Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 23.8s

Alternatives: 25

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. distribute-frac-neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. lower-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

4. Applied egg-rr 99.6%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 99.6

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 99.7

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot e^{\frac{r}{-s}}\\ t_1 := r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)\\ \mathbf{if}\;\frac{t\_0}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{3 \cdot \left(-s\right)}}}{t\_1} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{2 \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(\pi \cdot \left(s \cdot 6\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{s \cdot \left(r \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{-0.006172839506172839}{s}, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* 0.25 (exp (/ r (- s))))) (t_1 (* r (* s (* PI 6.0)))))
   (if (<=
        (+
         (/ t_0 (* r (* s (* 2.0 PI))))
         (/ (* 0.75 (exp (/ r (* 3.0 (- s))))) t_1))
        4.999999873689376e-5)
     (+
      (/ t_0 (* 2.0 (* r s)))
      (/
       (* 0.75 (exp (/ (* r -0.3333333333333333) s)))
       (* r (* PI (* s 6.0)))))
     (+
      (/ t_0 (* s (* r (* 2.0 PI))))
      (/
       (*
        0.75
        (fma
         (/ r s)
         (fma
          (/ r s)
          (fma r (/ -0.006172839506172839 s) 0.05555555555555555)
          -0.3333333333333333)
         1.0))
       t_1)))))

C

float code(float s, float r) {
	float t_0 = 0.25f * expf((r / -s));
	float t_1 = r * (s * (((float) M_PI) * 6.0f));
	float tmp;
	if (((t_0 / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * expf((r / (3.0f * -s)))) / t_1)) <= 4.999999873689376e-5f) {
		tmp = (t_0 / (2.0f * (r * s))) + ((0.75f * expf(((r * -0.3333333333333333f) / s))) / (r * (((float) M_PI) * (s * 6.0f))));
	} else {
		tmp = (t_0 / (s * (r * (2.0f * ((float) M_PI))))) + ((0.75f * fmaf((r / s), fmaf((r / s), fmaf(r, (-0.006172839506172839f / s), 0.05555555555555555f), -0.3333333333333333f), 1.0f)) / t_1);
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.25) * exp(Float32(r / Float32(-s))))
	t_1 = Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))
	tmp = Float32(0.0)
	if (Float32(Float32(t_0 / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(Float32(3.0) * Float32(-s))))) / t_1)) <= Float32(4.999999873689376e-5))
		tmp = Float32(Float32(t_0 / Float32(Float32(2.0) * Float32(r * s))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s))) / Float32(r * Float32(Float32(pi) * Float32(s * Float32(6.0))))));
	else
		tmp = Float32(Float32(t_0 / Float32(s * Float32(r * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * fma(Float32(r / s), fma(Float32(r / s), fma(r, Float32(Float32(-0.006172839506172839) / s), Float32(0.05555555555555555)), Float32(-0.3333333333333333)), Float32(1.0))) / t_1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 4.99999987e-5

   1. Initial program 99.7%

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

   4. Applied egg-rr 99.1%

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

      1. neg-mul-1 N/A

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

      2. times-frac N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f32 N/A

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

      7. lift-/.f32 99.1

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

   6. Applied egg-rr 99.1%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 99.1

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. lift-*.f32 99.1

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

   8. Applied egg-rr 99.1%

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

   if 4.99999987e-5 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 98.9%

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

   3. Taylor expanded in s around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in r around 0

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

   8. Simplified 73.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{s \cdot \left(r \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{-0.006172839506172839}{s}, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{3 \cdot \left(-s\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{2 \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(\pi \cdot \left(s \cdot 6\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{s \cdot \left(r \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{-0.006172839506172839}{s}, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot e^{\frac{r}{-s}}\\ t_1 := r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)\\ \mathbf{if}\;\frac{t\_0}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{3 \cdot \left(-s\right)}}}{t\_1} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.75 \cdot e^{\frac{r \cdot -0.3333333333333333}{s}}}{t\_1} + \frac{0.125}{\left(r \cdot s\right) \cdot e^{\frac{r}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{s \cdot \left(r \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{-0.006172839506172839}{s}, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* 0.25 (exp (/ r (- s))))) (t_1 (* r (* s (* PI 6.0)))))
   (if (<=
        (+
         (/ t_0 (* r (* s (* 2.0 PI))))
         (/ (* 0.75 (exp (/ r (* 3.0 (- s))))) t_1))
        4.999999873689376e-5)
     (+
      (/ (* 0.75 (exp (/ (* r -0.3333333333333333) s))) t_1)
      (/ 0.125 (* (* r s) (exp (/ r s)))))
     (+
      (/ t_0 (* s (* r (* 2.0 PI))))
      (/
       (*
        0.75
        (fma
         (/ r s)
         (fma
          (/ r s)
          (fma r (/ -0.006172839506172839 s) 0.05555555555555555)
          -0.3333333333333333)
         1.0))
       t_1)))))

C

float code(float s, float r) {
	float t_0 = 0.25f * expf((r / -s));
	float t_1 = r * (s * (((float) M_PI) * 6.0f));
	float tmp;
	if (((t_0 / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * expf((r / (3.0f * -s)))) / t_1)) <= 4.999999873689376e-5f) {
		tmp = ((0.75f * expf(((r * -0.3333333333333333f) / s))) / t_1) + (0.125f / ((r * s) * expf((r / s))));
	} else {
		tmp = (t_0 / (s * (r * (2.0f * ((float) M_PI))))) + ((0.75f * fmaf((r / s), fmaf((r / s), fmaf(r, (-0.006172839506172839f / s), 0.05555555555555555f), -0.3333333333333333f), 1.0f)) / t_1);
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.25) * exp(Float32(r / Float32(-s))))
	t_1 = Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))
	tmp = Float32(0.0)
	if (Float32(Float32(t_0 / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(Float32(3.0) * Float32(-s))))) / t_1)) <= Float32(4.999999873689376e-5))
		tmp = Float32(Float32(Float32(Float32(0.75) * exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s))) / t_1) + Float32(Float32(0.125) / Float32(Float32(r * s) * exp(Float32(r / s)))));
	else
		tmp = Float32(Float32(t_0 / Float32(s * Float32(r * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * fma(Float32(r / s), fma(Float32(r / s), fma(r, Float32(Float32(-0.006172839506172839) / s), Float32(0.05555555555555555)), Float32(-0.3333333333333333)), Float32(1.0))) / t_1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 4.99999987e-5

