Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 15.9s

Alternatives: 21

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

7. Applied rewrites 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.3%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around 0

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 99.3%

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

Alternative 4: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(r \cdot 0.125\right) \cdot \left(\frac{1}{s \cdot \pi} \cdot \left(e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}\right)\right)}{r \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   (* r 0.125)
   (* (/ 1.0 (* s PI)) (+ (exp (/ r (- s))) (exp (/ r (* s -3.0))))))
  (* r r)))

C

float code(float s, float r) {
	return ((r * 0.125f) * ((1.0f / (s * ((float) M_PI))) * (expf((r / -s)) + expf((r / (s * -3.0f)))))) / (r * r);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(r * Float32(0.125)) * Float32(Float32(Float32(1.0) / Float32(s * Float32(pi))) * Float32(exp(Float32(r / Float32(-s))) + exp(Float32(r / Float32(s * Float32(-3.0))))))) / Float32(r * r))
end

MATLAB

function tmp = code(s, r)
	tmp = ((r * single(0.125)) * ((single(1.0) / (s * single(pi))) * (exp((r / -s)) + exp((r / (s * single(-3.0))))))) / (r * r);
end

Derivation

1. Initial program 99.3%

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

7. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{\left(0.125 \cdot r\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{s \cdot \pi} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{s \cdot \pi}\right)}}{r \cdot r} \]
8. Step-by-step derivation

9. Applied rewrites 99.3%

   \[\leadsto \frac{\left(\frac{1}{s \cdot \pi} \cdot \left(e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}\right)\right) \cdot \color{blue}{\left(0.125 \cdot r\right)}}{r \cdot r} \]

10. Final simplification 99.3%

    \[\leadsto \frac{\left(r \cdot 0.125\right) \cdot \left(\frac{1}{s \cdot \pi} \cdot \left(e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}\right)\right)}{r \cdot r} \]
11. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / (s * single(-3.0)))) / r) + (exp((r / -s)) / r));
end

Derivation

1. Initial program 99.3%

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

4. Applied rewrites 99.3%

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

Alternative 6: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 98.1%

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

Alternative 7: 10.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

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

6. Final simplification 10.8%

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

Alternative 8: 10.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 10.8%

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

9. Final simplification 10.8%

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

Alternative 9: 10.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

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

6. Final simplification 10.8%

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

Alternative 10: 10.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.125, e^{\frac{r}{-s}} \cdot \frac{1}{s \cdot \left(r \cdot \pi\right)}, \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot \pi\right)}, 0.006944444444444444, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.125
  (* (exp (/ r (- s))) (/ 1.0 (* s (* r PI))))
  (/
   (+
    (/ 0.125 (* r PI))
    (fma
     (/ r (* s (* s PI)))
     0.006944444444444444
     (/ -0.041666666666666664 (* s PI))))
   s)))

C

float code(float s, float r) {
	return fmaf(0.125f, (expf((r / -s)) * (1.0f / (s * (r * ((float) M_PI))))), (((0.125f / (r * ((float) M_PI))) + fmaf((r / (s * (s * ((float) M_PI)))), 0.006944444444444444f, (-0.041666666666666664f / (s * ((float) M_PI))))) / s));
}

Julia

function code(s, r)
	return fma(Float32(0.125), Float32(exp(Float32(r / Float32(-s))) * Float32(Float32(1.0) / Float32(s * Float32(r * Float32(pi))))), Float32(Float32(Float32(Float32(0.125) / Float32(r * Float32(pi))) + fma(Float32(r / Float32(s * Float32(s * Float32(pi)))), Float32(0.006944444444444444), Float32(Float32(-0.041666666666666664) / Float32(s * Float32(pi))))) / s))
end

Derivation

1. Initial program 99.3%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 10.7%

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

7. Applied rewrites 10.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{1}{s \cdot \left(r \cdot \pi\right)} \cdot e^{\frac{r}{-s}}, \frac{\frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{\pi}, r \cdot \left(\frac{r}{s \cdot \pi} \cdot -0.0007716049382716049\right)\right)}{s \cdot s} + \left(\frac{-0.041666666666666664}{s \cdot \pi} + \frac{0.125}{r \cdot \pi}\right)}{s}\right)} \]

8. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-frac-neg N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-out N/A

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

   6. remove-double-neg N/A

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

   7. lower-/.f32 N/A

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

10. Applied rewrites 10.8%

    \[\leadsto \mathsf{fma}\left(0.125, \frac{1}{s \cdot \left(r \cdot \pi\right)} \cdot e^{\frac{r}{-s}}, \color{blue}{\frac{\mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot \pi\right)}, 0.006944444444444444, \frac{-0.041666666666666664}{s \cdot \pi}\right) + \frac{0.125}{r \cdot \pi}}{s}}\right) \]

