Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.6%

Time: 7.0s

Alternatives: 17

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: 99.5% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-*.f32 99.6

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

7. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around 0

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

   1. associate-/r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 99.5

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

7. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / ((((float) M_PI) + ((float) M_PI)) * (s * r))) + ((0.75f * expf((-0.3333333333333333f * (r / 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(pi) + Float32(pi)) * Float32(s * r))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-0.3333333333333333) * Float32(r / 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(pi) + single(pi)) * (s * r))) + ((single(0.75) * exp((single(-0.3333333333333333) * (r / 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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. count-2-rev N/A

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

   5. lower-+.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lift-/.f32 99.5

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

9. Applied rewrites 99.5%

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

Alternative 5: 99.5% 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. Add Preprocessing

Alternative 6: 10.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.75, \frac{\frac{\frac{\mathsf{fma}\left(\frac{r \cdot \frac{r}{s}}{\pi}, 0.00102880658436214, -0.009259259259259259 \cdot \frac{r}{\pi}\right)}{s} + \frac{0.05555555555555555}{\pi}}{s} - \frac{0.16666666666666666}{\pi \cdot r}}{-s}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.75
  (/
   (-
    (/
     (+
      (/
       (fma
        (/ (* r (/ r s)) PI)
        0.00102880658436214
        (* -0.009259259259259259 (/ r PI)))
       s)
      (/ 0.05555555555555555 PI))
     s)
    (/ 0.16666666666666666 (* PI r)))
   (- s))
  (* 0.25 (/ (exp (/ (- r) s)) (* (* (* PI 2.0) s) r)))))

C

float code(float s, float r) {
	return fmaf(0.75f, (((((fmaf(((r * (r / s)) / ((float) M_PI)), 0.00102880658436214f, (-0.009259259259259259f * (r / ((float) M_PI)))) / s) + (0.05555555555555555f / ((float) M_PI))) / s) - (0.16666666666666666f / (((float) M_PI) * r))) / -s), (0.25f * (expf((-r / s)) / (((((float) M_PI) * 2.0f) * s) * r))));
}

Julia

function code(s, r)
	return fma(Float32(0.75), Float32(Float32(Float32(Float32(Float32(fma(Float32(Float32(r * Float32(r / s)) / Float32(pi)), Float32(0.00102880658436214), Float32(Float32(-0.009259259259259259) * Float32(r / Float32(pi)))) / s) + Float32(Float32(0.05555555555555555) / Float32(pi))) / s) - Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * r))) / Float32(-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%

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

5. Taylor expanded in s around -inf

   \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{108} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{972} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{18} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

7. Applied rewrites 12.6%

   \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r \cdot \frac{r}{s}}{\pi}, 0.00102880658436214, -0.009259259259259259 \cdot \frac{r}{\pi}\right)}{s}\right) - \frac{0.05555555555555555}{\pi}}{s}\right) - \frac{0.16666666666666666}{\pi \cdot r}}{s}}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

8. Final simplification 12.6%

   \[\leadsto \mathsf{fma}\left(0.75, \frac{\frac{\frac{\mathsf{fma}\left(\frac{r \cdot \frac{r}{s}}{\pi}, 0.00102880658436214, -0.009259259259259259 \cdot \frac{r}{\pi}\right)}{s} + \frac{0.05555555555555555}{\pi}}{s} - \frac{0.16666666666666666}{\pi \cdot r}}{-s}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]
9. Add Preprocessing

Alternative 7: 10.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.75, \frac{\mathsf{fma}\left(0.009259259259259259, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.16666666666666666}{\pi \cdot r}\right) - \frac{0.05555555555555555}{\pi \cdot s}}{s}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.75
  (/
   (-
    (fma
     0.009259259259259259
     (/ r (* (* s s) PI))
     (/ 0.16666666666666666 (* PI r)))
    (/ 0.05555555555555555 (* PI s)))
   s)
  (* 0.25 (/ (exp (/ (- r) s)) (* (* (* PI 2.0) s) r)))))

C

float code(float s, float r) {
	return fmaf(0.75f, ((fmaf(0.009259259259259259f, (r / ((s * s) * ((float) M_PI))), (0.16666666666666666f / (((float) M_PI) * r))) - (0.05555555555555555f / (((float) M_PI) * s))) / s), (0.25f * (expf((-r / s)) / (((((float) M_PI) * 2.0f) * s) * r))));
}

Julia

function code(s, r)
	return fma(Float32(0.75), Float32(Float32(fma(Float32(0.009259259259259259), Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * r))) - Float32(Float32(0.05555555555555555) / Float32(Float32(pi) * s))) / 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%

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

5. Taylor expanded in s around inf

   \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \color{blue}{\frac{\left(\frac{1}{108} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{s \cdot \mathsf{PI}\left(\right)}}{s}}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{\left(\frac{1}{108} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{s}}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