   1. Initial program 99.7%

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

   4. Applied egg-rr 99.1%

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

      1. neg-mul-1 N/A

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

      2. times-frac N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f32 N/A

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

      7. lift-/.f32 99.1

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. exp-neg N/A

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

      3. associate-/l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-exp.f32 N/A

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

      11. lower-/.f32 99.1

          \[\leadsto \frac{0.125}{\left(s \cdot r\right) \cdot e^{\color{blue}{\frac{r}{s}}}} + \frac{0.75 \cdot e^{\frac{r \cdot -0.3333333333333333}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{0.125}{\left(s \cdot r\right) \cdot e^{\frac{r}{s}}}} + \frac{0.75 \cdot e^{\frac{r \cdot -0.3333333333333333}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   if 4.99999987e-5 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 98.9%

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

   3. Taylor expanded in s around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in r around 0

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

   8. Simplified 73.9%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{s \cdot \left(r \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{-0.006172839506172839}{s}, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{3 \cdot \left(-s\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.75 \cdot e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{\left(r \cdot s\right) \cdot e^{\frac{r}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{s \cdot \left(r \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{-0.006172839506172839}{s}, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 20.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(6 \cdot \left(r \cdot s\right)\right)\\ t_1 := s \cdot \left(r \cdot 0.75\right)\\ t_2 := 0.25 \cdot e^{\frac{r}{-s}}\\ t_3 := r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)\\ t_4 := \pi \cdot \left(r \cdot s\right)\\ t_5 := 36 \cdot \left(t\_4 \cdot t\_4\right)\\ \mathbf{if}\;\frac{t\_2}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{3 \cdot \left(-s\right)}}}{t\_3} \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.001953125, t\_0 \cdot t\_5, 0.421875 \cdot \left(\left(r \cdot s\right) \cdot \left(\left(r \cdot s\right) \cdot \left(r \cdot s\right)\right)\right)\right) \cdot \frac{1}{\left(r \cdot s\right) \cdot t\_0}}{\mathsf{fma}\left(0.015625, t\_5, t\_1 \cdot \left(t\_1 - \left(\pi \cdot \left(s \cdot 6\right)\right) \cdot \left(r \cdot 0.125\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{s \cdot \left(r \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{-0.006172839506172839}{s}, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* PI (* 6.0 (* r s))))
        (t_1 (* s (* r 0.75)))
        (t_2 (* 0.25 (exp (/ r (- s)))))
        (t_3 (* r (* s (* PI 6.0))))
        (t_4 (* PI (* r s)))
        (t_5 (* 36.0 (* t_4 t_4))))
   (if (<=
        (+
         (/ t_2 (* r (* s (* 2.0 PI))))
         (/ (* 0.75 (exp (/ r (* 3.0 (- s))))) t_3))
        2.0000000233721948e-7)
     (/
      (*
       (fma
        0.001953125
        (* t_0 t_5)
        (* 0.421875 (* (* r s) (* (* r s) (* r s)))))
       (/ 1.0 (* (* r s) t_0)))
      (fma 0.015625 t_5 (* t_1 (- t_1 (* (* PI (* s 6.0)) (* r 0.125))))))
     (+
      (/ t_2 (* s (* r (* 2.0 PI))))
      (/
       (*
        0.75
        (fma
         (/ r s)
         (fma
          (/ r s)
          (fma r (/ -0.006172839506172839 s) 0.05555555555555555)
          -0.3333333333333333)
         1.0))
       t_3)))))

C

float code(float s, float r) {
	float t_0 = ((float) M_PI) * (6.0f * (r * s));
	float t_1 = s * (r * 0.75f);
	float t_2 = 0.25f * expf((r / -s));
	float t_3 = r * (s * (((float) M_PI) * 6.0f));
	float t_4 = ((float) M_PI) * (r * s);
	float t_5 = 36.0f * (t_4 * t_4);
	float tmp;
	if (((t_2 / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * expf((r / (3.0f * -s)))) / t_3)) <= 2.0000000233721948e-7f) {
		tmp = (fmaf(0.001953125f, (t_0 * t_5), (0.421875f * ((r * s) * ((r * s) * (r * s))))) * (1.0f / ((r * s) * t_0))) / fmaf(0.015625f, t_5, (t_1 * (t_1 - ((((float) M_PI) * (s * 6.0f)) * (r * 0.125f)))));
	} else {
		tmp = (t_2 / (s * (r * (2.0f * ((float) M_PI))))) + ((0.75f * fmaf((r / s), fmaf((r / s), fmaf(r, (-0.006172839506172839f / s), 0.05555555555555555f), -0.3333333333333333f), 1.0f)) / t_3);
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(pi) * Float32(Float32(6.0) * Float32(r * s)))
	t_1 = Float32(s * Float32(r * Float32(0.75)))
	t_2 = Float32(Float32(0.25) * exp(Float32(r / Float32(-s))))
	t_3 = Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))
	t_4 = Float32(Float32(pi) * Float32(r * s))
	t_5 = Float32(Float32(36.0) * Float32(t_4 * t_4))
	tmp = Float32(0.0)
	if (Float32(Float32(t_2 / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(Float32(3.0) * Float32(-s))))) / t_3)) <= Float32(2.0000000233721948e-7))
		tmp = Float32(Float32(fma(Float32(0.001953125), Float32(t_0 * t_5), Float32(Float32(0.421875) * Float32(Float32(r * s) * Float32(Float32(r * s) * Float32(r * s))))) * Float32(Float32(1.0) / Float32(Float32(r * s) * t_0))) / fma(Float32(0.015625), t_5, Float32(t_1 * Float32(t_1 - Float32(Float32(Float32(pi) * Float32(s * Float32(6.0))) * Float32(r * Float32(0.125)))))));
	else
		tmp = Float32(Float32(t_2 / Float32(s * Float32(r * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * fma(Float32(r / s), fma(Float32(r / s), fma(r, Float32(Float32(-0.006172839506172839) / s), Float32(0.05555555555555555)), Float32(-0.3333333333333333)), Float32(1.0))) / t_3));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 2.00000002e-7

