Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.7s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(\pi \cdot 6\right) \cdot s}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right)} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{e^{\frac{\frac{-r}{3}}{s}} \cdot 0.75}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (exp (/ (- r) s)) (* (* PI s) r)) 0.125)
  (/ (* (exp (/ (/ (- r) 3.0) s)) 0.75) (* (* (* PI 6.0) s) r))))

C

float code(float s, float r) {
	return ((expf((-r / s)) / ((((float) M_PI) * s) * r)) * 0.125f) + ((expf(((-r / 3.0f) / s)) * 0.75f) / (((((float) M_PI) * 6.0f) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(pi) * s) * r)) * Float32(0.125)) + Float32(Float32(exp(Float32(Float32(Float32(-r) / Float32(3.0)) / s)) * Float32(0.75)) / Float32(Float32(Float32(Float32(pi) * Float32(6.0)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) / ((single(pi) * s) * r)) * single(0.125)) + ((exp(((-r / single(3.0)) / s)) * single(0.75)) / (((single(pi) * single(6.0)) * s) * r));
end

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-neg.f32 99.5

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

   13. lift-PI.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lift-PI.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125} + \frac{e^{\frac{\frac{-r}{3}}{s}} \cdot 0.75}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r} \]

8. Final simplification 99.5%

   \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{e^{\frac{\frac{-r}{3}}{s}} \cdot 0.75}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r} \]
9. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\pi \cdot 2\right) \cdot \left(s \cdot r\right)} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(\pi \cdot s\right) \cdot r\right) \cdot 6} \]

   7. lift-*.f32 99.5

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

7. Applied rewrites 99.5%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (exp (/ (- r) s)) (* (* PI s) r)) 0.125)
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((expf((-r / s)) / ((((float) M_PI) * s) * r)) * 0.125f) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(pi) * s) * r)) * Float32(0.125)) + 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 = ((exp((-r / s)) / ((single(pi) * s) * r)) * single(0.125)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 99.4

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Add Preprocessing

Alternative 6: 10.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\mathsf{fma}\left(0.0007716049382716049, \frac{\frac{r}{s} \cdot r}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} + \frac{0.041666666666666664}{\pi}}{s} - \frac{0.125}{\pi \cdot r}}{-s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/
   (-
    (/
     (+
      (/
       (fma
        0.0007716049382716049
        (/ (* (/ r s) r) PI)
        (* -0.006944444444444444 (/ r PI)))
       s)
      (/ 0.041666666666666664 PI))
     s)
    (/ 0.125 (* PI r)))
   (- s))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + (((((fmaf(0.0007716049382716049f, (((r / s) * r) / ((float) M_PI)), (-0.006944444444444444f * (r / ((float) M_PI)))) / s) + (0.041666666666666664f / ((float) M_PI))) / s) - (0.125f / (((float) M_PI) * r))) / -s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(Float32(Float32(fma(Float32(0.0007716049382716049), Float32(Float32(Float32(r / s) * r) / Float32(pi)), Float32(Float32(-0.006944444444444444) * Float32(r / Float32(pi)))) / s) + Float32(Float32(0.041666666666666664) / Float32(pi))) / s) - Float32(Float32(0.125) / Float32(Float32(pi) * r))) / Float32(-s)))
end

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-neg.f32 99.5

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

   13. lift-PI.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lift-PI.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 10.8%

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

8. Final simplification 10.8%

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

Alternative 7: 10.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\mathsf{fma}\left(0.0007716049382716049, \frac{\frac{r \cdot r}{s}}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} + \frac{0.041666666666666664}{\pi}}{s} - \frac{0.125}{\pi \cdot r}}{-s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/
   (-
    (/
     (+
      (/
       (fma
        0.0007716049382716049
        (/ (/ (* r r) s) PI)
        (* -0.006944444444444444 (/ r PI)))
       s)
      (/ 0.041666666666666664 PI))
     s)
    (/ 0.125 (* PI r)))
   (- s))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + (((((fmaf(0.0007716049382716049f, (((r * r) / s) / ((float) M_PI)), (-0.006944444444444444f * (r / ((float) M_PI)))) / s) + (0.041666666666666664f / ((float) M_PI))) / s) - (0.125f / (((float) M_PI) * r))) / -s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(Float32(Float32(fma(Float32(0.0007716049382716049), Float32(Float32(Float32(r * r) / s) / Float32(pi)), Float32(Float32(-0.006944444444444444) * Float32(r / Float32(pi)))) / s) + Float32(Float32(0.041666666666666664) / Float32(pi))) / s) - Float32(Float32(0.125) / Float32(Float32(pi) * r))) / Float32(-s)))
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 10.8%

