Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 7.5s

Alternatives: 14

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 4: 18.8% accurate, 1.3× speedup

Math

\[\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\left(\pi \cdot s\right) \cdot r}, 0.125, \frac{0.125}{\left(\pi \cdot s\right) \cdot \left(\left(1 + \frac{r}{s}\right) \cdot r\right)}\right) \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in s around inf

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\left(\pi \cdot s\right) \cdot r}, 0.125, \frac{0.125}{\left(\pi \cdot s\right) \cdot \left(\color{blue}{\left(1 + \frac{r}{s}\right)} \cdot r\right)}\right) \]
7. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lower-/.f32 18.8%

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

8. Applied rewrites 18.8%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\left(\pi \cdot s\right) \cdot r}, 0.125, \frac{0.125}{\left(\pi \cdot s\right) \cdot \left(\color{blue}{\left(1 + \frac{r}{s}\right)} \cdot r\right)}\right) \]
9. Add Preprocessing

Alternative 5: 12.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in r around 0

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\left(\pi \cdot s\right) \cdot r}, 0.125, \frac{0.125}{\color{blue}{r \cdot \left(r \cdot \pi + s \cdot \pi\right)}}\right) \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 12.4%

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

8. Applied rewrites 12.4%

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

Alternative 6: 9.4% accurate, 1.7× speedup

Math

\[\mathsf{fma}\left(\frac{0.053051647563049226}{r}, \frac{0.75}{s}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.053051647563049226 r)
  (/ 0.75 s)
  (/ (/ 0.125 (* (* PI s) (exp (/ r s)))) r)))

C

float code(float s, float r) {
	return fmaf((0.053051647563049226f / r), (0.75f / s), ((0.125f / ((((float) M_PI) * s) * expf((r / s)))) / r));
}

Julia

function code(s, r)
	return fma(Float32(Float32(0.053051647563049226) / r), Float32(Float32(0.75) / s), Float32(Float32(Float32(0.125) / Float32(Float32(Float32(pi) * s) * exp(Float32(r / s)))) / r))
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

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

4. Evaluated real constant 99.4%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{r \cdot \color{blue}{18.84955596923828}}, \frac{0.75}{s}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]

5. Taylor expanded in s around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{131072}{2470649}}{r}}, \frac{0.75}{s}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]
6. Step-by-step derivation

   1. lower-/.f32 9.4%

      \[\leadsto \mathsf{fma}\left(\frac{0.053051647563049226}{\color{blue}{r}}, \frac{0.75}{s}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]

7. Applied rewrites 9.4%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.053051647563049226}{r}}, \frac{0.75}{s}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]
8. Add Preprocessing

Alternative 7: 9.4% accurate, 1.7× speedup

Math

\[\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \frac{r}{s}, 0.75\right), \frac{0.16666666666666666}{\pi \cdot s}, \frac{0.125}{\left(\left(1 + \frac{r}{s}\right) \cdot \pi\right) \cdot s}\right)}{r} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   (fma -0.25 (/ r s) 0.75)
   (/ 0.16666666666666666 (* PI s))
   (/ 0.125 (* (* (+ 1.0 (/ r s)) PI) s)))
  r))

C

float code(float s, float r) {
	return fmaf(fmaf(-0.25f, (r / s), 0.75f), (0.16666666666666666f / (((float) M_PI) * s)), (0.125f / (((1.0f + (r / s)) * ((float) M_PI)) * s))) / r;
}

Julia

function code(s, r)
	return Float32(fma(fma(Float32(-0.25), Float32(r / s), Float32(0.75)), Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * s)), Float32(Float32(0.125) / Float32(Float32(Float32(Float32(1.0) + Float32(r / s)) * Float32(pi)) * s))) / r)
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + \frac{-1}{4} \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 9.6%

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

4. Applied rewrites 9.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

6. Applied rewrites 9.6%

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

8. Applied rewrites 9.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \frac{r}{s}, 0.75\right), \frac{0.16666666666666666}{\pi \cdot s}, \frac{0.125}{\left(e^{\frac{r}{s}} \cdot \pi\right) \cdot s}\right)}{r}} \]