   1. Initial program 99.7%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in r around 0

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

   7. Simplified 5.0%

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

   8. Taylor expanded in r around 0

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 4.9

         \[\leadsto \frac{0.125}{\color{blue}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. Simplified 4.9%

       \[\leadsto \color{blue}{\frac{0.125}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   12. Applied egg-rr 12.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.001953125, \left(\pi \cdot \left(6 \cdot \left(s \cdot r\right)\right)\right) \cdot \left(36 \cdot \left(\left(\pi \cdot \left(s \cdot r\right)\right) \cdot \left(\pi \cdot \left(s \cdot r\right)\right)\right)\right), 0.421875 \cdot \left(\left(s \cdot r\right) \cdot \left(\left(s \cdot r\right) \cdot \left(s \cdot r\right)\right)\right)\right) \cdot \frac{1}{\left(s \cdot r\right) \cdot \left(\pi \cdot \left(6 \cdot \left(s \cdot r\right)\right)\right)}}{\mathsf{fma}\left(0.015625, 36 \cdot \left(\left(\pi \cdot \left(s \cdot r\right)\right) \cdot \left(\pi \cdot \left(s \cdot r\right)\right)\right), \left(s \cdot \left(r \cdot 0.75\right)\right) \cdot \left(s \cdot \left(r \cdot 0.75\right) - \left(0.125 \cdot r\right) \cdot \left(\pi \cdot \left(s \cdot 6\right)\right)\right)\right)}} \]

   if 2.00000002e-7 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 98.9%

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

   3. Taylor expanded in s around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in r around 0

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

   8. Simplified 70.8%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{s \cdot \left(r \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{-0.006172839506172839}{s}, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{3 \cdot \left(-s\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.001953125, \left(\pi \cdot \left(6 \cdot \left(r \cdot s\right)\right)\right) \cdot \left(36 \cdot \left(\left(\pi \cdot \left(r \cdot s\right)\right) \cdot \left(\pi \cdot \left(r \cdot s\right)\right)\right)\right), 0.421875 \cdot \left(\left(r \cdot s\right) \cdot \left(\left(r \cdot s\right) \cdot \left(r \cdot s\right)\right)\right)\right) \cdot \frac{1}{\left(r \cdot s\right) \cdot \left(\pi \cdot \left(6 \cdot \left(r \cdot s\right)\right)\right)}}{\mathsf{fma}\left(0.015625, 36 \cdot \left(\left(\pi \cdot \left(r \cdot s\right)\right) \cdot \left(\pi \cdot \left(r \cdot s\right)\right)\right), \left(s \cdot \left(r \cdot 0.75\right)\right) \cdot \left(s \cdot \left(r \cdot 0.75\right) - \left(\pi \cdot \left(s \cdot 6\right)\right) \cdot \left(r \cdot 0.125\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{s \cdot \left(r \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{-0.006172839506172839}{s}, 0.05555555555555555\right), -0.3333333333333333\right), 1\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 20.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(6 \cdot \left(r \cdot s\right)\right)\\ t_1 := s \cdot \left(r \cdot 0.75\right)\\ t_2 := 0.25 \cdot e^{\frac{r}{-s}}\\ t_3 := r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)\\ t_4 := \pi \cdot \left(r \cdot s\right)\\ t_5 := 36 \cdot \left(t\_4 \cdot t\_4\right)\\ \mathbf{if}\;\frac{t\_2}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{3 \cdot \left(-s\right)}}}{t\_3} \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.001953125, t\_0 \cdot t\_5, 0.421875 \cdot \left(\left(r \cdot s\right) \cdot \left(\left(r \cdot s\right) \cdot \left(r \cdot s\right)\right)\right)\right) \cdot \frac{1}{\left(r \cdot s\right) \cdot t\_0}}{\mathsf{fma}\left(0.015625, t\_5, t\_1 \cdot \left(t\_1 - \left(\pi \cdot \left(s \cdot 6\right)\right) \cdot \left(r \cdot 0.125\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{s \cdot \left(r \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s}, -0.3333333333333333\right), 1\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* PI (* 6.0 (* r s))))
        (t_1 (* s (* r 0.75)))
        (t_2 (* 0.25 (exp (/ r (- s)))))
        (t_3 (* r (* s (* PI 6.0))))
        (t_4 (* PI (* r s)))
        (t_5 (* 36.0 (* t_4 t_4))))
   (if (<=
        (+
         (/ t_2 (* r (* s (* 2.0 PI))))
         (/ (* 0.75 (exp (/ r (* 3.0 (- s))))) t_3))
        2.0000000233721948e-7)
     (/
      (*
       (fma
        0.001953125
        (* t_0 t_5)
        (* 0.421875 (* (* r s) (* (* r s) (* r s)))))
       (/ 1.0 (* (* r s) t_0)))
      (fma 0.015625 t_5 (* t_1 (- t_1 (* (* PI (* s 6.0)) (* r 0.125))))))
     (+
      (/ t_2 (* s (* r (* 2.0 PI))))
      (/
       (*
        0.75
        (fma
         (/ r s)
         (fma r (/ 0.05555555555555555 s) -0.3333333333333333)
         1.0))
       t_3)))))

