Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.6%

Time: 11.9s

Alternatives: 12

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{r}{-3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.125 (exp (/ (- r) s))) (* (* PI s) r))
  (/ (* 0.75 (exp (/ r (* -3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.125f * expf((-r / s))) / ((((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(0.125) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(pi) * s) * r)) + Float32(Float32(Float32(0.75) * exp(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(0.125) * exp((-r / s))) / ((single(pi) * s) * r)) + ((single(0.75) * exp((r / (single(-3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

3. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. frac-2neg N/A

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

   5. lift-neg.f32 N/A

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

   6. remove-double-neg N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval 99.5

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{s \cdot e^{\frac{r}{s}}}}{\pi \cdot r} + \frac{0.75 \cdot e^{\frac{r}{-3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (/ 0.125 (* s (exp (/ r s)))) (* PI r))
  (/ (* 0.75 (exp (/ r (* -3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.125f / (s * expf((r / s)))) / (((float) M_PI) * 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(0.125) / Float32(s * exp(Float32(r / s)))) / Float32(Float32(pi) * r)) + Float32(Float32(Float32(0.75) * exp(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(0.125) / (s * exp((r / s)))) / (single(pi) * r)) + ((single(0.75) * exp((r / (single(-3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

3. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. frac-2neg N/A

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

   5. lift-neg.f32 N/A

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

   6. remove-double-neg N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. times-frac N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. frac-times N/A

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

   15. *-commutative N/A

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

   16. lower-/.f32 N/A

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

7. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

Alternative 4: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp(((r / s) * single(-0.3333333333333333))) + exp((-r / s))) * single(0.125)) / ((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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around 0

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

6. Applied rewrites 99.5%

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

Alternative 5: 43.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.9

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

4. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. add-log-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)} \]

   7. log-pow-rev N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{s}\right)} \]

   8. log-pow-rev N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{s}\right)}^{r}\right)} \]

   9. pow-unpow N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(r \cdot s\right)}\right)} \]

   11. pow-unpow N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right)}^{s}\right)} \]

   12. log-pow-rev N/A

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

   13. log-pow-rev N/A

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

   14. add-log-exp N/A

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

   15. lower-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 N/A

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

   18. lift-PI.f32 8.9

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

6. Applied rewrites 8.9%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. add-log-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot \left(r \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)} \]

   5. log-pow-rev N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right)} \]

   6. lower-log.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right)} \]

   7. pow-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot \log \left(e^{\mathsf{PI}\left(\right) \cdot r}\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot \log \left(e^{r \cdot \mathsf{PI}\left(\right)}\right)} \]

   9. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot \log \left(e^{r \cdot \mathsf{PI}\left(\right)}\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{s \cdot \log \left(e^{\mathsf{PI}\left(\right) \cdot r}\right)} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{s \cdot \log \left(e^{\mathsf{PI}\left(\right) \cdot r}\right)} \]

   12. lift-PI.f32 43.2

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

8. Applied rewrites 43.2%

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

Alternative 6: 10.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / log(exp(((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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.9

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

4. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. add-log-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)} \]

   10. log-pow-rev N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

   11. lower-log.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

   12. pow-exp N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}\right)} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}\right)} \]

   14. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r}\right)} \]

   16. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   17. lower-exp.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   18. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r}\right)} \]

   19. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right)} \]

   20. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right)} \]

   21. lift-PI.f32 N/A

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

   22. lift-*.f32 10.1

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

6. Applied rewrites 10.1%

   \[\leadsto \frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]
7. Add Preprocessing

Alternative 7: 8.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Taylor expanded in s around inf

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

   1. div-add N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{-1 \cdot r}{s} + \frac{\frac{-1}{3} \cdot r}{s}\right) - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(-1 \cdot \frac{r}{s} + \frac{\frac{-1}{3} \cdot r}{s}\right) - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(-1 \cdot \frac{r}{s} + \frac{-1}{3} \cdot \frac{r}{s}\right) - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   4. distribute-rgt-out N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{r}{s} \cdot \left(-1 + \frac{-1}{3}\right)\right) - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{r}{s} \cdot \frac{-4}{3}\right) - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{-4}{3} \cdot \frac{r}{s}\right) - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{\frac{-4}{3} \cdot r}{s} - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   8. frac-2neg N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{\mathsf{neg}\left(\frac{-4}{3} \cdot r\right)}{\mathsf{neg}\left(s\right)} - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

