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} + \left(\frac{1}{2} \cdot \frac{r}{{s}^{2}} + 2 \cdot \frac{1}{r}\right)\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right)} \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} + \left(\frac{1}{2} \cdot \frac{r}{{s}^{2}} + 2 \cdot \frac{1}{r}\right)\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right)} \]

   3. associate-+r+ N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\left(\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} + \frac{1}{2} \cdot \frac{r}{{s}^{2}}\right) + 2 \cdot \frac{1}{r}\right)} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\left(\color{blue}{\frac{r}{{s}^{2}} \cdot \left(\frac{1}{18} + \frac{1}{2}\right)} + 2 \cdot \frac{1}{r}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\left(\frac{r}{{s}^{2}} \cdot \color{blue}{\frac{5}{9}} + 2 \cdot \frac{1}{r}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\left(\color{blue}{\frac{r \cdot \frac{5}{9}}{{s}^{2}}} + 2 \cdot \frac{1}{r}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\left(\color{blue}{r \cdot \frac{\frac{5}{9}}{{s}^{2}}} + 2 \cdot \frac{1}{r}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\left(r \cdot \frac{\color{blue}{\frac{5}{9} \cdot 1}}{{s}^{2}} + 2 \cdot \frac{1}{r}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\left(r \cdot \color{blue}{\left(\frac{5}{9} \cdot \frac{1}{{s}^{2}}\right)} + 2 \cdot \frac{1}{r}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   10. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\mathsf{fma}\left(r, \frac{5}{9} \cdot \frac{1}{{s}^{2}}, 2 \cdot \frac{1}{r}\right)} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\mathsf{fma}\left(r, \color{blue}{\frac{\frac{5}{9} \cdot 1}{{s}^{2}}}, 2 \cdot \frac{1}{r}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\mathsf{fma}\left(r, \frac{\color{blue}{\frac{5}{9}}}{{s}^{2}}, 2 \cdot \frac{1}{r}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\mathsf{fma}\left(r, \color{blue}{\frac{\frac{5}{9}}{{s}^{2}}}, 2 \cdot \frac{1}{r}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\mathsf{fma}\left(r, \frac{\frac{5}{9}}{\color{blue}{s \cdot s}}, 2 \cdot \frac{1}{r}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\mathsf{fma}\left(r, \frac{\frac{5}{9}}{\color{blue}{s \cdot s}}, 2 \cdot \frac{1}{r}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   16. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\mathsf{fma}\left(r, \frac{\frac{5}{9}}{s \cdot s}, \color{blue}{\frac{2 \cdot 1}{r}}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\mathsf{fma}\left(r, \frac{\frac{5}{9}}{s \cdot s}, \frac{\color{blue}{2}}{r}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   18. /-lowering-/.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\mathsf{fma}\left(r, \frac{\frac{5}{9}}{s \cdot s}, \color{blue}{\frac{2}{r}}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

7. Simplified 12.3%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(\mathsf{fma}\left(r, \frac{0.5555555555555556}{s \cdot s}, \frac{2}{r}\right) + \frac{-1.3333333333333333}{s}\right)} \]
8. Add Preprocessing

Alternative 12: 9.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{\mathsf{fma}\left(-0.5555555555555556, \frac{r}{s}, 1.3333333333333333\right)}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (- (/ 2.0 r) (/ (fma -0.5555555555555556 (/ r s) 1.3333333333333333) s))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((2.0f / r) - (fmaf(-0.5555555555555556f, (r / s), 1.3333333333333333f) / s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(Float32(2.0) / r) - Float32(fma(Float32(-0.5555555555555556), Float32(r / s), Float32(1.3333333333333333)) / s)))
end

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. times-frac N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{1}{4}}{\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4}}{2}}{\mathsf{PI}\left(\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{1}{8}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{\frac{3}{4}}{6}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{3}{4}}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{3}{4}}{\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   10. distribute-lft-out N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(-1 \cdot \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s} + 2 \cdot \frac{1}{r}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} + -1 \cdot \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(2 \cdot \frac{1}{r} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right)\right)}\right) \]

   3. unsub-neg N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} - \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right)} \]