C

float code(float s, float r) {
	float t_0 = ((float) M_PI) * (6.0f * (r * s));
	float t_1 = s * (r * 0.75f);
	float t_2 = 0.25f * expf((r / -s));
	float t_3 = r * (s * (((float) M_PI) * 6.0f));
	float t_4 = ((float) M_PI) * (r * s);
	float t_5 = 36.0f * (t_4 * t_4);
	float tmp;
	if (((t_2 / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * expf((r / (3.0f * -s)))) / t_3)) <= 2.0000000233721948e-7f) {
		tmp = (fmaf(0.001953125f, (t_0 * t_5), (0.421875f * ((r * s) * ((r * s) * (r * s))))) * (1.0f / ((r * s) * t_0))) / fmaf(0.015625f, t_5, (t_1 * (t_1 - ((((float) M_PI) * (s * 6.0f)) * (r * 0.125f)))));
	} else {
		tmp = (t_2 / (s * (r * (2.0f * ((float) M_PI))))) + ((0.75f * fmaf((r / s), fmaf(r, (0.05555555555555555f / s), -0.3333333333333333f), 1.0f)) / t_3);
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(pi) * Float32(Float32(6.0) * Float32(r * s)))
	t_1 = Float32(s * Float32(r * Float32(0.75)))
	t_2 = Float32(Float32(0.25) * exp(Float32(r / Float32(-s))))
	t_3 = Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))
	t_4 = Float32(Float32(pi) * Float32(r * s))
	t_5 = Float32(Float32(36.0) * Float32(t_4 * t_4))
	tmp = Float32(0.0)
	if (Float32(Float32(t_2 / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(Float32(3.0) * Float32(-s))))) / t_3)) <= Float32(2.0000000233721948e-7))
		tmp = Float32(Float32(fma(Float32(0.001953125), Float32(t_0 * t_5), Float32(Float32(0.421875) * Float32(Float32(r * s) * Float32(Float32(r * s) * Float32(r * s))))) * Float32(Float32(1.0) / Float32(Float32(r * s) * t_0))) / fma(Float32(0.015625), t_5, Float32(t_1 * Float32(t_1 - Float32(Float32(Float32(pi) * Float32(s * Float32(6.0))) * Float32(r * Float32(0.125)))))));
	else
		tmp = Float32(Float32(t_2 / Float32(s * Float32(r * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * fma(Float32(r / s), fma(r, Float32(Float32(0.05555555555555555) / s), Float32(-0.3333333333333333)), Float32(1.0))) / t_3));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 2.00000002e-7

   1. Initial program 99.7%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in r around 0

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

   7. Simplified 5.0%

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

   8. Taylor expanded in r around 0

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 4.9

         \[\leadsto \frac{0.125}{\color{blue}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. Simplified 4.9%

       \[\leadsto \color{blue}{\frac{0.125}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   12. Applied egg-rr 12.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.001953125, \left(\pi \cdot \left(6 \cdot \left(s \cdot r\right)\right)\right) \cdot \left(36 \cdot \left(\left(\pi \cdot \left(s \cdot r\right)\right) \cdot \left(\pi \cdot \left(s \cdot r\right)\right)\right)\right), 0.421875 \cdot \left(\left(s \cdot r\right) \cdot \left(\left(s \cdot r\right) \cdot \left(s \cdot r\right)\right)\right)\right) \cdot \frac{1}{\left(s \cdot r\right) \cdot \left(\pi \cdot \left(6 \cdot \left(s \cdot r\right)\right)\right)}}{\mathsf{fma}\left(0.015625, 36 \cdot \left(\left(\pi \cdot \left(s \cdot r\right)\right) \cdot \left(\pi \cdot \left(s \cdot r\right)\right)\right), \left(s \cdot \left(r \cdot 0.75\right)\right) \cdot \left(s \cdot \left(r \cdot 0.75\right) - \left(0.125 \cdot r\right) \cdot \left(\pi \cdot \left(s \cdot 6\right)\right)\right)\right)}} \]

   if 2.00000002e-7 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 98.9%

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

   3. Taylor expanded in s around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in r around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. unpow2 N/A

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

      14. times-frac N/A

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

      15. distribute-rgt-out N/A

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

   8. Simplified 68.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{s \cdot \left(r \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s}, -0.3333333333333333\right), 1\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{3 \cdot \left(-s\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.001953125, \left(\pi \cdot \left(6 \cdot \left(r \cdot s\right)\right)\right) \cdot \left(36 \cdot \left(\left(\pi \cdot \left(r \cdot s\right)\right) \cdot \left(\pi \cdot \left(r \cdot s\right)\right)\right)\right), 0.421875 \cdot \left(\left(r \cdot s\right) \cdot \left(\left(r \cdot s\right) \cdot \left(r \cdot s\right)\right)\right)\right) \cdot \frac{1}{\left(r \cdot s\right) \cdot \left(\pi \cdot \left(6 \cdot \left(r \cdot s\right)\right)\right)}}{\mathsf{fma}\left(0.015625, 36 \cdot \left(\left(\pi \cdot \left(r \cdot s\right)\right) \cdot \left(\pi \cdot \left(r \cdot s\right)\right)\right), \left(s \cdot \left(r \cdot 0.75\right)\right) \cdot \left(s \cdot \left(r \cdot 0.75\right) - \left(\pi \cdot \left(s \cdot 6\right)\right) \cdot \left(r \cdot 0.125\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{s \cdot \left(r \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s}, -0.3333333333333333\right), 1\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. distribute-frac-neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. lower-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

4. Applied egg-rr 99.6%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 99.6

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 7: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((r / -s))) / (single(2.0) * (single(pi) * (r * s)))) + ((single(0.75) * exp(((r * single(-0.3333333333333333)) / s))) / (r * (s * (single(pi) * single(6.0)))));
end

Derivation

1. Initial program 99.6%

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

   1. lift-neg.f32 N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. neg-mul-1 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

4. Applied egg-rr 99.6%

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

   1. lift-PI.f32 N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 99.6

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 8: 10.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. sub-neg N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-neg-in N/A

