Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.3s

Alternatives: 19

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

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

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 99.6%

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

4. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return fma(Float32(0.75), Float32(exp(Float32(Float32(Float32(-r) / Float32(3.0)) / s)) / Float32(Float32(Float32(Float32(pi) * Float32(6.0)) * 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 99.6%

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

4. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r}, \frac{0.75}{s \cdot \left(\pi \cdot 6\right)}, \frac{\frac{e^{\frac{-r}{s}}}{\pi} \cdot 0.125}{r \cdot s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ (* -0.3333333333333333 r) s)) r)
  (/ 0.75 (* s (* PI 6.0)))
  (/ (* (/ (exp (/ (- r) s)) PI) 0.125) (* r s))))

C

float code(float s, float r) {
	return fmaf((expf(((-0.3333333333333333f * r) / s)) / r), (0.75f / (s * (((float) M_PI) * 6.0f))), (((expf((-r / s)) / ((float) M_PI)) * 0.125f) / (r * s)));
}

Julia

function code(s, r)
	return fma(Float32(exp(Float32(Float32(Float32(-0.3333333333333333) * r) / s)) / r), Float32(Float32(0.75) / Float32(s * Float32(Float32(pi) * Float32(6.0)))), Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(pi)) * Float32(0.125)) / Float32(r * s)))
end

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. lower-*.f32 99.6

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

7. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

9. Applied rewrites 99.6%

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

11. Applied rewrites 99.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r}, \frac{0.75}{s \cdot \left(\pi \cdot 6\right)}, \frac{\frac{e^{\frac{-r}{s}}}{\pi} \cdot 0.125}{r \cdot s}\right)} \]
12. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.75}{\left(\pi \cdot 6\right) \cdot s}, \frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r}, \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.75 (* (* PI 6.0) s))
  (/ (exp (/ (* -0.3333333333333333 r) s)) r)
  (* (/ (exp (/ (- r) s)) (* (* PI s) r)) 0.125)))

C

float code(float s, float r) {
	return fmaf((0.75f / ((((float) M_PI) * 6.0f) * s)), (expf(((-0.3333333333333333f * r) / s)) / r), ((expf((-r / s)) / ((((float) M_PI) * s) * r)) * 0.125f));
}

Julia

function code(s, r)
	return fma(Float32(Float32(0.75) / Float32(Float32(Float32(pi) * Float32(6.0)) * s)), Float32(exp(Float32(Float32(Float32(-0.3333333333333333) * r) / s)) / r), Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(pi) * s) * r)) * Float32(0.125)))
end

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. lower-*.f32 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 99.6%

    \[\leadsto \mathsf{fma}\left(\frac{0.75}{\left(\pi \cdot 6\right) \cdot s}, \frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r}, \color{blue}{\frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125}\right) \]
11. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.125}{\pi \cdot s}, \frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.125 (* PI s))
  (/ (exp (/ (* -0.3333333333333333 r) s)) r)
  (* 0.25 (/ (exp (/ (- r) s)) (* (* (* PI 2.0) s) r)))))

C

float code(float s, float r) {
	return fmaf((0.125f / (((float) M_PI) * s)), (expf(((-0.3333333333333333f * r) / s)) / r), (0.25f * (expf((-r / s)) / (((((float) M_PI) * 2.0f) * s) * r))));
}

Julia

function code(s, r)
	return fma(Float32(Float32(0.125) / Float32(Float32(pi) * s)), Float32(exp(Float32(Float32(Float32(-0.3333333333333333) * r) / 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 99.6%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. lower-*.f32 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 99.5

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\pi \cdot s}, \frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

10. Applied rewrites 99.5%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.125}{\pi \cdot s}}, \frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]
11. Add Preprocessing

Alternative 6: 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 99.6%

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

3. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6

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

5. Applied rewrites 99.6%

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

Alternative 7: 10.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return fma(Float32(Float32(0.75) / Float32(Float32(Float32(pi) * Float32(6.0)) * s)), fma(Float32(fma(Float32(fma(Float32(Float32(r * r) / s), Float32(-0.006172839506172839), Float32(Float32(0.05555555555555555) * r)) / s), Float32(-1.0), Float32(0.3333333333333333)) / s), Float32(-1.0), Float32(Float32(1.0) / r)), Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) * Float32(0.25)) / Float32(Float32(pi) * Float32(2.0))) / Float32(r * s)))
end

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. lower-*.f32 99.6

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

7. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

9. Applied rewrites 99.6%

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

10. Taylor expanded in s around -inf

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

    1. *-commutative N/A

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

    2. lower-fma.f32 N/A

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

12. Applied rewrites 10.6%

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

Alternative 8: 10.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.75 (* (* PI 6.0) s))
  (fma
   (/
    (fma
     (/ (fma (/ (* r r) s) -0.006172839506172839 (* 0.05555555555555555 r)) s)
     -1.0
     0.3333333333333333)
    s)
   -1.0
   (/ 1.0 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)), fmaf((fmaf((fmaf(((r * r) / s), -0.006172839506172839f, (0.05555555555555555f * r)) / s), -1.0f, 0.3333333333333333f) / s), -1.0f, (1.0f / 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)), fma(Float32(fma(Float32(fma(Float32(Float32(r * r) / s), Float32(-0.006172839506172839), Float32(Float32(0.05555555555555555) * r)) / s), Float32(-1.0), Float32(0.3333333333333333)) / s), Float32(-1.0), Float32(Float32(1.0) / 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 99.6%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around -inf

