Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 13.9s

Alternatives: 19

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f32 N/A

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

   7. remove-double-neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp(((single(-0.3333333333333333) * r) / s)) + exp((-r / s))) * single(0.125)) / ((single(pi) * s) * r);
end

Derivation

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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f32 N/A

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

   7. remove-double-neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / ((single(pi) * s) * r)) * (exp(((single(-0.3333333333333333) * r) / s)) + exp((-r / s)));
end

Derivation

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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f32 N/A

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

   7. remove-double-neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.3%

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

Alternative 4: 11.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi \cdot r} + \frac{\frac{-0.041666666666666664}{\pi} - \frac{\mathsf{fma}\left(0.0007716049382716049, \frac{r}{\pi \cdot s} \cdot r, \frac{r}{\pi} \cdot -0.006944444444444444\right)}{s}}{s}}{s} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/
   (+
    (/ 0.125 (* PI r))
    (/
     (-
      (/ -0.041666666666666664 PI)
      (/
       (fma
        0.0007716049382716049
        (* (/ r (* PI s)) r)
        (* (/ r PI) -0.006944444444444444))
       s))
     s))
   s)
  (/ (* (exp (/ (- r) s)) 0.25) (* (* (* PI 2.0) s) r))))

C

float code(float s, float r) {
	return (((0.125f / (((float) M_PI) * r)) + (((-0.041666666666666664f / ((float) M_PI)) - (fmaf(0.0007716049382716049f, ((r / (((float) M_PI) * s)) * r), ((r / ((float) M_PI)) * -0.006944444444444444f)) / s)) / s)) / s) + ((expf((-r / s)) * 0.25f) / (((((float) M_PI) * 2.0f) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(0.125) / Float32(Float32(pi) * r)) + Float32(Float32(Float32(Float32(-0.041666666666666664) / Float32(pi)) - Float32(fma(Float32(0.0007716049382716049), Float32(Float32(r / Float32(Float32(pi) * s)) * r), Float32(Float32(r / Float32(pi)) * Float32(-0.006944444444444444))) / s)) / s)) / s) + Float32(Float32(exp(Float32(Float32(-r) / s)) * Float32(0.25)) / Float32(Float32(Float32(Float32(pi) * Float32(2.0)) * s) * r)))
end

Derivation

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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f32 N/A

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

   7. remove-double-neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{1296} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

7. Applied rewrites 10.1%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\frac{-0.041666666666666664}{\pi} - \frac{\mathsf{fma}\left(0.0007716049382716049, r \cdot \frac{r}{\pi \cdot s}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s}}{s} + \frac{0.125}{\pi \cdot r}}{s}} \]

8. Final simplification 10.1%

   \[\leadsto \frac{\frac{0.125}{\pi \cdot r} + \frac{\frac{-0.041666666666666664}{\pi} - \frac{\mathsf{fma}\left(0.0007716049382716049, \frac{r}{\pi \cdot s} \cdot r, \frac{r}{\pi} \cdot -0.006944444444444444\right)}{s}}{s}}{s} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r} \]
9. Add Preprocessing

Alternative 5: 11.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\frac{-0.041666666666666664}{\pi} - \frac{\mathsf{fma}\left(0.0007716049382716049, \frac{r}{\pi \cdot s} \cdot r, \frac{r}{\pi} \cdot -0.006944444444444444\right)}{s}}{s \cdot s} - \frac{-0.125}{\left(\pi \cdot s\right) \cdot r}\right) + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (-
   (/
    (-
     (/ -0.041666666666666664 PI)
     (/
      (fma
       0.0007716049382716049
       (* (/ r (* PI s)) r)
       (* (/ r PI) -0.006944444444444444))
      s))
    (* s s))
   (/ -0.125 (* (* PI s) r)))
  (/ (* (exp (/ (- r) s)) 0.25) (* (* (* PI 2.0) s) r))))

C

float code(float s, float r) {
	return ((((-0.041666666666666664f / ((float) M_PI)) - (fmaf(0.0007716049382716049f, ((r / (((float) M_PI) * s)) * r), ((r / ((float) M_PI)) * -0.006944444444444444f)) / s)) / (s * s)) - (-0.125f / ((((float) M_PI) * s) * r))) + ((expf((-r / s)) * 0.25f) / (((((float) M_PI) * 2.0f) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(-0.041666666666666664) / Float32(pi)) - Float32(fma(Float32(0.0007716049382716049), Float32(Float32(r / Float32(Float32(pi) * s)) * r), Float32(Float32(r / Float32(pi)) * Float32(-0.006944444444444444))) / s)) / Float32(s * s)) - Float32(Float32(-0.125) / Float32(Float32(Float32(pi) * s) * r))) + Float32(Float32(exp(Float32(Float32(-r) / s)) * Float32(0.25)) / Float32(Float32(Float32(Float32(pi) * Float32(2.0)) * s) * r)))
end

Derivation

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

3. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{1296} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 10.1%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{-0.041666666666666664}{\pi} - \frac{\mathsf{fma}\left(0.0007716049382716049, \frac{r}{\pi \cdot s} \cdot r, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s}}{s \cdot s} - \frac{-0.125}{\left(\pi \cdot s\right) \cdot r}\right)} \]