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

6. Final simplification 10.8%

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

Alternative 8: 11.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.25}{\pi \cdot 2}, \frac{e^{\frac{-r}{s}}}{s \cdot r}, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.25 (* PI 2.0))
  (/ (exp (/ (- r) s)) (* s r))
  (/
   (fma (/ (fma (* r (/ r s)) -0.041666666666666664 (* r 0.25)) s) -1.0 0.75)
   (* (* (* PI 6.0) s) r))))

C

float code(float s, float r) {
	return fmaf((0.25f / (((float) M_PI) * 2.0f)), (expf((-r / s)) / (s * r)), (fmaf((fmaf((r * (r / s)), -0.041666666666666664f, (r * 0.25f)) / s), -1.0f, 0.75f) / (((((float) M_PI) * 6.0f) * s) * r)));
}

Julia

function code(s, r)
	return fma(Float32(Float32(0.25) / Float32(Float32(pi) * Float32(2.0))), Float32(exp(Float32(Float32(-r) / s)) / Float32(s * r)), Float32(fma(Float32(fma(Float32(r * Float32(r / s)), Float32(-0.041666666666666664), Float32(r * Float32(0.25))) / s), Float32(-1.0), Float32(0.75)) / Float32(Float32(Float32(Float32(pi) * Float32(6.0)) * s) * r)))
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + -1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{-1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} + \color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} \cdot -1 + \frac{3}{4}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, \color{blue}{-1}, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{{r}^{2}}{s} \cdot \frac{-1}{24} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. lower-*.f32 10.4

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 10.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Applied rewrites 10.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.25}{\pi \cdot 2}, \frac{e^{\frac{-r}{s}}}{s \cdot r}, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}\right)} \]
8. Add Preprocessing

Alternative 9: 11.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/
   (fma (/ (fma (/ (* r r) s) -0.041666666666666664 (* r 0.25)) s) -1.0 0.75)
   (* (* PI 6.0) (* s r)))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + (fmaf((fmaf(((r * r) / s), -0.041666666666666664f, (r * 0.25f)) / s), -1.0f, 0.75f) / ((((float) M_PI) * 6.0f) * (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(fma(Float32(fma(Float32(Float32(r * r) / s), Float32(-0.041666666666666664), Float32(r * Float32(0.25))) / s), Float32(-1.0), Float32(0.75)) / Float32(Float32(Float32(pi) * Float32(6.0)) * Float32(s * r))))
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + -1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{-1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} + \color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} \cdot -1 + \frac{3}{4}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, \color{blue}{-1}, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{{r}^{2}}{s} \cdot \frac{-1}{24} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. lower-*.f32 10.4

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 10.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right)} \cdot r} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\color{blue}{\left(6 \cdot \pi\right) \cdot \left(s \cdot r\right)}} \]

   4. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(6 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(s \cdot r\right)} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(s \cdot r\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 6\right)} \cdot \left(s \cdot r\right)} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 6\right)} \cdot \left(s \cdot r\right)} \]

   8. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\color{blue}{\pi} \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\color{blue}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)}} \]

   10. lift-*.f32 10.4

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\pi \cdot 6\right) \cdot \color{blue}{\left(s \cdot r\right)}} \]

7. Applied rewrites 10.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\color{blue}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)}} \]
8. Add Preprocessing

Alternative 10: 10.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(\pi \cdot s\right) \cdot r\right) \cdot 6} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/
   (fma (/ (fma (/ (* r r) s) -0.041666666666666664 (* r 0.25)) s) -1.0 0.75)
   (* (* (* PI s) r) 6.0))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + (fmaf((fmaf(((r * r) / s), -0.041666666666666664f, (r * 0.25f)) / s), -1.0f, 0.75f) / (((((float) M_PI) * s) * r) * 6.0f));
}

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(fma(Float32(fma(Float32(Float32(r * r) / s), Float32(-0.041666666666666664), Float32(r * Float32(0.25))) / s), Float32(-1.0), Float32(0.75)) / Float32(Float32(Float32(Float32(pi) * s) * r) * Float32(6.0))))
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + -1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{-1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} + \color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} \cdot -1 + \frac{3}{4}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, \color{blue}{-1}, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{{r}^{2}}{s} \cdot \frac{-1}{24} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. lower-*.f32 10.4

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 10.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

6. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\color{blue}{6 \cdot \left(r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{6}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{6}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r\right) \cdot 6} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\right) \cdot 6} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\right) \cdot 6} \]

   6. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(\pi \cdot s\right) \cdot r\right) \cdot 6} \]

   7. lift-*.f32 10.4

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(\pi \cdot s\right) \cdot r\right) \cdot 6} \]

8. Applied rewrites 10.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\color{blue}{\left(\left(\pi \cdot s\right) \cdot r\right) \cdot 6}} \]
9. Add Preprocessing

Alternative 11: 11.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.25, \frac{e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot \left(\pi \cdot 2\right)}, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.25
  (/ (exp (/ (- r) s)) (* (* s r) (* PI 2.0)))
  (/
   (fma (/ (fma (* r (/ r s)) -0.041666666666666664 (* r 0.25)) s) -1.0 0.75)
   (* (* (* PI 6.0) s) r))))

C

float code(float s, float r) {
	return fmaf(0.25f, (expf((-r / s)) / ((s * r) * (((float) M_PI) * 2.0f))), (fmaf((fmaf((r * (r / s)), -0.041666666666666664f, (r * 0.25f)) / s), -1.0f, 0.75f) / (((((float) M_PI) * 6.0f) * s) * r)));
}

Julia

function code(s, r)
	return fma(Float32(0.25), Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(s * r) * Float32(Float32(pi) * Float32(2.0)))), Float32(fma(Float32(fma(Float32(r * Float32(r / s)), Float32(-0.041666666666666664), Float32(r * Float32(0.25))) / s), Float32(-1.0), Float32(0.75)) / Float32(Float32(Float32(Float32(pi) * Float32(6.0)) * s) * r)))
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + -1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{-1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} + \color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} \cdot -1 + \frac{3}{4}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, \color{blue}{-1}, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{{r}^{2}}{s} \cdot \frac{-1}{24} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. lower-*.f32 10.4

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 10.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Applied rewrites 10.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot \left(\pi \cdot 2\right)}, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}\right)} \]
8. Add Preprocessing

Alternative 12: 11.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.041666666666666664, r, s \cdot 0.25\right) \cdot r}{s \cdot s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/
   (fma (/ (* (fma -0.041666666666666664 r (* s 0.25)) r) (* s s)) -1.0 0.75)
   (* (* (* 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)) + (fmaf(((fmaf(-0.041666666666666664f, r, (s * 0.25f)) * r) / (s * s)), -1.0f, 0.75f) / (((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(fma(Float32(Float32(fma(Float32(-0.041666666666666664), r, Float32(s * Float32(0.25))) * r) / Float32(s * s)), Float32(-1.0), Float32(0.75)) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + -1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{-1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} + \color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} \cdot -1 + \frac{3}{4}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, \color{blue}{-1}, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{{r}^{2}}{s} \cdot \frac{-1}{24} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. lower-*.f32 10.4

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 10.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lift-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{r \cdot r}{s} \cdot \frac{-1}{24} + r \cdot \frac{1}{4}}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. div-add N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{r \cdot r}{s} \cdot \frac{-1}{24}}{s} + \frac{r \cdot \frac{1}{4}}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{r \cdot r}{s}}{s} + \frac{r \cdot \frac{1}{4}}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{r \cdot r}{s}}{s} + \frac{r \cdot \frac{1}{4}}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{r \cdot r}{s}}{s} + \frac{r \cdot \frac{1}{4}}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s}}{s} + \frac{r \cdot \frac{1}{4}}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s}}{s} + \frac{\frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. frac-add N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\left(\frac{-1}{24} \cdot \frac{{r}^{2}}{s}\right) \cdot s + s \cdot \left(\frac{1}{4} \cdot r\right)}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. unpow2 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\left(\frac{-1}{24} \cdot \frac{{r}^{2}}{s}\right) \cdot s + s \cdot \left(\frac{1}{4} \cdot r\right)}{{s}^{2}}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   12. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\left(\frac{-1}{24} \cdot \frac{{r}^{2}}{s}\right) \cdot s + s \cdot \left(\frac{1}{4} \cdot r\right)}{{s}^{2}}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Applied rewrites 10.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(r \cdot \frac{r}{s}\right) \cdot -0.041666666666666664, s, s \cdot \left(r \cdot 0.25\right)\right)}{s \cdot s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r \cdot \left(\frac{-1}{24} \cdot r + \frac{1}{4} \cdot s\right)}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\left(\frac{-1}{24} \cdot r + \frac{1}{4} \cdot s\right) \cdot r}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\left(\frac{-1}{24} \cdot r + \frac{1}{4} \cdot s\right) \cdot r}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-1}{24}, r, \frac{1}{4} \cdot s\right) \cdot r}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-1}{24}, r, s \cdot \frac{1}{4}\right) \cdot r}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lower-*.f32 10.4