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. unpow2 N/A

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

   14. times-frac N/A

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

   15. distribute-rgt-out N/A

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

8. Simplified 12.0%

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

9. Final simplification 12.0%

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

Alternative 9: 9.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 10.3

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

8. Simplified 10.3%

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

9. Final simplification 10.3%

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

Alternative 10: 9.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 10.3

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

8. Simplified 10.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(r, \frac{-0.3333333333333333}{s}, 1\right)}}{s \cdot \left(6 \cdot \left(\pi \cdot r\right)\right)} \]
9. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. distribute-rgt-in N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. associate-*r* N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 10.3

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

10. Applied egg-rr 10.3%

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

    1. lift-PI.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. lift-*.f32 N/A

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

    6. associate-*r* N/A

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

    7. associate-*l* N/A

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

    8. *-commutative N/A

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

    9. lower-*.f32 N/A

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

    10. lower-*.f32 N/A

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

    11. lower-*.f32 10.3

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

12. Applied egg-rr 10.3%

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

13. Final simplification 10.3%

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

Alternative 11: 9.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 10.3

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

8. Simplified 10.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(r, \frac{-0.3333333333333333}{s}, 1\right)}}{s \cdot \left(6 \cdot \left(\pi \cdot r\right)\right)} \]
9. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. distribute-rgt-in N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. associate-*r* N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 10.3

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

10. Applied egg-rr 10.3%

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

11. Final simplification 10.3%

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

Alternative 12: 9.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

8. Simplified 9.9%

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

9. Final simplification 9.9%

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

Alternative 13: 9.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 10.3

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

8. Simplified 10.3%

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

9. Taylor expanded in r around 0

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

11. Simplified 9.9%

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

12. Final simplification 9.9%

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

Alternative 14: 9.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \left(\frac{\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(\frac{r}{s}, -0.16666666666666666, 0.5\right)}{s} - r}{s} + 1\right)}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(r, \frac{-0.3333333333333333}{s}, 1\right)}{s \cdot \left(6 \cdot \left(r \cdot \pi\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/
   (*
    0.25
    (+
     (/ (- (/ (* (* r r) (fma (/ r s) -0.16666666666666666 0.5)) s) r) s)
     1.0))
   (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (fma r (/ -0.3333333333333333 s) 1.0)) (* s (* 6.0 (* r PI))))))

C

float code(float s, float r) {
	return ((0.25f * ((((((r * r) * fmaf((r / s), -0.16666666666666666f, 0.5f)) / s) - r) / s) + 1.0f)) / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * fmaf(r, (-0.3333333333333333f / s), 1.0f)) / (s * (6.0f * (r * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * Float32(Float32(Float32(Float32(Float32(Float32(r * r) * fma(Float32(r / s), Float32(-0.16666666666666666), Float32(0.5))) / s) - r) / s) + Float32(1.0))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * fma(r, Float32(Float32(-0.3333333333333333) / s), Float32(1.0))) / Float32(s * Float32(Float32(6.0) * Float32(r * Float32(pi))))))
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 10.3

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

8. Simplified 10.3%

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

9. Taylor expanded in s around -inf

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f32 N/A

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

    4. lower-/.f32 N/A

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

11. Simplified 9.7%

    \[\leadsto \frac{0.25 \cdot \color{blue}{\left(1 - \frac{r - \frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(\frac{r}{s}, -0.16666666666666666, 0.5\right)}{s}}{s}\right)}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(r, \frac{-0.3333333333333333}{s}, 1\right)}{s \cdot \left(6 \cdot \left(\pi \cdot r\right)\right)} \]

12. Final simplification 9.7%

    \[\leadsto \frac{0.25 \cdot \left(\frac{\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(\frac{r}{s}, -0.16666666666666666, 0.5\right)}{s} - r}{s} + 1\right)}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot \mathsf{fma}\left(r, \frac{-0.3333333333333333}{s}, 1\right)}{s \cdot \left(6 \cdot \left(r \cdot \pi\right)\right)} \]
13. Add Preprocessing

Alternative 15: 9.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{r}{s}, -0.25, 0.75\right)}{s \cdot \left(6 \cdot \left(r \cdot \pi\right)\right)} + \frac{0.25 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (fma (/ r s) -0.25 0.75) (* s (* 6.0 (* r PI))))
  (/
   (*
    0.25
    (fma
     (/ r s)
     (fma (/ r s) (fma (/ r s) -0.16666666666666666 0.5) -1.0)
     1.0))
   (* r (* s (* 2.0 PI))))))

C

float code(float s, float r) {
	return (fmaf((r / s), -0.25f, 0.75f) / (s * (6.0f * (r * ((float) M_PI))))) + ((0.25f * fmaf((r / s), fmaf((r / s), fmaf((r / s), -0.16666666666666666f, 0.5f), -1.0f), 1.0f)) / (r * (s * (2.0f * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(r / s), Float32(-0.25), Float32(0.75)) / Float32(s * Float32(Float32(6.0) * Float32(r * Float32(pi))))) + Float32(Float32(Float32(0.25) * fma(Float32(r / s), fma(Float32(r / s), fma(Float32(r / s), Float32(-0.16666666666666666), Float32(0.5)), Float32(-1.0)), Float32(1.0))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))))
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 10.3

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

8. Simplified 10.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(r, \frac{-0.3333333333333333}{s}, 1\right)}}{s \cdot \left(6 \cdot \left(\pi \cdot r\right)\right)} \]
9. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. distribute-rgt-in N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. associate-*r* N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 10.3

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

10. Applied egg-rr 10.3%

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

11. Taylor expanded in r around 0

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

13. Simplified 9.7%

    \[\leadsto \frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s}, -0.25, 0.75\right)}{s \cdot \left(6 \cdot \left(\pi \cdot r\right)\right)} \]

14. Final simplification 9.7%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s}, -0.25, 0.75\right)}{s \cdot \left(6 \cdot \left(r \cdot \pi\right)\right)} + \frac{0.25 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{r}{s}, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} \]
15. Add Preprocessing

