Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 19.0s

Alternatives: 11

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp((r / (s * single(-3.0))))) / (s * (single(pi) * (r * single(6.0))))) + ((single(0.25) * exp(-(r / s))) / ((single(pi) * (r * s)) * single(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 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 99.7

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

5. Simplified 99.7%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 99.7

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.7

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

7. Applied egg-rr 99.7%

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

   1. lift-*.f32 N/A

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

   2. distribute-frac-neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. lower-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.7

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.75 \cdot e^{\frac{-r}{s \cdot 3}}\\ t_1 := 0.25 \cdot e^{-\frac{r}{s}}\\ t_2 := r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)\\ \mathbf{if}\;\frac{t\_1}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{t\_0}{t\_2} \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_1}{\left(r \cdot s\right) \cdot 2} + \frac{t\_0}{s \cdot \left(\pi \cdot \left(r \cdot 6\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(\pi \cdot \left(r \cdot s\right)\right) \cdot 2} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.006172839506172839, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* 0.75 (exp (/ (- r) (* s 3.0)))))
        (t_1 (* 0.25 (exp (- (/ r s)))))
        (t_2 (* r (* s (* PI 6.0)))))
   (if (<= (+ (/ t_1 (* r (* s (* PI 2.0)))) (/ t_0 t_2)) 4.999999987376214e-7)
     (+ (/ t_1 (* (* r s) 2.0)) (/ t_0 (* s (* PI (* r 6.0)))))
     (+
      (/ t_1 (* (* PI (* r s)) 2.0))
      (/
       (*
        0.75
        (fma
         (/ r s)
         (fma
          (/ r s)
          (fma (/ r s) -0.006172839506172839 0.05555555555555555)
          -0.3333333333333333)
         1.0))
       t_2)))))

C

float code(float s, float r) {
	float t_0 = 0.75f * expf((-r / (s * 3.0f)));
	float t_1 = 0.25f * expf(-(r / s));
	float t_2 = r * (s * (((float) M_PI) * 6.0f));
	float tmp;
	if (((t_1 / (r * (s * (((float) M_PI) * 2.0f)))) + (t_0 / t_2)) <= 4.999999987376214e-7f) {
		tmp = (t_1 / ((r * s) * 2.0f)) + (t_0 / (s * (((float) M_PI) * (r * 6.0f))));
	} else {
		tmp = (t_1 / ((((float) M_PI) * (r * s)) * 2.0f)) + ((0.75f * fmaf((r / s), fmaf((r / s), fmaf((r / s), -0.006172839506172839f, 0.05555555555555555f), -0.3333333333333333f), 1.0f)) / t_2);
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(s * Float32(3.0)))))
	t_1 = Float32(Float32(0.25) * exp(Float32(-Float32(r / s))))
	t_2 = Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))
	tmp = Float32(0.0)
	if (Float32(Float32(t_1 / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(t_0 / t_2)) <= Float32(4.999999987376214e-7))
		tmp = Float32(Float32(t_1 / Float32(Float32(r * s) * Float32(2.0))) + Float32(t_0 / Float32(s * Float32(Float32(pi) * Float32(r * Float32(6.0))))));
	else
		tmp = Float32(Float32(t_1 / Float32(Float32(Float32(pi) * Float32(r * s)) * Float32(2.0))) + Float32(Float32(Float32(0.75) * fma(Float32(r / s), fma(Float32(r / s), fma(Float32(r / s), Float32(-0.006172839506172839), Float32(0.05555555555555555)), Float32(-0.3333333333333333)), Float32(1.0))) / t_2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 4.99999999e-7

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.7%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lower-*.f32 99.7

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

   6. Applied egg-rr 99.7%

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

   if 4.99999999e-7 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 97.4%

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

   3. Taylor expanded in s around 0

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

      1. lower-*.f32 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in r around 0