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.041666666666666664, r, s \cdot 0.25\right) \cdot r}{s \cdot s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

10. Applied rewrites 10.4%

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.041666666666666664, r, s \cdot 0.25\right) \cdot r}{s \cdot s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
11. Add Preprocessing

Alternative 13: 11.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.041666666666666664, r, s \cdot 0.25\right) \cdot r}{s \cdot s}, -1, 0.75\right)}{\left(\left(6 \cdot s\right) \cdot \pi\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/
   (fma (/ (* (fma -0.041666666666666664 r (* s 0.25)) r) (* s s)) -1.0 0.75)
   (* (* (* 6.0 s) PI) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + (fmaf(((fmaf(-0.041666666666666664f, r, (s * 0.25f)) * r) / (s * s)), -1.0f, 0.75f) / (((6.0f * s) * ((float) M_PI)) * 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(fma(Float32(Float32(fma(Float32(-0.041666666666666664), r, Float32(s * Float32(0.25))) * r) / Float32(s * s)), Float32(-1.0), Float32(0.75)) / Float32(Float32(Float32(Float32(6.0) * s) * Float32(pi)) * r)))
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + -1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{-1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} + \color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} \cdot -1 + \frac{3}{4}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, \color{blue}{-1}, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{{r}^{2}}{s} \cdot \frac{-1}{24} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. lower-*.f32 10.4

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 10.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lift-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{r \cdot r}{s} \cdot \frac{-1}{24} + r \cdot \frac{1}{4}}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. div-add N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{r \cdot r}{s} \cdot \frac{-1}{24}}{s} + \frac{r \cdot \frac{1}{4}}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{r \cdot r}{s}}{s} + \frac{r \cdot \frac{1}{4}}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{r \cdot r}{s}}{s} + \frac{r \cdot \frac{1}{4}}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{r \cdot r}{s}}{s} + \frac{r \cdot \frac{1}{4}}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s}}{s} + \frac{r \cdot \frac{1}{4}}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s}}{s} + \frac{\frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. frac-add N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\left(\frac{-1}{24} \cdot \frac{{r}^{2}}{s}\right) \cdot s + s \cdot \left(\frac{1}{4} \cdot r\right)}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. unpow2 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\left(\frac{-1}{24} \cdot \frac{{r}^{2}}{s}\right) \cdot s + s \cdot \left(\frac{1}{4} \cdot r\right)}{{s}^{2}}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   12. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\left(\frac{-1}{24} \cdot \frac{{r}^{2}}{s}\right) \cdot s + s \cdot \left(\frac{1}{4} \cdot r\right)}{{s}^{2}}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Applied rewrites 10.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(r \cdot \frac{r}{s}\right) \cdot -0.041666666666666664, s, s \cdot \left(r \cdot 0.25\right)\right)}{s \cdot s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(r \cdot \frac{r}{s}\right) \cdot \frac{-1}{24}, s, s \cdot \left(r \cdot \frac{1}{4}\right)\right)}{s \cdot s}, -1, \frac{3}{4}\right)}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right)} \cdot r} \]

   2. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(r \cdot \frac{r}{s}\right) \cdot \frac{-1}{24}, s, s \cdot \left(r \cdot \frac{1}{4}\right)\right)}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot s\right) \cdot r} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(r \cdot \frac{r}{s}\right) \cdot \frac{-1}{24}, s, s \cdot \left(r \cdot \frac{1}{4}\right)\right)}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right)} \cdot s\right) \cdot r} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(r \cdot \frac{r}{s}\right) \cdot \frac{-1}{24}, s, s \cdot \left(r \cdot \frac{1}{4}\right)\right)}{s \cdot s}, -1, \frac{3}{4}\right)}{\color{blue}{\left(6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)\right)} \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(r \cdot \frac{r}{s}\right) \cdot \frac{-1}{24}, s, s \cdot \left(r \cdot \frac{1}{4}\right)\right)}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(6 \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot r} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(r \cdot \frac{r}{s}\right) \cdot \frac{-1}{24}, s, s \cdot \left(r \cdot \frac{1}{4}\right)\right)}{s \cdot s}, -1, \frac{3}{4}\right)}{\color{blue}{\left(\left(6 \cdot s\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot r} \]