7. Applied rewrites 12.6%

   \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{\frac{\mathsf{fma}\left(0.009259259259259259, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.16666666666666666}{\pi \cdot r}\right) - \frac{0.05555555555555555}{\pi \cdot s}}{s}}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]
8. Add Preprocessing

Alternative 8: 10.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. associate-/r* N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower--.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 12.5

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

7. Applied rewrites 12.5%

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

Alternative 9: 10.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 12.5

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

5. Applied rewrites 12.5%

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

6. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. pow2 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 12.5

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

8. Applied rewrites 12.5%

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

Alternative 10: 10.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return fma(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(Float32(pi) * Float32(2.0)) * s) * r)), Float32(0.25), Float32(fma(Float32(fma(Float32(r * Float32(r / s)), Float32(-0.041666666666666664), Float32(Float32(0.25) * r)) / s), Float32(-1.0), Float32(0.75)) / Float32(Float32(Float32(Float32(pi) * Float32(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. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 12.5

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

5. Applied rewrites 12.5%

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

7. Applied rewrites 12.5%

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

Alternative 11: 10.7% accurate, 1.4× speedup

Math

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

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 12.5

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

5. Applied rewrites 12.5%

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

Alternative 12: 10.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(fma(Float32(Float32(fma(Float32(r / s), Float32(-0.041666666666666664), Float32(0.25)) * r) / s), Float32(-1.0), 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 s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 12.5

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

5. Applied rewrites 12.5%

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

6. Taylor expanded in r around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-/.f32 12.5

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

8. Applied rewrites 12.5%

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

Alternative 13: 10.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\left(s \cdot s\right) \cdot \pi}\\ \frac{\mathsf{fma}\left(t\_0, 0.006944444444444444, \mathsf{fma}\left(0.0625, t\_0, \frac{0.25}{\pi \cdot r}\right)\right) - \frac{0.16666666666666666}{\pi \cdot s}}{s} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ r (* (* s s) PI))))
   (/
    (-
     (fma t_0 0.006944444444444444 (fma 0.0625 t_0 (/ 0.25 (* PI r))))
     (/ 0.16666666666666666 (* PI s)))
    s)))

C

float code(float s, float r) {
	float t_0 = r / ((s * s) * ((float) M_PI));
	return (fmaf(t_0, 0.006944444444444444f, fmaf(0.0625f, t_0, (0.25f / (((float) M_PI) * r)))) - (0.16666666666666666f / (((float) M_PI) * s))) / s;
}

Julia

function code(s, r)
	t_0 = Float32(r / Float32(Float32(s * s) * Float32(pi)))
	return Float32(Float32(fma(t_0, Float32(0.006944444444444444), fma(Float32(0.0625), t_0, Float32(Float32(0.25) / Float32(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. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right) - \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

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

5. Applied rewrites 11.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \pi}, 0.006944444444444444, \mathsf{fma}\left(0.0625, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.25}{\pi \cdot r}\right)\right) - \frac{0.16666666666666666}{\pi \cdot s}}{s}} \]
6. Add Preprocessing

Alternative 14: 10.1% 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 11.8%

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

6. Final simplification 11.8%

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

Alternative 15: 10.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\frac{r}{\pi} \cdot 0.8333333333333334}{s \cdot s}, 0.08333333333333333, \frac{0.25}{\pi \cdot r}\right) - \frac{0.16666666666666666}{\pi \cdot s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (fma
    (/ (* (/ r PI) 0.8333333333333334) (* s s))
    0.08333333333333333
    (/ 0.25 (* PI r)))
   (/ 0.16666666666666666 (* PI s)))
  s))

C

float code(float s, float r) {
	return (fmaf((((r / ((float) M_PI)) * 0.8333333333333334f) / (s * s)), 0.08333333333333333f, (0.25f / (((float) M_PI) * r))) - (0.16666666666666666f / (((float) M_PI) * s))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(Float32(Float32(r / Float32(pi)) * Float32(0.8333333333333334)) / Float32(s * s)), Float32(0.08333333333333333), Float32(Float32(0.25) / Float32(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. Add Preprocessing

3. Taylor expanded in s around inf

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

5. Applied rewrites 11.4%

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

7. Applied rewrites 11.3%

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

8. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{12} \cdot \frac{\frac{1}{12} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{3}{4} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{{s}^{2}} + \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

10. Applied rewrites 11.8%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{r}{\pi} \cdot 0.8333333333333334}{s \cdot s}, 0.08333333333333333, \frac{0.25}{\pi \cdot r}\right) - \frac{0.16666666666666666}{\pi \cdot s}}{s}} \]
11. Add Preprocessing

Alternative 16: 9.1% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 10.9%

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

Alternative 17: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / ((s * r) * 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 10.9

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

5. Applied rewrites 10.9%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
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-PI.f32 10.9

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

7. Applied rewrites 10.9%

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

Reproduce

herbie shell --seed 2025064 
(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))))