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \color{blue}{\left(1 + r \cdot \left(r \cdot \left(\frac{-1}{162} \cdot \frac{r}{{s}^{3}} + \frac{1}{18} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{3} \cdot \frac{1}{s}\right)\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   8. Simplified 74.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{2 \cdot \left(\pi \cdot \left(s \cdot r\right)\right)} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.006172839506172839, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{-\frac{r}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25 \cdot e^{-\frac{r}{s}}}{\left(r \cdot s\right) \cdot 2} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{s \cdot \left(\pi \cdot \left(r \cdot 6\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot e^{-\frac{r}{s}}}{\left(\pi \cdot \left(r \cdot s\right)\right) \cdot 2} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.006172839506172839, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot e^{-\frac{r}{s}}\\ t_1 := r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)\\ \mathbf{if}\;\frac{t\_0}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{t\_1} \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_0}{\left(r \cdot s\right) \cdot 2} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\pi \cdot \left(r \cdot s\right)\right) \cdot 2} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.006172839506172839, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* 0.25 (exp (- (/ r s))))) (t_1 (* r (* s (* PI 6.0)))))
   (if (<=
        (+
         (/ t_0 (* r (* s (* PI 2.0))))
         (/ (* 0.75 (exp (/ (- r) (* s 3.0)))) t_1))
        4.999999987376214e-7)
     (+
      (/ t_0 (* (* r s) 2.0))
      (/ (* 0.75 (exp (* -0.3333333333333333 (/ r s)))) t_1))
     (+
      (/ t_0 (* (* PI (* r s)) 2.0))
      (/
       (*
        0.75
        (fma
         (/ r s)
         (fma
          (/ r s)
          (fma (/ r s) -0.006172839506172839 0.05555555555555555)
          -0.3333333333333333)
         1.0))
       t_1)))))

C

float code(float s, float r) {
	float t_0 = 0.25f * expf(-(r / s));
	float t_1 = r * (s * (((float) M_PI) * 6.0f));
	float tmp;
	if (((t_0 / (r * (s * (((float) M_PI) * 2.0f)))) + ((0.75f * expf((-r / (s * 3.0f)))) / t_1)) <= 4.999999987376214e-7f) {
		tmp = (t_0 / ((r * s) * 2.0f)) + ((0.75f * expf((-0.3333333333333333f * (r / s)))) / t_1);
	} else {
		tmp = (t_0 / ((((float) M_PI) * (r * s)) * 2.0f)) + ((0.75f * fmaf((r / s), fmaf((r / s), fmaf((r / s), -0.006172839506172839f, 0.05555555555555555f), -0.3333333333333333f), 1.0f)) / t_1);
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.25) * exp(Float32(-Float32(r / s))))
	t_1 = Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))
	tmp = Float32(0.0)
	if (Float32(Float32(t_0 / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(s * Float32(3.0))))) / t_1)) <= Float32(4.999999987376214e-7))
		tmp = Float32(Float32(t_0 / Float32(Float32(r * s) * Float32(2.0))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-0.3333333333333333) * Float32(r / s)))) / t_1));
	else
		tmp = Float32(Float32(t_0 / Float32(Float32(Float32(pi) * Float32(r * s)) * Float32(2.0))) + Float32(Float32(Float32(0.75) * fma(Float32(r / s), fma(Float32(r / s), fma(Float32(r / s), Float32(-0.006172839506172839), Float32(0.05555555555555555)), Float32(-0.3333333333333333)), Float32(1.0))) / t_1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 4.99999999e-7

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.7%

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

      1. neg-mul-1 N/A

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

      2. times-frac N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 99.7

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

   6. Applied egg-rr 99.7%

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

   if 4.99999999e-7 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 97.4%

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

   3. Taylor expanded in s around 0

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

      1. lower-*.f32 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in r around 0

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \color{blue}{\left(1 + r \cdot \left(r \cdot \left(\frac{-1}{162} \cdot \frac{r}{{s}^{3}} + \frac{1}{18} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{3} \cdot \frac{1}{s}\right)\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   8. Simplified 74.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{2 \cdot \left(\pi \cdot \left(s \cdot r\right)\right)} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.006172839506172839, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{-\frac{r}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25 \cdot e^{-\frac{r}{s}}}{\left(r \cdot s\right) \cdot 2} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot e^{-\frac{r}{s}}}{\left(\pi \cdot \left(r \cdot s\right)\right) \cdot 2} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.006172839506172839, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp(-(r / s))) / (s * (single(pi) * (r * single(2.0))))) + ((single(0.75) * exp(((r * single(-0.3333333333333333)) / s))) / (r * (s * (single(pi) * single(6.0)))));
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. Step-by-step derivation