   7. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(r \cdot \frac{r}{s}\right) \cdot \frac{-1}{24}, s, s \cdot \left(r \cdot \frac{1}{4}\right)\right)}{s \cdot s}, -1, \frac{3}{4}\right)}{\color{blue}{\left(\left(6 \cdot s\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot r} \]

   8. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(r \cdot \frac{r}{s}\right) \cdot \frac{-1}{24}, s, s \cdot \left(r \cdot \frac{1}{4}\right)\right)}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\color{blue}{\left(6 \cdot s\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot r} \]

   9. lift-PI.f32 10.3

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(r \cdot \frac{r}{s}\right) \cdot -0.041666666666666664, s, s \cdot \left(r \cdot 0.25\right)\right)}{s \cdot s}, -1, 0.75\right)}{\left(\left(6 \cdot s\right) \cdot \color{blue}{\pi}\right) \cdot r} \]

9. Applied rewrites 10.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(r \cdot \frac{r}{s}\right) \cdot -0.041666666666666664, s, s \cdot \left(r \cdot 0.25\right)\right)}{s \cdot s}, -1, 0.75\right)}{\color{blue}{\left(\left(6 \cdot s\right) \cdot \pi\right)} \cdot r} \]

10. Taylor expanded in r around 0

    \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r \cdot \left(\frac{-1}{24} \cdot r + \frac{1}{4} \cdot s\right)}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot s\right) \cdot \pi\right) \cdot r} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\left(\frac{-1}{24} \cdot r + \frac{1}{4} \cdot s\right) \cdot r}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot s\right) \cdot \pi\right) \cdot r} \]

    2. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\left(\frac{-1}{24} \cdot r + \frac{1}{4} \cdot s\right) \cdot r}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot s\right) \cdot \pi\right) \cdot r} \]

    3. lower-fma.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-1}{24}, r, \frac{1}{4} \cdot s\right) \cdot r}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot s\right) \cdot \pi\right) \cdot r} \]

    4. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-1}{24}, r, s \cdot \frac{1}{4}\right) \cdot r}{s \cdot s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot s\right) \cdot \pi\right) \cdot r} \]

    5. lower-*.f32 10.3

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.041666666666666664, r, s \cdot 0.25\right) \cdot r}{s \cdot s}, -1, 0.75\right)}{\left(\left(6 \cdot s\right) \cdot \pi\right) \cdot r} \]

12. Applied rewrites 10.3%

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.041666666666666664, r, s \cdot 0.25\right) \cdot r}{s \cdot s}, -1, 0.75\right)}{\left(\left(6 \cdot s\right) \cdot \pi\right) \cdot r} \]
13. Add Preprocessing

Alternative 14: 10.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, \frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{-s}\right)}{s} + \frac{0.16666666666666666}{\pi}}{s} - \frac{0.25}{\pi \cdot r}}{-s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/
    (+
     (/
      (fma
       (/ r PI)
       -0.06944444444444445
       (/ (* (/ (* r r) PI) -0.021604938271604937) (- s)))
      s)
     (/ 0.16666666666666666 PI))
    s)
   (/ 0.25 (* PI r)))
  (- s)))

C

float code(float s, float r) {
	return ((((fmaf((r / ((float) M_PI)), -0.06944444444444445f, ((((r * r) / ((float) M_PI)) * -0.021604938271604937f) / -s)) / s) + (0.16666666666666666f / ((float) M_PI))) / s) - (0.25f / (((float) M_PI) * r))) / -s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(fma(Float32(r / Float32(pi)), Float32(-0.06944444444444445), Float32(Float32(Float32(Float32(r * r) / Float32(pi)) * Float32(-0.021604938271604937)) / Float32(-s))) / s) + Float32(Float32(0.16666666666666666) / Float32(pi))) / s) - Float32(Float32(0.25) / Float32(Float32(pi) * r))) / Float32(-s))
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 9.6%

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

6. Final simplification 9.6%

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

Alternative 15: 10.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-/.f32 N/A

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

5. Applied rewrites 9.5%

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

6. Final simplification 9.5%

   \[\leadsto \frac{\frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s} - \frac{0.25}{\pi \cdot r}}{-s} \]
7. Add Preprocessing

Alternative 16: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.5

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

5. Applied rewrites 8.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 8.5

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

7. Applied rewrites 8.5%

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

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 8.5

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

9. Applied rewrites 8.5%

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

Reproduce

herbie shell --seed 2025084 
(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))))