Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.6%

Time: 4.1s

Alternatives: 18

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

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 PI) 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) + ((float) M_PI)) * 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(pi)) * s) * r))))
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 99.6

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

5. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

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 PI) 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) + ((float) M_PI)) * 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(-r) / Float32(Float32(3.0) * s))) / r), Float32(Float32(0.25) * Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(Float32(pi) + Float32(pi)) * s) * r))))
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

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 PI) 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) + ((float) M_PI)) * s) * r))));
}

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 99.6

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\pi \cdot s}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

8. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.125}{\pi \cdot s}}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

9. Taylor expanded in s around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. mul-1-neg N/A

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

    4. distribute-frac-neg N/A

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

    5. lower-/.f32 N/A

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

    6. lift-neg.f32 N/A

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

    7. lift-/.f32 N/A

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

    8. lift-exp.f32 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f32 N/A

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

    11. lower-*.f32 N/A

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

    12. lift-PI.f32 99.6

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

11. Applied rewrites 99.6%

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

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (((exp((-r / s)) + exp(((r / s) * single(-0.3333333333333333)))) / (single(pi) * r)) / s) * single(0.125);
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. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi \cdot r}}{s} \cdot \color{blue}{0.125} \]
7. Add Preprocessing

Alternative 6: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp(((r / s) * single(-0.3333333333333333))) + exp((-r / s))) / ((s * single(pi)) * r)) * single(0.125);
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. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

10. Applied rewrites 99.5%

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

Alternative 7: 42.4% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (s r) :precision binary32 (/ (/ 0.25 (log (pow (exp PI) r))) s))

C

float code(float s, float r) {
	return (0.25f / logf(powf(expf(((float) M_PI)), r))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / log((exp(Float32(pi)) ^ r))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / log((exp(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. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 9.1

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

7. Applied rewrites 9.1%

   \[\leadsto \frac{\frac{0.25}{\pi \cdot r}}{s} \]
8. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lift-PI.f32 42.4

      \[\leadsto \frac{\frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right)}}{s} \]

9. Applied rewrites 42.4%

   \[\leadsto \frac{\frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right)}}{s} \]
10. Add Preprocessing

Alternative 8: 10.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / log((exp(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. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
3. 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 9.1

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

4. Applied rewrites 9.1%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
5. 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. add-log-exp N/A

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

   8. log-pow-rev N/A

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

   9. lower-log.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lower-*.f32 10.1

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

6. Applied rewrites 10.1%

   \[\leadsto \frac{0.25}{\log \left({\left(e^{\pi}\right)}^{\left(r \cdot s\right)}\right)} \]
7. Add Preprocessing

Alternative 9: 10.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(fma(r, Float32(Float32(Float32(Float32(r / Float32(Float32(pi) * s)) * Float32(0.06944444444444445)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / Float32(s * s)), Float32(Float32(0.25) / Float32(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. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{r}} \]

4. Applied rewrites 8.8%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. pow2 N/A

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

   14. lift-*.f32 10.2

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

7. Applied rewrites 10.2%

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

Alternative 10: 10.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(0.06944444444444445), Float32(Float32(Float32(0.25) / Float32(pi)) / r)) - Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * s))) / 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. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{r}} \]

4. Applied rewrites 8.8%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 10.2%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \pi}, 0.06944444444444445, \frac{0.25}{\pi \cdot r}\right) - \frac{0.16666666666666666}{\pi \cdot s}}{\color{blue}{s}} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f32 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 10.2

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

9. Applied rewrites 10.2%

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

Alternative 11: 10.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(-Float32(Float32(Float32(fma(Float32(r / Float32(Float32(pi) * s)), Float32(-0.06944444444444445), Float32(Float32(0.16666666666666666) / Float32(pi))) / s) - Float32(Float32(0.25) / Float32(Float32(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. Taylor expanded in s around -inf

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

4. Applied rewrites 10.0%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 10.2

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

7. Applied rewrites 10.2%

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

Alternative 12: 9.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = -r * (((single(0.16666666666666666) / ((single(pi) * s) * r)) - (single(0.25) / ((r * r) * single(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. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{r}} \]

4. Applied rewrites 8.8%

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

5. Taylor expanded in r around -inf

   \[\leadsto -1 \cdot \color{blue}{\left(r \cdot \left(-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}}{r} - \frac{5}{72} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)} \]

7. Applied rewrites 8.8%

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

8. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

10. Applied rewrites 9.2%

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

Alternative 13: 9.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp = code(s, r)
	tmp = -((-((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. Taylor expanded in s around -inf

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

4. Applied rewrites 10.0%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lift-PI.f32 9.2

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

7. Applied rewrites 9.2%

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

Alternative 14: 9.2% accurate, 3.3× 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 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 9.2%

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

Alternative 15: 9.1% accurate, 5.3× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{r}} \]

4. Applied rewrites 8.8%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-/.f32 N/A

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

7. Applied rewrites 10.2%

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

8. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lift-PI.f32 9.1

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

10. Applied rewrites 9.1%

    \[\leadsto \frac{-\frac{\frac{-0.25}{\pi}}{s}}{r} \]
11. Add Preprocessing

Alternative 16: 9.1% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / r) / (single(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. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
3. 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 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 9.1

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

6. Applied rewrites 9.1%

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

Alternative 17: 9.1% accurate, 6.4× 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 99.5%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
3. 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 9.1

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

4. Applied rewrites 9.1%

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

Alternative 18: 9.1% accurate, 6.4× 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. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
3. 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 9.1

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

4. Applied rewrites 9.1%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
5. 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 9.1

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

6. Applied rewrites 9.1%

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

Reproduce

herbie shell --seed 2025121 
(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))))