   4. --lowering--.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} - \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right)} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\frac{2 \cdot 1}{r}} - \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\color{blue}{2}}{r} - \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\frac{2}{r}} - \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \color{blue}{\frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}}\right) \]

   9. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\color{blue}{-1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s} + \frac{4}{3}}}{s}\right) \]

   10. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}\right)\right)} + \frac{4}{3}}{s}\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\color{blue}{\frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{\mathsf{neg}\left(s\right)}} + \frac{4}{3}}{s}\right) \]

   12. distribute-rgt-out N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\frac{\color{blue}{r \cdot \left(\frac{1}{18} + \frac{1}{2}\right)}}{\mathsf{neg}\left(s\right)} + \frac{4}{3}}{s}\right) \]

   13. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\frac{r \cdot \color{blue}{\frac{5}{9}}}{\mathsf{neg}\left(s\right)} + \frac{4}{3}}{s}\right) \]

   14. *-commutative N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\frac{\color{blue}{\frac{5}{9} \cdot r}}{\mathsf{neg}\left(s\right)} + \frac{4}{3}}{s}\right) \]

   15. neg-mul-1 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\frac{\frac{5}{9} \cdot r}{\color{blue}{-1 \cdot s}} + \frac{4}{3}}{s}\right) \]

   16. times-frac N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\color{blue}{\frac{\frac{5}{9}}{-1} \cdot \frac{r}{s}} + \frac{4}{3}}{s}\right) \]

   17. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{5}{9}}{-1}, \frac{r}{s}, \frac{4}{3}\right)}}{s}\right) \]

   18. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\mathsf{fma}\left(\color{blue}{\frac{-5}{9}}, \frac{r}{s}, \frac{4}{3}\right)}{s}\right) \]

   19. /-lowering-/.f32 12.2

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{\mathsf{fma}\left(-0.5555555555555556, \color{blue}{\frac{r}{s}}, 1.3333333333333333\right)}{s}\right) \]

7. Simplified 12.2%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(\frac{2}{r} - \frac{\mathsf{fma}\left(-0.5555555555555556, \frac{r}{s}, 1.3333333333333333\right)}{s}\right)} \]
8. Add Preprocessing

Alternative 13: 8.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, \frac{0.25}{\pi}\right)}{r}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Simplified 12.0%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(r, \frac{0.06944444444444445}{\pi}, \frac{-0.021604938271604937 \cdot \left(r \cdot r\right)}{s \cdot \pi}\right)}{s \cdot s} + \left(\frac{0.25}{r \cdot \pi} + \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s}} \]

6. Taylor expanded in r around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{6} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}}{s} \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{6} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}}{s} \]

   2. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}}{r}}{s} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{r}{s \cdot \mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{r}}{s} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{\color{blue}{s \cdot \mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{r}}{s} \]

   5. PI-lowering-PI.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \color{blue}{\mathsf{PI}\left(\right)}}, \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{r}}{s} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \mathsf{PI}\left(\right)}, \color{blue}{\frac{\frac{1}{4} \cdot 1}{\mathsf{PI}\left(\right)}}\right)}{r}}{s} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{\color{blue}{\frac{1}{4}}}{\mathsf{PI}\left(\right)}\right)}{r}}{s} \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{r}{s \cdot \mathsf{PI}\left(\right)}, \color{blue}{\frac{\frac{1}{4}}{\mathsf{PI}\left(\right)}}\right)}{r}}{s} \]

   9. PI-lowering-PI.f32 11.0

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

8. Simplified 11.0%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, \frac{0.25}{\pi}\right)}{r}}}{s} \]
9. Add Preprocessing

Alternative 14: 8.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} + \frac{-1.3333333333333333}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (/ 0.125 (* s PI)) (+ (/ 2.0 r) (/ -1.3333333333333333 s))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((2.0f / r) + (-1.3333333333333333f / s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(Float32(2.0) / r) + Float32(Float32(-1.3333333333333333) / s)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((single(2.0) / r) + (single(-1.3333333333333333) / s));
end

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. times-frac N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{1}{4}}{\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4}}{2}}{\mathsf{PI}\left(\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{1}{8}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{\frac{3}{4}}{6}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{3}{4}}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{3}{4}}{\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   10. distribute-lft-out N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} - \frac{4}{3} \cdot \frac{1}{s}\right)} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right)} \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right)} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\frac{2 \cdot 1}{r}} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\color{blue}{2}}{r} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\frac{2}{r}} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{s}\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{4}{3} \cdot 1}{s}}\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{4}{3}}}{s}\right)\right)\right) \]