   1. lift-neg.f32 N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. neg-mul-1 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.5

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

4. Applied egg-rr 99.5%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 99.5

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 10.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{-\frac{r}{s}}}{\left(\pi \cdot \left(r \cdot s\right)\right) \cdot 2} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s}, -0.3333333333333333\right), 1\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (- (/ r s)))) (* (* PI (* r s)) 2.0))
  (/
   (*
    0.75
    (fma (/ r s) (fma r (/ 0.05555555555555555 s) -0.3333333333333333) 1.0))
   (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	return ((0.25f * expf(-(r / s))) / ((((float) M_PI) * (r * s)) * 2.0f)) + ((0.75f * fmaf((r / s), fmaf(r, (0.05555555555555555f / s), -0.3333333333333333f), 1.0f)) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(-Float32(r / s)))) / Float32(Float32(Float32(pi) * Float32(r * s)) * Float32(2.0))) + Float32(Float32(Float32(0.75) * fma(Float32(r / s), fma(r, Float32(Float32(0.05555555555555555) / s), Float32(-0.3333333333333333)), Float32(1.0))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
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 0

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

   1. lower-*.f32 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \color{blue}{\left(r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) + 1\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. sub-neg N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(r \cdot \color{blue}{\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)\right)} + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\color{blue}{\left(r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}}\right) + r \cdot \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)\right)} + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\left(\color{blue}{\left(r \cdot \frac{1}{18}\right) \cdot \frac{r}{{s}^{2}}} + r \cdot \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)\right) + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\left(\color{blue}{\left(\frac{1}{18} \cdot r\right)} \cdot \frac{r}{{s}^{2}} + r \cdot \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)\right) + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. associate-/l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\left(\color{blue}{\frac{\left(\frac{1}{18} \cdot r\right) \cdot r}{{s}^{2}}} + r \cdot \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)\right) + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   7. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\left(\frac{\left(\frac{1}{18} \cdot r\right) \cdot r}{\color{blue}{s \cdot s}} + r \cdot \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)\right) + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   8. times-frac N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\left(\color{blue}{\frac{\frac{1}{18} \cdot r}{s} \cdot \frac{r}{s}} + r \cdot \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)\right) + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\left(\frac{\frac{1}{18} \cdot r}{s} \cdot \frac{r}{s} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right) \cdot r}\right) + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\left(\frac{\frac{1}{18} \cdot r}{s} \cdot \frac{r}{s} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}\right)} \cdot r\right) + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\left(\frac{\frac{1}{18} \cdot r}{s} \cdot \frac{r}{s} + \left(\color{blue}{\frac{-1}{3}} \cdot \frac{1}{s}\right) \cdot r\right) + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\left(\frac{\frac{1}{18} \cdot r}{s} \cdot \frac{r}{s} + \color{blue}{\frac{-1}{3} \cdot \left(\frac{1}{s} \cdot r\right)}\right) + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   13. associate-*l/ N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\left(\frac{\frac{1}{18} \cdot r}{s} \cdot \frac{r}{s} + \frac{-1}{3} \cdot \color{blue}{\frac{1 \cdot r}{s}}\right) + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   14. *-lft-identity N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\left(\frac{\frac{1}{18} \cdot r}{s} \cdot \frac{r}{s} + \frac{-1}{3} \cdot \frac{\color{blue}{r}}{s}\right) + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   15. distribute-rgt-out N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \left(\color{blue}{\frac{r}{s} \cdot \left(\frac{\frac{1}{18} \cdot r}{s} + \frac{-1}{3}\right)} + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   16. lower-fma.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} + \frac{\frac{3}{4} \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \frac{\frac{1}{18} \cdot r}{s} + \frac{-1}{3}, 1\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

8. Simplified 10.0%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{2 \cdot \left(\pi \cdot \left(s \cdot r\right)\right)} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s}, -0.3333333333333333\right), 1\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

9. Final simplification 10.0%

   \[\leadsto \frac{0.25 \cdot e^{-\frac{r}{s}}}{\left(\pi \cdot \left(r \cdot s\right)\right) \cdot 2} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s}, -0.3333333333333333\right), 1\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
10. Add Preprocessing