Alternative 16: 8.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.75 \cdot \mathsf{fma}\left(r, \frac{-0.3333333333333333}{s}, 1\right)}{s \cdot \left(6 \cdot \left(r \cdot \pi\right)\right)} + \frac{0.25 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{0.5}{s}, -1\right), 1\right)}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.75 (fma r (/ -0.3333333333333333 s) 1.0)) (* s (* 6.0 (* r PI))))
  (/
   (* 0.25 (fma (/ r s) (fma r (/ 0.5 s) -1.0) 1.0))
   (* r (* s (* 2.0 PI))))))

C

float code(float s, float r) {
	return ((0.75f * fmaf(r, (-0.3333333333333333f / s), 1.0f)) / (s * (6.0f * (r * ((float) M_PI))))) + ((0.25f * fmaf((r / s), fmaf(r, (0.5f / s), -1.0f), 1.0f)) / (r * (s * (2.0f * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.75) * fma(r, Float32(Float32(-0.3333333333333333) / s), Float32(1.0))) / Float32(s * Float32(Float32(6.0) * Float32(r * Float32(pi))))) + Float32(Float32(Float32(0.25) * fma(Float32(r / s), fma(r, Float32(Float32(0.5) / s), Float32(-1.0)), Float32(1.0))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))))
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 10.3

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

8. Simplified 10.3%

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

9. Taylor expanded in r around 0

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

    1. +-commutative N/A

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

    2. sub-neg N/A

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

    3. distribute-lft-in N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. associate-/l* N/A

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

    7. unpow2 N/A

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

    8. times-frac N/A

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

    9. distribute-rgt-neg-out N/A

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

    10. neg-mul-1 N/A

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

    11. associate-*r/ N/A

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

    12. *-rgt-identity N/A

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

    13. distribute-rgt-out N/A

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

    14. lower-fma.f32 N/A

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

11. Simplified 9.5%

    \[\leadsto \frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{0.5}{s}, -1\right), 1\right)}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(r, \frac{-0.3333333333333333}{s}, 1\right)}{s \cdot \left(6 \cdot \left(\pi \cdot r\right)\right)} \]

12. Final simplification 9.5%

    \[\leadsto \frac{0.75 \cdot \mathsf{fma}\left(r, \frac{-0.3333333333333333}{s}, 1\right)}{s \cdot \left(6 \cdot \left(r \cdot \pi\right)\right)} + \frac{0.25 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{0.5}{s}, -1\right), 1\right)}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} \]
13. Add Preprocessing

Alternative 17: 8.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{r}{s}, -0.25, 0.75\right)}{s \cdot \left(6 \cdot \left(r \cdot \pi\right)\right)} + \frac{0.25 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{0.5}{s}, -1\right), 1\right)}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (fma (/ r s) -0.25 0.75) (* s (* 6.0 (* r PI))))
  (/
   (* 0.25 (fma (/ r s) (fma r (/ 0.5 s) -1.0) 1.0))
   (* r (* s (* 2.0 PI))))))

C

float code(float s, float r) {
	return (fmaf((r / s), -0.25f, 0.75f) / (s * (6.0f * (r * ((float) M_PI))))) + ((0.25f * fmaf((r / s), fmaf(r, (0.5f / s), -1.0f), 1.0f)) / (r * (s * (2.0f * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(r / s), Float32(-0.25), Float32(0.75)) / Float32(s * Float32(Float32(6.0) * Float32(r * Float32(pi))))) + Float32(Float32(Float32(0.25) * fma(Float32(r / s), fma(r, Float32(Float32(0.5) / s), Float32(-1.0)), Float32(1.0))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))))
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 10.3

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

8. Simplified 10.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(r, \frac{-0.3333333333333333}{s}, 1\right)}}{s \cdot \left(6 \cdot \left(\pi \cdot r\right)\right)} \]
9. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. distribute-rgt-in N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. associate-*r* N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 10.3

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

10. Applied egg-rr 10.3%

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

11. Taylor expanded in r around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-out-- N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. associate-/l* N/A

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

    6. unpow2 N/A

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

    7. *-commutative N/A

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

    8. associate-*r/ N/A

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

    9. *-rgt-identity N/A

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

    10. unsub-neg N/A

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

    11. mul-1-neg N/A

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

    12. associate-*r/ N/A

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

    13. unpow2 N/A

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

    14. associate-*r* N/A

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

    15. unpow2 N/A

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

    16. times-frac N/A

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

    17. distribute-rgt-out N/A

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

13. Simplified 9.5%

    \[\leadsto \frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{0.5}{s}, -1\right), 1\right)}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{r}{s}, -0.25, 0.75\right)}{s \cdot \left(6 \cdot \left(\pi \cdot r\right)\right)} \]

14. Final simplification 9.5%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s}, -0.25, 0.75\right)}{s \cdot \left(6 \cdot \left(r \cdot \pi\right)\right)} + \frac{0.25 \cdot \mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(r, \frac{0.5}{s}, -1\right), 1\right)}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} \]
15. Add Preprocessing

Alternative 18: 9.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 10.3

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

8. Simplified 10.3%

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

9. Taylor expanded in r around 0

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f32 N/A

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

    4. lower-/.f32 9.3

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

11. Simplified 9.3%

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

12. Final simplification 9.3%

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

Alternative 19: 9.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 10.3

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

8. Simplified 10.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(r, \frac{-0.3333333333333333}{s}, 1\right)}}{s \cdot \left(6 \cdot \left(\pi \cdot r\right)\right)} \]
9. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. distribute-rgt-in N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. associate-*r* N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 10.3

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

10. Applied egg-rr 10.3%

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

11. Taylor expanded in r around 0

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f32 N/A

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

    4. lower-/.f32 9.3

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

13. Simplified 9.3%

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

14. Final simplification 9.3%

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

Alternative 20: 8.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 10.3

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

8. Simplified 10.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(r, \frac{-0.3333333333333333}{s}, 1\right)}}{s \cdot \left(6 \cdot \left(\pi \cdot r\right)\right)} \]
9. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. distribute-rgt-in N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. associate-*r* N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 10.3

