Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 16.3s

Alternatives: 21

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(r \cdot 0.125\right) \cdot \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{s}, \frac{1}{\pi}, \frac{e^{-\frac{r}{s}}}{s \cdot \pi}\right)\right) \cdot \frac{1}{r \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (*
   (* r 0.125)
   (fma
    (/ (exp (/ r (* s -3.0))) s)
    (/ 1.0 PI)
    (/ (exp (- (/ r s))) (* s PI))))
  (/ 1.0 (* r r))))

C

float code(float s, float r) {
	return ((r * 0.125f) * fmaf((expf((r / (s * -3.0f))) / s), (1.0f / ((float) M_PI)), (expf(-(r / s)) / (s * ((float) M_PI))))) * (1.0f / (r * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(r * Float32(0.125)) * fma(Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / s), Float32(Float32(1.0) / Float32(pi)), Float32(exp(Float32(-Float32(r / s))) / Float32(s * Float32(pi))))) * Float32(Float32(1.0) / Float32(r * r)))
end

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

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

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

7. Simplified 99.6%

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

   1. +-commutative N/A

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

   2. associate-/r* N/A

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

   3. div-inv N/A

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

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

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

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

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

    \[\leadsto \left(\left(r \cdot 0.125\right) \cdot \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{s}, \frac{1}{\pi}, \frac{e^{-\frac{r}{s}}}{s \cdot \pi}\right)\right) \cdot \frac{1}{r \cdot r} \]
11. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ \frac{1}{r \cdot r} \cdot \mathsf{fma}\left(e^{\frac{r}{s \cdot -3}} \cdot t\_0, r, r \cdot \left(t\_0 \cdot e^{-\frac{r}{s}}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* s PI))))
   (*
    (/ 1.0 (* r r))
    (fma (* (exp (/ r (* s -3.0))) t_0) r (* r (* t_0 (exp (- (/ r s)))))))))

C

float code(float s, float r) {
	float t_0 = 0.125f / (s * ((float) M_PI));
	return (1.0f / (r * r)) * fmaf((expf((r / (s * -3.0f))) * t_0), r, (r * (t_0 * expf(-(r / s)))));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(s * Float32(pi)))
	return Float32(Float32(Float32(1.0) / Float32(r * r)) * fma(Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) * t_0), r, Float32(r * Float32(t_0 * exp(Float32(-Float32(r / s)))))))
end

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

   \[\leadsto \frac{1}{r \cdot r} \cdot \mathsf{fma}\left(e^{\frac{r}{s \cdot -3}} \cdot \frac{0.125}{s \cdot \pi}, r, r \cdot \left(\frac{0.125}{s \cdot \pi} \cdot e^{-\frac{r}{s}}\right)\right) \]
6. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

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

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

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

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

7. Simplified 99.6%

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

   1. +-commutative N/A

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

   2. associate-/r* N/A

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

   3. div-inv N/A

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

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

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

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

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

9. Applied egg-rr 99.7%

   \[\leadsto \left(\left(r \cdot 0.125\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{s}, \frac{1}{\pi}, \frac{e^{\frac{-r}{s}}}{s \cdot \pi}\right)}\right) \cdot \frac{1}{r \cdot r} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. associate-*l* N/A

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

    3. associate-*r* N/A

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

    4. inv-pow N/A

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

    5. pow2 N/A

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

    6. pow-pow N/A

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

    7. pow-plus N/A

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

    8. metadata-eval N/A

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

    9. metadata-eval N/A

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

    10. inv-pow N/A

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

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

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(r \cdot 0.125\right) \cdot \frac{\frac{1}{s \cdot \pi} \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{-\frac{r}{s}}\right)}{r \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (* r 0.125)
  (/
   (* (/ 1.0 (* s PI)) (+ (exp (/ r (* s -3.0))) (exp (- (/ r s)))))
   (* r r))))

C

float code(float s, float r) {
	return (r * 0.125f) * (((1.0f / (s * ((float) M_PI))) * (expf((r / (s * -3.0f))) + expf(-(r / s)))) / (r * r));
}

Julia

function code(s, r)
	return Float32(Float32(r * Float32(0.125)) * Float32(Float32(Float32(Float32(1.0) / Float32(s * Float32(pi))) * Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) + exp(Float32(-Float32(r / s))))) / Float32(r * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (r * single(0.125)) * (((single(1.0) / (s * single(pi))) * (exp((r / (s * single(-3.0)))) + exp(-(r / s)))) / (r * r));
end

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

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

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

7. Simplified 99.6%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

9. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{s \cdot \pi} \cdot \left(e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}\right)}{r \cdot r} \cdot \left(r \cdot 0.125\right)} \]

10. Final simplification 99.6%

    \[\leadsto \left(r \cdot 0.125\right) \cdot \frac{\frac{1}{s \cdot \pi} \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{-\frac{r}{s}}\right)}{r \cdot r} \]
11. Add Preprocessing

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

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 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

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

5. Final simplification 99.6%

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

Alternative 7: 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 99.6%

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

   1. div-inv N/A

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

4. Applied egg-rr 98.9%

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

   1. distribute-rgt-out N/A

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

   2. associate-*l/ N/A

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

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

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 8: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. div-inv N/A

