Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.9s

Alternatives: 13

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 99.5%

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

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* 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)) + ((0.75f * expf((-r / (3.0f * s)))) / ((((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(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(pi) * Float32(6.0)) * Float32(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(pi) * single(6.0)) * (s * r)));
end

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(0.125) * Float32(exp(Float32(Float32(-r) / s)) / Float32(r * Float32(s * Float32(pi))))))
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.5

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

7. Applied rewrites 99.5%

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

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

   1. lower-*.f32 N/A

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

   2. mul-1-neg N/A

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

   3. sinh---cosh N/A

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

   4. lower-/.f32 N/A

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

   5. sinh---cosh N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift-PI.f32 99.5

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

10. Applied rewrites 99.5%

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

Alternative 4: 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 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 10.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.75 (* (* PI 6.0) s))
  (-
   (fma (/ r (* s s)) 0.05555555555555555 (/ 1.0 r))
   (/ 0.3333333333333333 s))
  (* 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((r / (s * s)), 0.05555555555555555f, (1.0f / r)) - (0.3333333333333333f / s)), (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(r / Float32(s * s)), Float32(0.05555555555555555), Float32(Float32(1.0) / r)) - Float32(Float32(0.3333333333333333) / s)), 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 99.5%

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

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

   1. lower--.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 10.8

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

7. Applied rewrites 10.8%

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

Alternative 6: 10.5% 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 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 10.8

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

7. Applied rewrites 10.8%

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

Alternative 7: 10.5% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 10.7

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

5. Applied rewrites 10.7%

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

Alternative 8: 9.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(-0.06944444444444445, \frac{r}{\left(s \cdot s\right) \cdot \pi}, 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}\right), -0.25 \cdot \frac{1}{\pi}\right)}{r}}{-s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (/
   (fma
    r
    (fma
     -0.06944444444444445
     (/ r (* (* s s) PI))
     (* 0.16666666666666666 (/ 1.0 (* s PI))))
    (* -0.25 (/ 1.0 PI)))
   r)
  (- s)))

C

float code(float s, float r) {
	return (fmaf(r, fmaf(-0.06944444444444445f, (r / ((s * s) * ((float) M_PI))), (0.16666666666666666f * (1.0f / (s * ((float) M_PI))))), (-0.25f * (1.0f / ((float) M_PI)))) / r) / -s;
}

Julia

function code(s, r)
	return Float32(Float32(fma(r, fma(Float32(-0.06944444444444445), Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(Float32(0.16666666666666666) * Float32(Float32(1.0) / Float32(s * Float32(pi))))), Float32(Float32(-0.25) * Float32(Float32(1.0) / Float32(pi)))) / r) / Float32(-s))
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 10.0%

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

6. Taylor expanded in r around 0

   \[\leadsto -\frac{\frac{r \cdot \left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{s} \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

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

8. Applied rewrites 10.0%

   \[\leadsto -\frac{\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(-0.06944444444444445, \frac{r}{\left(s \cdot s\right) \cdot \pi}, 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}\right), -0.25 \cdot \frac{1}{\pi}\right)}{r}}{s} \]

9. Final simplification 10.0%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(-0.06944444444444445, \frac{r}{\left(s \cdot s\right) \cdot \pi}, 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}\right), -0.25 \cdot \frac{1}{\pi}\right)}{r}}{-s} \]
10. Add Preprocessing

Alternative 9: 9.9% 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 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 10.0%

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

6. Final simplification 10.0%

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

Alternative 10: 9.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.06944444444444445, \frac{r}{\left(s \cdot s\right) \cdot \pi}, 0.25 \cdot \frac{1}{r \cdot \pi}\right) - \frac{0.16666666666666666}{s \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (fma 0.06944444444444445 (/ r (* (* s s) PI)) (* 0.25 (/ 1.0 (* r PI))))
   (/ 0.16666666666666666 (* s PI)))
  s))

C

float code(float s, float r) {
	return (fmaf(0.06944444444444445f, (r / ((s * s) * ((float) M_PI))), (0.25f * (1.0f / (r * ((float) M_PI))))) - (0.16666666666666666f / (s * ((float) M_PI)))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(0.06944444444444445), Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(r * Float32(pi))))) - Float32(Float32(0.16666666666666666) / Float32(s * Float32(pi)))) / s)
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 10.0%

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

6. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

8. Applied rewrites 10.0%

   \[\leadsto \frac{\mathsf{fma}\left(0.06944444444444445, \frac{r}{\left(s \cdot s\right) \cdot \pi}, 0.25 \cdot \frac{1}{r \cdot \pi}\right) - \frac{0.16666666666666666}{s \cdot \pi}}{\color{blue}{s}} \]
9. Add Preprocessing

Alternative 11: 9.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((((0.06944444444444445f * (r / (s * ((float) M_PI)))) - (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(0.06944444444444445) * Float32(r / Float32(s * Float32(pi)))) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s) + Float32(Float32(0.25) / Float32(Float32(pi) * r))) / s)
end

MATLAB

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 10.0%

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

6. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-PI.f32 10.0

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

8. Applied rewrites 10.0%

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

9. Final simplification 10.0%

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

Alternative 12: 8.9% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 9.2

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

5. Applied rewrites 9.2%

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

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 9.3

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

7. Applied rewrites 9.3%

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

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-PI.f32 9.3

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

9. Applied rewrites 9.3%

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

Alternative 13: 8.9% 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 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 9.2

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

5. Applied rewrites 9.2%

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

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 9.3

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

7. Applied rewrites 9.3%

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

Reproduce

herbie shell --seed 2025072 
(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))))