   8. distribute-neg-frac N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} + \color{blue}{\frac{\mathsf{neg}\left(\frac{4}{3}\right)}{s}}\right) \]

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} + \frac{\color{blue}{\frac{-4}{3}}}{s}\right) \]

   10. /-lowering-/.f32 11.0

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} + \color{blue}{\frac{-1.3333333333333333}{s}}\right) \]

7. Simplified 11.0%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(\frac{2}{r} + \frac{-1.3333333333333333}{s}\right)} \]
8. Add Preprocessing

Alternative 15: 8.8% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r}} + \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. times-frac N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{1}{4}}{\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{\frac{1}{4}}{2}}{\mathsf{PI}\left(\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{1}{8}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\color{blue}{\frac{\frac{3}{4}}{6}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \color{blue}{\frac{\frac{3}{4}}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{\frac{3}{4}}{\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s}} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r} \]

   10. distribute-lft-out N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{3}{4}}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \left(\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{r} + \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]
5. Step-by-step derivation

   1. frac-add N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{e^{\frac{r}{s \cdot -3}} \cdot r + r \cdot e^{0 - \frac{r}{s}}}{r \cdot r}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{\frac{e^{\frac{r}{s \cdot -3}} \cdot r + r \cdot e^{0 - \frac{r}{s}}}{r}}{r}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{\frac{e^{\frac{r}{s \cdot -3}} \cdot r + r \cdot e^{0 - \frac{r}{s}}}{r}}{r}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{\color{blue}{\frac{e^{\frac{r}{s \cdot -3}} \cdot r + r \cdot e^{0 - \frac{r}{s}}}{r}}}{r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{\frac{\color{blue}{r \cdot e^{\frac{r}{s \cdot -3}}} + r \cdot e^{0 - \frac{r}{s}}}{r}}{r} \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(r, e^{\frac{r}{s \cdot -3}}, r \cdot e^{0 - \frac{r}{s}}\right)}}{r}}{r} \]

   7. exp-lowering-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{\frac{\mathsf{fma}\left(r, \color{blue}{e^{\frac{r}{s \cdot -3}}}, r \cdot e^{0 - \frac{r}{s}}\right)}{r}}{r} \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{\frac{\mathsf{fma}\left(r, e^{\color{blue}{\frac{r}{s \cdot -3}}}, r \cdot e^{0 - \frac{r}{s}}\right)}{r}}{r} \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{\frac{\mathsf{fma}\left(r, e^{\frac{r}{\color{blue}{s \cdot -3}}}, r \cdot e^{0 - \frac{r}{s}}\right)}{r}}{r} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{\frac{\mathsf{fma}\left(r, e^{\frac{r}{s \cdot -3}}, \color{blue}{r \cdot e^{0 - \frac{r}{s}}}\right)}{r}}{r} \]

   11. exp-lowering-exp.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{\frac{\mathsf{fma}\left(r, e^{\frac{r}{s \cdot -3}}, r \cdot \color{blue}{e^{0 - \frac{r}{s}}}\right)}{r}}{r} \]

   12. --lowering--.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{\frac{\mathsf{fma}\left(r, e^{\frac{r}{s \cdot -3}}, r \cdot e^{\color{blue}{0 - \frac{r}{s}}}\right)}{r}}{r} \]

   13. /-lowering-/.f32 99.8

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{\frac{\mathsf{fma}\left(r, e^{\frac{r}{s \cdot -3}}, r \cdot e^{0 - \color{blue}{\frac{r}{s}}}\right)}{r}}{r} \]

6. Applied egg-rr 99.8%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(r, e^{\frac{r}{s \cdot -3}}, r \cdot e^{0 - \frac{r}{s}}\right)}{r}}{r}} \]

7. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}\right)} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)}} \]

   3. div-sub N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)} - \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)}} \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)\right)} - \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)} \]

   5. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{\frac{1}{6} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}}}{s}\right)\right) - \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)} \]