Alternative 6: 9.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \left(1 + \frac{\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(r, \frac{-0.16666666666666666}{s}, 0.5\right)}{s} - r}{s}\right)}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.006172839506172839, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{s \cdot \left(\pi \cdot \left(r \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/
   (*
    0.25
    (+
     1.0
     (/ (- (/ (* (* r r) (fma r (/ -0.16666666666666666 s) 0.5)) s) r) s)))
   (* r (* s (* PI 2.0))))
  (/
   (*
    0.75
    (fma
     (/ r s)
     (fma
      (/ r s)
      (fma (/ r s) -0.006172839506172839 0.05555555555555555)
      -0.3333333333333333)
     1.0))
   (* s (* PI (* r 6.0))))))

C

float code(float s, float r) {
	return ((0.25f * (1.0f + (((((r * r) * fmaf(r, (-0.16666666666666666f / s), 0.5f)) / s) - r) / s))) / (r * (s * (((float) M_PI) * 2.0f)))) + ((0.75f * fmaf((r / s), fmaf((r / s), fmaf((r / s), -0.006172839506172839f, 0.05555555555555555f), -0.3333333333333333f), 1.0f)) / (s * (((float) M_PI) * (r * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * Float32(Float32(1.0) + Float32(Float32(Float32(Float32(Float32(r * r) * fma(r, Float32(Float32(-0.16666666666666666) / s), Float32(0.5))) / s) - r) / s))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(0.75) * fma(Float32(r / s), fma(Float32(r / s), fma(Float32(r / s), Float32(-0.006172839506172839), Float32(0.05555555555555555)), Float32(-0.3333333333333333)), Float32(1.0))) / Float32(s * Float32(Float32(pi) * Float32(r * Float32(6.0))))))
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 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \color{blue}{\left(1 + r \cdot \left(r \cdot \left(\frac{-1}{162} \cdot \frac{r}{{s}^{3}} + \frac{1}{18} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{3} \cdot \frac{1}{s}\right)\right)}}{s \cdot \left(\mathsf{PI}\left(\right) \cdot \left(6 \cdot r\right)\right)} \]

8. Simplified 9.9%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.006172839506172839, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}}{s \cdot \left(\pi \cdot \left(6 \cdot r\right)\right)} \]

9. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(1 + -1 \cdot \frac{r + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{r}^{3}}{s} + \frac{1}{2} \cdot {r}^{2}}{s}}{s}\right)}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \frac{-1}{162}, \frac{1}{18}\right), \frac{-1}{3}\right), 1\right)}{s \cdot \left(\mathsf{PI}\left(\right) \cdot \left(6 \cdot r\right)\right)} \]
10. Step-by-step derivation

    1. lower-+.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(1 + -1 \cdot \frac{r + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{r}^{3}}{s} + \frac{1}{2} \cdot {r}^{2}}{s}}{s}\right)}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \frac{-1}{162}, \frac{1}{18}\right), \frac{-1}{3}\right), 1\right)}{s \cdot \left(\mathsf{PI}\left(\right) \cdot \left(6 \cdot r\right)\right)} \]

    2. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(1 + \color{blue}{\frac{-1 \cdot \left(r + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{r}^{3}}{s} + \frac{1}{2} \cdot {r}^{2}}{s}\right)}{s}}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \frac{-1}{162}, \frac{1}{18}\right), \frac{-1}{3}\right), 1\right)}{s \cdot \left(\mathsf{PI}\left(\right) \cdot \left(6 \cdot r\right)\right)} \]

    3. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(1 + \frac{\color{blue}{\mathsf{neg}\left(\left(r + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{r}^{3}}{s} + \frac{1}{2} \cdot {r}^{2}}{s}\right)\right)}}{s}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \frac{-1}{162}, \frac{1}{18}\right), \frac{-1}{3}\right), 1\right)}{s \cdot \left(\mathsf{PI}\left(\right) \cdot \left(6 \cdot r\right)\right)} \]

    4. +-commutative N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(1 + \frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{r}^{3}}{s} + \frac{1}{2} \cdot {r}^{2}}{s} + r\right)}\right)}{s}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \frac{-1}{162}, \frac{1}{18}\right), \frac{-1}{3}\right), 1\right)}{s \cdot \left(\mathsf{PI}\left(\right) \cdot \left(6 \cdot r\right)\right)} \]