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

10. Applied egg-rr 10.3%

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

11. Taylor expanded in r around 0

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

13. Simplified 8.2%

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

14. Final simplification 8.2%

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

Alternative 21: 6.9% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Add Preprocessing

4. Applied egg-rr 91.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(s \cdot r\right) \cdot 2}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(s \cdot r\right) \cdot 2} + \frac{\frac{3}{4} \cdot \color{blue}{1}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Simplified 7.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot 2} + \frac{0.75 \cdot \color{blue}{1}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r \cdot s}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
9. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r \cdot s}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{\color{blue}{s \cdot r}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-*.f32 7.4

      \[\leadsto \frac{0.125}{\color{blue}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

10. Simplified 7.4%

    \[\leadsto \color{blue}{\frac{0.125}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

12. Applied egg-rr 7.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, r, \frac{0.125 \cdot \left(s \cdot r\right)}{\pi \cdot s}\right)}{\left(s \cdot r\right) \cdot r}} \]

13. Final simplification 7.4%

    \[\leadsto \frac{\mathsf{fma}\left(0.125, r, \frac{\left(r \cdot s\right) \cdot 0.125}{s \cdot \pi}\right)}{r \cdot \left(r \cdot s\right)} \]
14. Add Preprocessing

Alternative 22: 6.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{r \cdot s} + \frac{\frac{0.16666666666666666}{\pi}}{s \cdot \left(r \cdot 1.3333333333333333\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ 0.125 (* r s))
  (/ (/ 0.16666666666666666 PI) (* s (* r 1.3333333333333333)))))

C

float code(float s, float r) {
	return (0.125f / (r * s)) + ((0.16666666666666666f / ((float) M_PI)) / (s * (r * 1.3333333333333333f)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(r * s)) + Float32(Float32(Float32(0.16666666666666666) / Float32(pi)) / Float32(s * Float32(r * Float32(1.3333333333333333)))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (r * s)) + ((single(0.16666666666666666) / single(pi)) / (s * (r * single(1.3333333333333333))));
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Add Preprocessing

4. Applied egg-rr 91.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(s \cdot r\right) \cdot 2}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(s \cdot r\right) \cdot 2} + \frac{\frac{3}{4} \cdot \color{blue}{1}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Simplified 7.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot 2} + \frac{0.75 \cdot \color{blue}{1}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r \cdot s}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
9. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r \cdot s}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{\color{blue}{s \cdot r}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-*.f32 7.4

      \[\leadsto \frac{0.125}{\color{blue}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

10. Simplified 7.4%

    \[\leadsto \color{blue}{\frac{0.125}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

12. Applied egg-rr 7.4%

    \[\leadsto \frac{0.125}{s \cdot r} + \color{blue}{\frac{\frac{0.16666666666666666}{\pi}}{s \cdot \left(r \cdot 1.3333333333333333\right)}} \]

13. Final simplification 7.4%

    \[\leadsto \frac{0.125}{r \cdot s} + \frac{\frac{0.16666666666666666}{\pi}}{s \cdot \left(r \cdot 1.3333333333333333\right)} \]
14. Add Preprocessing

Alternative 23: 6.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{r \cdot s} + \frac{0.75}{\pi \cdot \left(6 \cdot \left(r \cdot s\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+ (/ 0.125 (* r s)) (/ 0.75 (* PI (* 6.0 (* r s))))))

C

float code(float s, float r) {
	return (0.125f / (r * s)) + (0.75f / (((float) M_PI) * (6.0f * (r * s))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(r * s)) + Float32(Float32(0.75) / Float32(Float32(pi) * Float32(Float32(6.0) * Float32(r * s)))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (r * s)) + (single(0.75) / (single(pi) * (single(6.0) * (r * s))));
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Add Preprocessing

4. Applied egg-rr 91.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(s \cdot r\right) \cdot 2}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(s \cdot r\right) \cdot 2} + \frac{\frac{3}{4} \cdot \color{blue}{1}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Simplified 7.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot 2} + \frac{0.75 \cdot \color{blue}{1}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r \cdot s}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
9. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r \cdot s}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{\color{blue}{s \cdot r}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-*.f32 7.4

      \[\leadsto \frac{0.125}{\color{blue}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

10. Simplified 7.4%

    \[\leadsto \color{blue}{\frac{0.125}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
11. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\color{blue}{\frac{3}{4} \cdot 1}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

    2. lift-PI.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot s\right) \cdot r} \]

    3. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{\color{blue}{\left(6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)\right)} \cdot r} \]

    4. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{\color{blue}{6 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\right)}} \]

    5. *-commutative N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{6 \cdot \left(\color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot r\right)} \]

    6. associate-*r* N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{6 \cdot \color{blue}{\left(s \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)\right)}} \]

    7. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{6 \cdot \left(s \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot r\right)}\right)} \]

    8. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{6 \cdot \color{blue}{\left(s \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)\right)}} \]

    9. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{6 \cdot \color{blue}{\left(s \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)\right)}} \]

    10. lift-*.f32 N/A

        \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{6 \cdot \left(s \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot r\right)}\right)} \]

    11. associate-*r* N/A

        \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{6 \cdot \color{blue}{\left(\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r\right)}} \]

    12. *-commutative N/A

        \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{6 \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot s\right)} \cdot r\right)} \]

    13. associate-*l* N/A

        \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{\color{blue}{\left(6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)\right) \cdot r}} \]

    14. associate-*l* N/A

        \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{\color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)} \cdot r} \]

    15. lift-*.f32 N/A

        \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{\left(\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right)} \cdot s\right) \cdot r} \]

    16. lift-*.f32 N/A

        \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{\color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)} \cdot r} \]

    17. lift-*.f32 N/A

        \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{3}{4} \cdot 1}{\color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