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

7. Applied rewrites 10.6%

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

Alternative 9: 10.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.75 (* (* PI 6.0) s))
  (/
   (fma
    (/ (fma (/ (* r r) s) -0.05555555555555555 (* 0.3333333333333333 r)) s)
    -1.0
    1.0)
   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)), (fmaf((fmaf(((r * r) / s), -0.05555555555555555f, (0.3333333333333333f * r)) / s), -1.0f, 1.0f) / 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(fma(Float32(fma(Float32(Float32(r * r) / s), Float32(-0.05555555555555555), Float32(Float32(0.3333333333333333) * r)) / s), Float32(-1.0), Float32(1.0)) / 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 99.6%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around -inf

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

   1. associate-/r* N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 10.7

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

7. Applied rewrites 10.7%

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

Alternative 10: 10.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((((0.125f / (((float) M_PI) * r)) + ((r / ((s * s) * ((float) M_PI))) * 0.006944444444444444f)) - (0.041666666666666664f / (((float) M_PI) * s))) / 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(0.125) / Float32(Float32(pi) * r)) + Float32(Float32(r / Float32(Float32(s * s) * Float32(pi))) * Float32(0.006944444444444444))) - Float32(Float32(0.041666666666666664) / Float32(Float32(pi) * s))) / s))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((((single(0.125) / (single(pi) * r)) + ((r / ((s * s) * single(pi))) * single(0.006944444444444444))) - (single(0.041666666666666664) / (single(pi) * s))) / s);
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 10.7%

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

   1. lift-fma.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. pow2 N/A

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

   15. lower-+.f32 N/A

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

7. Applied rewrites 10.7%

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

Alternative 11: 10.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((fmaf((r / ((s * s) * ((float) M_PI))), 0.006944444444444444f, (0.125f / (((float) M_PI) * r))) - (0.041666666666666664f / (((float) M_PI) * s))) / 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(fma(Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(0.006944444444444444), Float32(Float32(0.125) / Float32(Float32(pi) * r))) - Float32(Float32(0.041666666666666664) / Float32(Float32(pi) * s))) / s))
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 10.7%

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

Alternative 12: 10.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.75 (* (* PI 6.0) s))
  (fma
   (/ (fma -0.05555555555555555 (/ r s) 0.3333333333333333) s)
   -1.0
   (/ 1.0 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)), fmaf((fmaf(-0.05555555555555555f, (r / s), 0.3333333333333333f) / s), -1.0f, (1.0f / 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)), fma(Float32(fma(Float32(-0.05555555555555555), Float32(r / s), Float32(0.3333333333333333)) / s), Float32(-1.0), Float32(Float32(1.0) / 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 99.6%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around -inf

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 10.7

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

7. Applied rewrites 10.7%

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

Alternative 13: 10.7% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 9.5%

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

6. Taylor expanded in s around inf

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. pow2 N/A

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

   13. lift-*.f32 10.6

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

8. Applied rewrites 10.6%

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

9. Taylor expanded in s around 0

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

    1. lower-/.f32 N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f32 N/A

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

    5. lower-fma.f32 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 N/A

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

    8. *-commutative N/A

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

    9. pow2 N/A

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

    10. lift-*.f32 N/A

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

    11. lift-*.f32 N/A

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

    12. pow2 N/A

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

    13. lift-*.f32 10.7

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

11. Applied rewrites 10.7%

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

Alternative 14: 10.7% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 9.5%

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

6. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 9.5

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

8. Applied rewrites 9.5%

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

9. Taylor expanded in r around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. lower--.f32 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. pow2 N/A

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

    9. lift-*.f32 N/A

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

    10. associate-*r/ N/A

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

    11. metadata-eval N/A

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

    12. lower-/.f32 10.7

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

11. Applied rewrites 10.7%

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

Alternative 15: 10.1% accurate, 3.3× speedup

Math

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

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(-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))) / 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 99.6%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
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 10.1%

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

Alternative 16: 9.1% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 9.1%

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

Alternative 17: 9.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 9.1%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{\pi \cdot r} - \frac{0.16666666666666666}{\pi \cdot s}}{s}} \]
6. Add Preprocessing

Alternative 18: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. 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 9.0

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

5. Applied rewrites 9.0%

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

Alternative 19: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in 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 9.0

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

5. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
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. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 9.0

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

7. Applied rewrites 9.0%

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

Reproduce

herbie shell --seed 2025091 
(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))))