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

4. Applied egg-rr 98.9%

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

   1. distribute-rgt-out N/A

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

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

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

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

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

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

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

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

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

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

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

   9. +-commutative N/A

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

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

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

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

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

   12. associate-/r* N/A

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

   13. div-inv N/A

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

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

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

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

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

   16. metadata-eval N/A

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

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

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

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

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

   19. neg-lowering-neg.f32 98.8

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 9: 10.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.9%

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

   1. times-frac N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. distribute-frac-neg N/A

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

   8. distribute-frac-neg2 N/A

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

   9. times-frac N/A

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

   10. associate-*r* N/A

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

   11. associate-/r* N/A

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

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

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

7. Applied egg-rr 8.9%

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

8. Final simplification 8.9%

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

Alternative 10: 10.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.9%

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

6. Final simplification 8.9%

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

Alternative 11: 10.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{1}{s}, \frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right), \frac{e^{-\frac{r}{s}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 8\right)}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 1.0 s)
  (+
   (/ 0.125 (* r PI))
   (fma
    r
    (/ 0.006944444444444444 (* s (* s PI)))
    (/ -0.041666666666666664 (* s PI))))
  (/ (exp (- (/ r s))) (* r (* (* s PI) 8.0)))))

C

float code(float s, float r) {
	return fmaf((1.0f / s), ((0.125f / (r * ((float) M_PI))) + fmaf(r, (0.006944444444444444f / (s * (s * ((float) M_PI)))), (-0.041666666666666664f / (s * ((float) M_PI))))), (expf(-(r / s)) / (r * ((s * ((float) M_PI)) * 8.0f))));
}

Julia

function code(s, r)
	return fma(Float32(Float32(1.0) / s), Float32(Float32(Float32(0.125) / Float32(r * Float32(pi))) + fma(r, Float32(Float32(0.006944444444444444) / Float32(s * Float32(s * Float32(pi)))), Float32(Float32(-0.041666666666666664) / Float32(s * Float32(pi))))), Float32(exp(Float32(-Float32(r / s))) / Float32(r * Float32(Float32(s * Float32(pi)) * Float32(8.0)))))
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.9%

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

7. Applied egg-rr 8.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{s}, \frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right), \frac{e^{\frac{r}{-s}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 8\right)}\right)} \]

8. Final simplification 8.9%

   \[\leadsto \mathsf{fma}\left(\frac{1}{s}, \frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right), \frac{e^{-\frac{r}{s}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 8\right)}\right) \]
9. Add Preprocessing

Alternative 12: 10.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.9%

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

7. Applied egg-rr 8.9%

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

8. Final simplification 8.9%

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

Alternative 13: 10.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.9%

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

6. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + s \cdot \left(s \cdot \left(\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{{s}^{3}}} \]
7. Step-by-step derivation

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

      \[\leadsto \color{blue}{\frac{\frac{1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + s \cdot \left(s \cdot \left(\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{{s}^{3}}} \]

8. Simplified 8.9%

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

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

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. distribute-frac-neg N/A

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

   6. div-inv N/A

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

   7. rec-exp N/A

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

   8. distribute-frac-neg2 N/A

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

   9. frac-2neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

10. Applied egg-rr 8.9%

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

11. Final simplification 8.9%

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

Alternative 14: 9.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. div-inv N/A

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in r around 0

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

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

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

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

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

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

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

   4. PI-lowering-PI.f32 8.4

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

7. Simplified 8.4%

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

   1. distribute-frac-neg2 N/A

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

   2. distribute-frac-neg N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft1-in N/A

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

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

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

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

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

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

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

   9. distribute-frac-neg N/A

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

   10. distribute-frac-neg2 N/A

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

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

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

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

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

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

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

   16. *-commutative N/A

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

   17. associate-*r* N/A

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

9. Applied egg-rr 8.4%

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

10. Final simplification 8.4%

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

Alternative 15: 9.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 7.7%

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

6. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{\color{blue}{\frac{5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)}}}{s \cdot s} + \left(\frac{\frac{1}{4}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. PI-lowering-PI.f32 8.3

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

8. Simplified 8.3%

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

Alternative 16: 9.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.7%

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

6. Taylor expanded in s around inf

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

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

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

8. Simplified 8.3%

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

Alternative 17: 9.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 7.7%

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

6. Taylor expanded in s around inf

   \[\leadsto \frac{\color{blue}{\left(\frac{5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}}}{s} \]

8. Simplified 8.3%

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

Alternative 18: 9.8% accurate, 4.0× speedup

Math

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

FPCore

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

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 / (r * (s * ((float) M_PI))));
}

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.3%

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

Alternative 19: 8.8% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.0

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

5. Simplified 8.0%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

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

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

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

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

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

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

   6. PI-lowering-PI.f32 8.0

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

7. Applied egg-rr 8.0%

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

Alternative 20: 8.8% accurate, 10.6× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.0

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

5. Simplified 8.0%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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. associate-/r* N/A

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

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

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

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

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

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

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

   6. PI-lowering-PI.f32 8.0

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

7. Applied egg-rr 8.0%

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

Alternative 21: 8.8% 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 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.0

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

5. Simplified 8.0%

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

Reproduce

herbie shell --seed 2024205 
(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))))