12. Applied egg-rr 7.4%

    \[\leadsto \frac{0.125}{s \cdot r} + \color{blue}{\frac{0.75}{\pi \cdot \left(6 \cdot \left(s \cdot r\right)\right)}} \]

13. Final simplification 7.4%

    \[\leadsto \frac{0.125}{r \cdot s} + \frac{0.75}{\pi \cdot \left(6 \cdot \left(r \cdot s\right)\right)} \]
14. Add Preprocessing

Alternative 24: 6.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{r \cdot s} + \frac{0.125}{s \cdot \left(r \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+ (/ 0.125 (* r s)) (/ 0.125 (* s (* r PI)))))

C

float code(float s, float r) {
	return (0.125f / (r * s)) + (0.125f / (s * (r * ((float) M_PI))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(r * s)) + Float32(Float32(0.125) / Float32(s * Float32(r * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (r * s)) + (single(0.125) / (s * (r * single(pi))));
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Add Preprocessing

4. Applied egg-rr 91.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(s \cdot r\right) \cdot 2}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(s \cdot r\right) \cdot 2} + \frac{\frac{3}{4} \cdot \color{blue}{1}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Simplified 7.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot 2} + \frac{0.75 \cdot \color{blue}{1}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r \cdot s}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
9. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r \cdot s}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{\color{blue}{s \cdot r}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-*.f32 7.4

      \[\leadsto \frac{0.125}{\color{blue}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

10. Simplified 7.4%

    \[\leadsto \color{blue}{\frac{0.125}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

11. Taylor expanded in s around 0

    \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \color{blue}{\frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
12. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \color{blue}{\frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{1}{8}}{\color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r}} \]

    3. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{1}{8}}{\color{blue}{s \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)}} \]

    4. *-commutative N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{1}{8}}{s \cdot \color{blue}{\left(r \cdot \mathsf{PI}\left(\right)\right)}} \]

    5. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{1}{8}}{\color{blue}{s \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)}} \]

    6. *-commutative N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{1}{8}}{s \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot r\right)}} \]

    7. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot r} + \frac{\frac{1}{8}}{s \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot r\right)}} \]

    8. lower-PI.f32 7.4

       \[\leadsto \frac{0.125}{s \cdot r} + \frac{0.125}{s \cdot \left(\color{blue}{\pi} \cdot r\right)} \]

13. Simplified 7.4%

    \[\leadsto \frac{0.125}{s \cdot r} + \color{blue}{\frac{0.125}{s \cdot \left(\pi \cdot r\right)}} \]

14. Final simplification 7.4%

    \[\leadsto \frac{0.125}{r \cdot s} + \frac{0.125}{s \cdot \left(r \cdot \pi\right)} \]
15. Add Preprocessing

Alternative 25: 6.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 + \frac{0.125}{\pi}}{r \cdot s} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ (+ 0.125 (/ 0.125 PI)) (* r s)))

C

float code(float s, float r) {
	return (0.125f + (0.125f / ((float) M_PI))) / (r * s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) + Float32(Float32(0.125) / Float32(pi))) / Float32(r * s))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) + (single(0.125) / single(pi))) / (r * s);
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Add Preprocessing

4. Applied egg-rr 91.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(s \cdot r\right) \cdot 2}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(s \cdot r\right) \cdot 2} + \frac{\frac{3}{4} \cdot \color{blue}{1}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Simplified 7.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot 2} + \frac{0.75 \cdot \color{blue}{1}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r \cdot s}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
9. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r \cdot s}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{\color{blue}{s \cdot r}} + \frac{\frac{3}{4} \cdot 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-*.f32 7.4

      \[\leadsto \frac{0.125}{\color{blue}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

10. Simplified 7.4%

    \[\leadsto \color{blue}{\frac{0.125}{s \cdot r}} + \frac{0.75 \cdot 1}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

11. Taylor expanded in s around 0

    \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{1}{r} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
12. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r} + \frac{1}{8} \cdot \color{blue}{\frac{\frac{1}{r}}{\mathsf{PI}\left(\right)}}}{s} \]

    2. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r} + \color{blue}{\frac{\frac{1}{8} \cdot \frac{1}{r}}{\mathsf{PI}\left(\right)}}}{s} \]

    3. associate-*l/ N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r} + \color{blue}{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)} \cdot \frac{1}{r}}}{s} \]

    4. distribute-rgt-in N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{r} \cdot \left(\frac{1}{8} + \frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}\right)}}{s} \]

    5. associate-*l/ N/A

       \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(\frac{1}{8} + \frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}\right)}{r}}}{s} \]

    6. *-lft-identity N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{8} + \frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}}{r}}{s} \]

    7. associate-/l/ N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{8} + \frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{s \cdot r}} \]

    8. *-commutative N/A

       \[\leadsto \frac{\frac{1}{8} + \frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{\color{blue}{r \cdot s}} \]

    9. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{8} + \frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{r \cdot s}} \]

    10. lower-+.f32 N/A

        \[\leadsto \frac{\color{blue}{\frac{1}{8} + \frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}}{r \cdot s} \]

    11. lower-/.f32 N/A

        \[\leadsto \frac{\frac{1}{8} + \color{blue}{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}}{r \cdot s} \]

    12. lower-PI.f32 N/A

        \[\leadsto \frac{\frac{1}{8} + \frac{\frac{1}{8}}{\color{blue}{\mathsf{PI}\left(\right)}}}{r \cdot s} \]

    13. *-commutative N/A

        \[\leadsto \frac{\frac{1}{8} + \frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{\color{blue}{s \cdot r}} \]

    14. lower-*.f32 7.4

        \[\leadsto \frac{0.125 + \frac{0.125}{\pi}}{\color{blue}{s \cdot r}} \]

13. Simplified 7.4%

    \[\leadsto \color{blue}{\frac{0.125 + \frac{0.125}{\pi}}{s \cdot r}} \]

14. Final simplification 7.4%

    \[\leadsto \frac{0.125 + \frac{0.125}{\pi}}{r \cdot s} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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))))