9. Taylor expanded in s around inf

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \frac{r}{s}, 0.75\right), \frac{0.16666666666666666}{\pi \cdot s}, \frac{0.125}{\left(\color{blue}{\left(1 + \frac{r}{s}\right)} \cdot \pi\right) \cdot s}\right)}{r} \]
10. Step-by-step derivation

    1. lower-+.f32 N/A

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

    2. lower-/.f32 9.4%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \frac{r}{s}, 0.75\right), \frac{0.16666666666666666}{\pi \cdot s}, \frac{0.125}{\left(\left(1 + \frac{r}{\color{blue}{s}}\right) \cdot \pi\right) \cdot s}\right)}{r} \]

11. Applied rewrites 9.4%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \frac{r}{s}, 0.75\right), \frac{0.16666666666666666}{\pi \cdot s}, \frac{0.125}{\left(\color{blue}{\left(1 + \frac{r}{s}\right)} \cdot \pi\right) \cdot s}\right)}{r} \]
12. Add Preprocessing

Alternative 8: 9.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 98.1%

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

4. Taylor expanded in r around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-PI.f32 9.0%

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

6. Applied rewrites 9.0%

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

Alternative 9: 9.0% accurate, 2.6× speedup

Math

\[\frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * (single(1.0) / (r * single(pi)))) - (single(0.16666666666666666) * (single(1.0) / (s * single(pi))))) / s;
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \pi} - \frac{1}{6} \cdot \frac{1}{s \cdot \pi}}{s}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.0%

       \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s} \]

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s}} \]
5. Add Preprocessing

Alternative 10: 9.0% accurate, 4.5× speedup

Math

\[\frac{\frac{0.25}{s} \cdot \frac{1}{\pi}}{r} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0%

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

4. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s \cdot r}}{\color{blue}{\pi}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s \cdot r}}{\color{blue}{\pi}} \]

   9. lower-/.f32 9.0%

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s \cdot r}}{\color{blue}{\pi}} \]

   2. mult-flip N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot r} \cdot \color{blue}{\frac{1}{\pi}} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot r} \cdot \frac{\color{blue}{1}}{\pi} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot r} \cdot \frac{1}{\pi} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s}}{r} \cdot \frac{\color{blue}{1}}{\pi} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s} \cdot \frac{1}{\pi}}{\color{blue}{r}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s} \cdot \frac{1}{\pi}}{\color{blue}{r}} \]

   8. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s} \cdot \frac{1}{\pi}}{r} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s} \cdot \frac{1}{\pi}}{r} \]

   10. lower-/.f32 9.0%

       \[\leadsto \frac{\frac{0.25}{s} \cdot \frac{1}{\pi}}{r} \]

8. Applied rewrites 9.0%

   \[\leadsto \frac{\frac{0.25}{s} \cdot \frac{1}{\pi}}{\color{blue}{r}} \]
9. Add Preprocessing

Alternative 11: 9.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0%

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

4. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi \cdot s}}{\color{blue}{r}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi \cdot s}}{\color{blue}{r}} \]

   9. lower-/.f32 9.0%

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi \cdot s}}{r} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi \cdot s}}{r} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s \cdot \pi}}{r} \]

   4. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{\frac{1}{4}}{s}}{\pi}}{r} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{\frac{1}{4}}{s}}{\pi}}{r} \]

   6. lower-/.f32 9.0%

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

8. Applied rewrites 9.0%

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

Alternative 12: 9.0% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0%

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

4. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l* N/A

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

   14. lift-*.f32 N/A

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

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

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

   16. lower-*.f32 N/A

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

   17. lower-neg.f32 9.0%

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

6. Applied rewrites 9.0%

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

Alternative 13: 9.0% accurate, 6.4× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0%

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

4. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi \cdot s}}{\color{blue}{r}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi \cdot s}}{\color{blue}{r}} \]

   9. lower-/.f32 9.0%

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi \cdot s}}{\color{blue}{r}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi \cdot s}}{r} \]

   3. associate-/l/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lower-/.f32 9.0%

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

8. Applied rewrites 9.0%

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

Alternative 14: 9.0% accurate, 6.4× speedup

Math

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

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 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0%

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

4. Applied rewrites 9.0%

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

Reproduce

herbie shell --seed 2025179 
(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))))