    5. distribute-neg-in N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(1 + \frac{\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{r}^{3}}{s} + \frac{1}{2} \cdot {r}^{2}}{s}\right)\right) + \left(\mathsf{neg}\left(r\right)\right)}}{s}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \frac{-1}{162}, \frac{1}{18}\right), \frac{-1}{3}\right), 1\right)}{s \cdot \left(\mathsf{PI}\left(\right) \cdot \left(6 \cdot r\right)\right)} \]

    6. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(1 + \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-1}{6} \cdot \frac{{r}^{3}}{s} + \frac{1}{2} \cdot {r}^{2}}{s}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(r\right)\right)}{s}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \frac{-1}{162}, \frac{1}{18}\right), \frac{-1}{3}\right), 1\right)}{s \cdot \left(\mathsf{PI}\left(\right) \cdot \left(6 \cdot r\right)\right)} \]

    7. remove-double-neg N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(1 + \frac{\color{blue}{\frac{\frac{-1}{6} \cdot \frac{{r}^{3}}{s} + \frac{1}{2} \cdot {r}^{2}}{s}} + \left(\mathsf{neg}\left(r\right)\right)}{s}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \frac{-1}{162}, \frac{1}{18}\right), \frac{-1}{3}\right), 1\right)}{s \cdot \left(\mathsf{PI}\left(\right) \cdot \left(6 \cdot r\right)\right)} \]

    8. sub-neg N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(1 + \frac{\color{blue}{\frac{\frac{-1}{6} \cdot \frac{{r}^{3}}{s} + \frac{1}{2} \cdot {r}^{2}}{s} - r}}{s}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \frac{-1}{162}, \frac{1}{18}\right), \frac{-1}{3}\right), 1\right)}{s \cdot \left(\mathsf{PI}\left(\right) \cdot \left(6 \cdot r\right)\right)} \]

    9. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(1 + \color{blue}{\frac{\frac{\frac{-1}{6} \cdot \frac{{r}^{3}}{s} + \frac{1}{2} \cdot {r}^{2}}{s} - r}{s}}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \frac{-1}{162}, \frac{1}{18}\right), \frac{-1}{3}\right), 1\right)}{s \cdot \left(\mathsf{PI}\left(\right) \cdot \left(6 \cdot r\right)\right)} \]

11. Simplified 9.1%

    \[\leadsto \frac{0.25 \cdot \color{blue}{\left(1 + \frac{\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(r, \frac{-0.16666666666666666}{s}, 0.5\right)}{s} - r}{s}\right)}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.006172839506172839, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{s \cdot \left(\pi \cdot \left(6 \cdot r\right)\right)} \]

12. Final simplification 9.1%

    \[\leadsto \frac{0.25 \cdot \left(1 + \frac{\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(r, \frac{-0.16666666666666666}{s}, 0.5\right)}{s} - r}{s}\right)}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.006172839506172839, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{s \cdot \left(\pi \cdot \left(r \cdot 6\right)\right)} \]
13. Add Preprocessing

Alternative 7: 8.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \left(1 - \frac{r}{s}\right)}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{r}{s}, 1\right)}{s \cdot \left(\pi \cdot \left(r \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (- 1.0 (/ r s))) (* r (* s (* PI 2.0))))
  (/ (* 0.75 (fma -0.3333333333333333 (/ r s) 1.0)) (* s (* PI (* r 6.0))))))

C

float code(float s, float r) {
	return ((0.25f * (1.0f - (r / s))) / (r * (s * (((float) M_PI) * 2.0f)))) + ((0.75f * fmaf(-0.3333333333333333f, (r / s), 1.0f)) / (s * (((float) M_PI) * (r * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * Float32(Float32(1.0) - Float32(r / s))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(0.75) * fma(Float32(-0.3333333333333333), Float32(r / s), Float32(1.0))) / Float32(s * Float32(Float32(pi) * Float32(r * Float32(6.0))))))
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 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 9.0

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

8. Simplified 9.0%

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

9. Taylor expanded in r around 0

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f32 N/A

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

    4. lower-/.f32 8.4

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

11. Simplified 8.4%

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

12. Final simplification 8.4%

    \[\leadsto \frac{0.25 \cdot \left(1 - \frac{r}{s}\right)}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{r}{s}, 1\right)}{s \cdot \left(\pi \cdot \left(r \cdot 6\right)\right)} \]
13. Add Preprocessing