11. Final simplification 10.8%

    \[\leadsto \mathsf{fma}\left(0.125, e^{\frac{r}{-s}} \cdot \frac{1}{s \cdot \left(r \cdot \pi\right)}, \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot \pi\right)}, 0.006944444444444444, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}\right) \]
12. Add Preprocessing

Alternative 11: 10.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 10.7%

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

7. Applied rewrites 10.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{1}{s \cdot \left(r \cdot \pi\right)} \cdot e^{\frac{r}{-s}}, \frac{\frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{\pi}, r \cdot \left(\frac{r}{s \cdot \pi} \cdot -0.0007716049382716049\right)\right)}{s \cdot s} + \left(\frac{-0.041666666666666664}{s \cdot \pi} + \frac{0.125}{r \cdot \pi}\right)}{s}\right)} \]

9. Applied rewrites 10.6%

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

10. Taylor expanded in s around -inf

    \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} + \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}\right) \]
11. Step-by-step derivation

12. Applied rewrites 10.8%

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

Alternative 12: 10.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 10.7%

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

7. Applied rewrites 10.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{1}{s \cdot \left(r \cdot \pi\right)} \cdot e^{\frac{r}{-s}}, \frac{\frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{\pi}, r \cdot \left(\frac{r}{s \cdot \pi} \cdot -0.0007716049382716049\right)\right)}{s \cdot s} + \left(\frac{-0.041666666666666664}{s \cdot \pi} + \frac{0.125}{r \cdot \pi}\right)}{s}\right)} \]

9. Applied rewrites 10.6%

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

10. Taylor expanded in s around inf

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

    1. lower-/.f32 N/A

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

12. Applied rewrites 10.8%

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

Alternative 13: 10.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in r around 0

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

5. Applied rewrites 10.1%

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

Alternative 14: 10.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 10.1%

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

Alternative 15: 9.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \frac{0.25}{r \cdot s} \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sqrt PI)))) (* (/ 0.25 (* r s)) (* t_0 t_0))))

C

float code(float s, float r) {
	float t_0 = 1.0f / sqrtf(((float) M_PI));
	return (0.25f / (r * s)) * (t_0 * t_0);
}

Julia

function code(s, r)
	t_0 = Float32(Float32(1.0) / sqrt(Float32(pi)))
	return Float32(Float32(Float32(0.25) / Float32(r * s)) * Float32(t_0 * t_0))
end

MATLAB

function tmp = code(s, r)
	t_0 = single(1.0) / sqrt(single(pi));
	tmp = (single(0.25) / (r * s)) * (t_0 * t_0);
end

Derivation

1. Initial program 99.3%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.9

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

5. Applied rewrites 8.9%

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

7. Applied rewrites 8.9%

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

9. Applied rewrites 9.0%

   \[\leadsto \frac{0.25}{r \cdot s} \cdot \color{blue}{\left(\frac{1}{\sqrt{\pi}} \cdot \frac{1}{\sqrt{\pi}}\right)} \]
10. Add Preprocessing

Alternative 16: 9.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\pi}} \cdot \frac{0.25}{\left(r \cdot s\right) \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (/ 1.0 (sqrt PI)) (/ 0.25 (* (* r s) (sqrt PI)))))

C

float code(float s, float r) {
	return (1.0f / sqrtf(((float) M_PI))) * (0.25f / ((r * s) * sqrtf(((float) M_PI))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(Float32(0.25) / Float32(Float32(r * s) * sqrt(Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / sqrt(single(pi))) * (single(0.25) / ((r * s) * sqrt(single(pi))));
end

Derivation

1. Initial program 99.3%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.9

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

5. Applied rewrites 8.9%

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

7. Applied rewrites 8.9%

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

9. Applied rewrites 9.0%

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\frac{0.25}{\left(r \cdot s\right) \cdot \sqrt{\pi}}} \]
10. Add Preprocessing

Alternative 17: 9.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.9

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

5. Applied rewrites 8.9%

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

7. Applied rewrites 8.9%

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

9. Applied rewrites 8.9%

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

10. Final simplification 8.9%

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

Alternative 18: 9.2% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.9

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

5. Applied rewrites 8.9%

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

7. Applied rewrites 8.9%

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

Alternative 19: 9.2% accurate, 13.5× speedup

Math

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

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(s * Float32(r * Float32(pi))))
end

MATLAB

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

Derivation

1. Initial program 99.3%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.9

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

5. Applied rewrites 8.9%

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

7. Applied rewrites 8.9%

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

8. Final simplification 8.9%

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

Alternative 20: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.9

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

5. Applied rewrites 8.9%

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

7. Applied rewrites 8.9%

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

8. Final simplification 8.9%

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

Alternative 21: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.9

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

5. Applied rewrites 8.9%

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

Reproduce

herbie shell --seed 2024228 
(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))))