6. Final simplification 10.1%

   \[\leadsto \left(\frac{\frac{-0.041666666666666664}{\pi} - \frac{\mathsf{fma}\left(0.0007716049382716049, \frac{r}{\pi \cdot s} \cdot r, \frac{r}{\pi} \cdot -0.006944444444444444\right)}{s}}{s \cdot s} - \frac{-0.125}{\left(\pi \cdot s\right) \cdot r}\right) + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r} \]
7. Add Preprocessing

Alternative 6: 11.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((fmaf((r / ((s * s) * ((float) M_PI))), 0.006944444444444444f, (-0.041666666666666664f / (((float) M_PI) * s))) - (-0.125f / (((float) M_PI) * r))) / s) + ((expf((-r / s)) * 0.25f) / (((((float) M_PI) * 2.0f) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(fma(Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(0.006944444444444444), Float32(Float32(-0.041666666666666664) / Float32(Float32(pi) * s))) - Float32(Float32(-0.125) / Float32(Float32(pi) * r))) / s) + Float32(Float32(exp(Float32(Float32(-r) / s)) * Float32(0.25)) / Float32(Float32(Float32(Float32(pi) * Float32(2.0)) * s) * r)))
end

Derivation

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 10.0%

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

6. Final simplification 10.0%

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

Alternative 7: 11.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((fmaf((r / ((s * s) * ((float) M_PI))), 0.006944444444444444f, (0.125f / (((float) M_PI) * r))) - (0.041666666666666664f / (((float) M_PI) * s))) / s) + ((expf((-r / s)) / ((((float) M_PI) * s) * r)) * 0.125f);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(fma(Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(0.006944444444444444), Float32(Float32(0.125) / Float32(Float32(pi) * r))) - Float32(Float32(0.041666666666666664) / Float32(Float32(pi) * s))) / s) + Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(pi) * s) * r)) * Float32(0.125)))
end

Derivation

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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. distribute-frac-neg N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 10.0%

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

11. Final simplification 10.0%

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

Alternative 8: 11.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return fmaf((0.125f / ((((float) M_PI) * s) * r)), expf((-r / s)), ((fmaf((r / ((s * s) * ((float) M_PI))), 0.006944444444444444f, (0.125f / (((float) M_PI) * r))) - (0.041666666666666664f / (((float) M_PI) * s))) / s));
}

Julia

function code(s, r)
	return fma(Float32(Float32(0.125) / Float32(Float32(Float32(pi) * s) * r)), exp(Float32(Float32(-r) / s)), Float32(Float32(fma(Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(0.006944444444444444), Float32(Float32(0.125) / Float32(Float32(pi) * r))) - Float32(Float32(0.041666666666666664) / Float32(Float32(pi) * s))) / s))
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

5. Applied rewrites 10.0%

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

   1. lift-+.f32 N/A

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

7. Applied rewrites 10.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{\left(\pi \cdot s\right) \cdot r}, e^{\frac{-r}{s}}, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.006944444444444444, \frac{r}{\pi \cdot s}, \frac{-0.041666666666666664}{\pi}\right)}{s \cdot s}, r, \frac{0.125}{\pi \cdot s}\right)}{r}\right)} \]

8. Taylor expanded in s around inf

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

10. Applied rewrites 10.0%

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

Alternative 9: 10.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f32 N/A

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

   7. remove-double-neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 9.5%

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

9. Applied rewrites 9.5%

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

Alternative 10: 10.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f32 N/A

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

   7. remove-double-neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 9.5%

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

9. Applied rewrites 9.5%

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

Alternative 11: 10.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f32 N/A

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

   7. remove-double-neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 9.5%

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

9. Applied rewrites 9.5%

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

Alternative 12: 10.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f32 N/A

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

   7. remove-double-neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 9.5%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 9.5%

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

11. Final simplification 9.5%

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

Alternative 13: 10.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f32 N/A

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

   7. remove-double-neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 9.5%

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

Alternative 14: 10.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 9.5%

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

Alternative 15: 9.8% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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

3. Taylor expanded in r around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f32 N/A

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

5. Applied rewrites 8.9%

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

Alternative 16: 9.8% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{-1}{48} \cdot \frac{{r}^{2}}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \left(\frac{-1}{1296} \cdot \frac{{r}^{2}}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right)\right)\right) - \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 7.8%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 8.8%

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

Alternative 17: 9.7% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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

3. Taylor expanded in s around inf

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. associate-/l* N/A

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

   4. associate-/l/ N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. +-commutative N/A

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

   8. remove-double-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f32 N/A

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

5. Applied rewrites 8.8%

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

Alternative 18: 9.6% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 8.8

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

5. Applied rewrites 8.8%

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

7. Applied rewrites 8.8%

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

Alternative 19: 9.6% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 8.8

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

5. Applied rewrites 8.8%

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

7. Applied rewrites 8.8%

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

Reproduce

herbie shell --seed 2024235 
(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))))