   6. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\frac{\color{blue}{\frac{1}{6}}}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)\right) - \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)} \]

   7. associate-/l/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{6}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)\right) - \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)} \]

   8. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{6} \cdot 1}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)\right) - \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)} \]

   9. associate-*l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\frac{1}{6} \cdot 1}{\color{blue}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}}\right)\right) - \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)} \]

   10. unpow2 N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{\frac{1}{6} \cdot 1}{\color{blue}{{s}^{2}} \cdot \mathsf{PI}\left(\right)}\right)\right) - \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)} \]

   11. associate-*r/ N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}}\right)\right) - \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)} \]

   12. distribute-neg-frac2 N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}\right)\right)} \]

   13. --lowering--.f32 N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right)\right) - \left(\mathsf{neg}\left(\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}\right)\right)} \]

9. Simplified 10.9%

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

Alternative 16: 8.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \frac{-0.16666666666666666}{s \cdot \left(s \cdot \pi\right)} + \frac{0.25}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]
4. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} - \frac{\frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{\frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)\right)} \]

   3. associate-/l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot \frac{\frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}}\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \color{blue}{\frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{\color{blue}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{\color{blue}{{s}^{2}} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right)\right)} \]

5. Simplified 10.9%

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

6. Final simplification 10.9%

   \[\leadsto \frac{-0.16666666666666666}{s \cdot \left(s \cdot \pi\right)} + \frac{0.25}{r \cdot \left(s \cdot \pi\right)} \]
7. Add Preprocessing

Alternative 17: 8.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \pi} \cdot \frac{0.25}{r} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (* (/ 1.0 (* s PI)) (/ 0.25 r)))

C

float code(float s, float r) {
	return (1.0f / (s * ((float) M_PI))) * (0.25f / r);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(1.0) / Float32(s * Float32(pi))) * Float32(Float32(0.25) / r))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / (s * single(pi))) * (single(0.25) / r);
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
4. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   4. PI-lowering-PI.f32 10.1

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

5. Simplified 10.1%

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

   1. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{r}}{s \cdot \mathsf{PI}\left(\right)}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r}} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)} \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r} \cdot \color{blue}{\frac{1}{s \cdot \mathsf{PI}\left(\right)}} \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r} \cdot \frac{1}{\color{blue}{s \cdot \mathsf{PI}\left(\right)}} \]

   7. PI-lowering-PI.f32 10.1

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

7. Applied egg-rr 10.1%

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

8. Final simplification 10.1%

   \[\leadsto \frac{1}{s \cdot \pi} \cdot \frac{0.25}{r} \]
9. Add Preprocessing

Alternative 18: 8.9% accurate, 10.6× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
4. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   4. PI-lowering-PI.f32 10.1

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

5. Simplified 10.1%

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

   1. associate-*r* N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(s \cdot r\right)} \cdot \mathsf{PI}\left(\right)} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)}} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(s \cdot r\right)} \cdot \mathsf{PI}\left(\right)} \]

   5. PI-lowering-PI.f32 10.1

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

7. Applied egg-rr 10.1%

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

   1. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}} \]

   2. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{\mathsf{PI}\left(\right)}}{s \cdot r}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{\mathsf{PI}\left(\right)}}{s \cdot r}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{\mathsf{PI}\left(\right)}}}{s \cdot r} \]

   5. PI-lowering-PI.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right)}}}{s \cdot r} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\mathsf{PI}\left(\right)}}{\color{blue}{r \cdot s}} \]

   7. *-lowering-*.f32 10.1

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

9. Applied egg-rr 10.1%

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

Alternative 19: 8.9% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
4. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   4. PI-lowering-PI.f32 10.1

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

5. Simplified 10.1%

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

   1. associate-*r* N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(s \cdot r\right)} \cdot \mathsf{PI}\left(\right)} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)}} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(s \cdot r\right)} \cdot \mathsf{PI}\left(\right)} \]

   5. PI-lowering-PI.f32 10.1

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

7. Applied egg-rr 10.1%

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

8. Final simplification 10.1%

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

Alternative 20: 8.9% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
4. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   4. PI-lowering-PI.f32 10.1

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

5. Simplified 10.1%

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

Reproduce

herbie shell --seed 2024197 
(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))))