Alternative 8: 7.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.75 \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{r}{s}, 1\right)}{s \cdot \left(\pi \cdot \left(r \cdot 6\right)\right)} + \frac{0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.75 (fma -0.3333333333333333 (/ r s) 1.0)) (* s (* PI (* r 6.0))))
  (/ 0.25 (* r (* s (* PI 2.0))))))

C

float code(float s, float r) {
	return ((0.75f * fmaf(-0.3333333333333333f, (r / s), 1.0f)) / (s * (((float) M_PI) * (r * 6.0f)))) + (0.25f / (r * (s * (((float) M_PI) * 2.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.75) * fma(Float32(-0.3333333333333333), Float32(r / s), Float32(1.0))) / Float32(s * Float32(Float32(pi) * Float32(r * Float32(6.0))))) + Float32(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 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 9.0

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

8. Simplified 9.0%

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

9. Taylor expanded in r around 0

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

11. Simplified 7.5%

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

12. Final simplification 7.5%

    \[\leadsto \frac{0.75 \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{r}{s}, 1\right)}{s \cdot \left(\pi \cdot \left(r \cdot 6\right)\right)} + \frac{0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} \]
13. Add Preprocessing

Alternative 9: 6.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.125, r, \frac{r \cdot s}{s \cdot \left(\pi \cdot 8\right)}\right)}{r \cdot \left(r \cdot s\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (fma 0.125 r (/ (* r s) (* s (* PI 8.0)))) (* r (* r s))))

C

float code(float s, float r) {
	return fmaf(0.125f, r, ((r * s) / (s * (((float) M_PI) * 8.0f)))) / (r * (r * s));
}

Julia

function code(s, r)
	return Float32(fma(Float32(0.125), r, Float32(Float32(r * s) / Float32(s * Float32(Float32(pi) * Float32(8.0))))) / Float32(r * 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 94.0%

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

5. Taylor expanded in r around 0

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

7. Simplified 6.8%

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

8. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 6.7

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

10. Simplified 6.7%

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

12. Applied egg-rr 6.7%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, r, \frac{r \cdot s}{s \cdot \left(\pi \cdot 8\right)}\right)}{\left(r \cdot s\right) \cdot r}} \]

13. Final simplification 6.7%

    \[\leadsto \frac{\mathsf{fma}\left(0.125, r, \frac{r \cdot s}{s \cdot \left(\pi \cdot 8\right)}\right)}{r \cdot \left(r \cdot s\right)} \]
14. Add Preprocessing

Alternative 10: 6.8% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / s) + (single(0.125) / (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

4. Applied egg-rr 94.0%

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

5. Taylor expanded in r around 0

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

7. Simplified 6.8%

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

8. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 6.7

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

10. Simplified 6.7%

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

11. Taylor expanded in r around 0

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

    1. lower-/.f32 N/A

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

    2. lower-+.f32 N/A

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

    5. lower-/.f32 N/A

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

    6. associate-*r/ N/A

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

    7. metadata-eval N/A

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

    8. lower-/.f32 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f32 N/A

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

    11. lower-PI.f32 6.7

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

13. Simplified 6.7%

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

14. Final simplification 6.7%

    \[\leadsto \frac{\frac{0.125}{s} + \frac{0.125}{s \cdot \pi}}{r} \]
15. Add Preprocessing

Alternative 11: 6.8% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (r * (s * single(pi)))) - (single(-0.125) / (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 94.0%

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

5. Taylor expanded in r around 0

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

7. Simplified 6.8%

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

8. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 6.7

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

10. Simplified 6.7%

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

11. Taylor expanded in s around 0

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

    1. +-commutative N/A

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

    2. remove-double-neg N/A

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

    3. unsub-neg N/A

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

    4. div-sub N/A

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

    5. associate-*r/ N/A

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

    6. metadata-eval N/A

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

    7. associate-/r* N/A

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

    8. associate-*r* N/A

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

    9. *-commutative N/A

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

    10. distribute-neg-frac N/A

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

    11. lower--.f32 N/A

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

13. Simplified 6.7%

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

14. Final simplification 6.7%

    \[\leadsto \frac{0.125}{r \cdot \left(s \cdot \pi\right)} - \frac{-0.125}{r \cdot s} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))