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

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right) \cdot r}{\mathsf{neg}\left(s\right)} - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{-1}{8} \cdot \frac{\frac{4}{3} \cdot r}{\mathsf{neg}\left(s\right)} - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{-1}{8} \cdot \frac{\left(\frac{1}{3} + 1\right) \cdot r}{\mathsf{neg}\left(s\right)} - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   12. distribute-rgt1-in N/A

       \[\leadsto \frac{\frac{-1}{8} \cdot \frac{r + \frac{1}{3} \cdot r}{\mathsf{neg}\left(s\right)} - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   13. associate-/l* N/A

       \[\leadsto \frac{\frac{\frac{-1}{8} \cdot \left(r + \frac{1}{3} \cdot r\right)}{\mathsf{neg}\left(s\right)} - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   14. distribute-frac-neg2 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{\frac{-1}{8} \cdot \left(r + \frac{1}{3} \cdot r\right)}{s}\right)\right) - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   15. associate-/l* N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{r + \frac{1}{3} \cdot r}{s}\right)\right) - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   16. distribute-lft-neg-out N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right) \cdot \frac{r + \frac{1}{3} \cdot r}{s} - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

   17. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \frac{r + \frac{1}{3} \cdot r}{s} - \frac{1}{4}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

6. Applied rewrites 8.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1.3333333333333333 \cdot \frac{r}{s}, 0.125, -0.25\right)}}{\left(s \cdot r\right) \cdot \left(-\pi\right)} \]

7. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 8.9

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

9. Applied rewrites 8.9%

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

Alternative 8: 8.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-1.3333333333333333, \frac{r}{s}, 2\right) \cdot \frac{0.125}{\left(s \cdot r\right) \cdot \pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (fma -1.3333333333333333 (/ r s) 2.0) (/ 0.125 (* (* s r) PI))))

C

float code(float s, float r) {
	return fmaf(-1.3333333333333333f, (r / s), 2.0f) * (0.125f / ((s * r) * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(fma(Float32(-1.3333333333333333), Float32(r / s), Float32(2.0)) * Float32(Float32(0.125) / Float32(Float32(s * r) * Float32(pi))))
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 97.8

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\left(2 + \left(-1 \cdot \frac{r}{s} + \frac{-1}{3} \cdot \frac{r}{s}\right)\right)} \cdot \frac{\frac{1}{8}}{\left(s \cdot r\right) \cdot \pi} \]
9. Step-by-step derivation

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

      \[\leadsto \left(2 + \frac{r}{s} \cdot \frac{-4}{3}\right) \cdot \frac{\frac{1}{8}}{\left(s \cdot r\right) \cdot \pi} \]

   3. *-commutative N/A

      \[\leadsto \left(2 + \frac{-4}{3} \cdot \color{blue}{\frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{\left(s \cdot r\right) \cdot \pi} \]

   4. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right) \cdot \frac{r}{s}}\right) \cdot \frac{\frac{1}{8}}{\left(s \cdot r\right) \cdot \pi} \]

   5. metadata-eval N/A

      \[\leadsto \left(2 - \frac{4}{3} \cdot \frac{\color{blue}{r}}{s}\right) \cdot \frac{\frac{1}{8}}{\left(s \cdot r\right) \cdot \pi} \]

   6. associate-/l* N/A

      \[\leadsto \left(2 - \frac{\frac{4}{3} \cdot r}{\color{blue}{s}}\right) \cdot \frac{\frac{1}{8}}{\left(s \cdot r\right) \cdot \pi} \]

   7. metadata-eval N/A

      \[\leadsto \left(2 - \frac{\left(\frac{1}{3} + 1\right) \cdot r}{s}\right) \cdot \frac{\frac{1}{8}}{\left(s \cdot r\right) \cdot \pi} \]

   8. distribute-rgt1-in N/A

      \[\leadsto \left(2 - \frac{r + \frac{1}{3} \cdot r}{s}\right) \cdot \frac{\frac{1}{8}}{\left(s \cdot r\right) \cdot \pi} \]

   9. *-lft-identity N/A

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

   10. fp-cancel-sub-sign-inv N/A

       \[\leadsto \left(2 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{r + \frac{1}{3} \cdot r}{s}}\right) \cdot \frac{\frac{1}{8}}{\left(s \cdot r\right) \cdot \pi} \]

   11. metadata-eval N/A

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

   12. fp-cancel-sign-sub-inv N/A

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

   13. metadata-eval N/A

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

   14. *-lft-identity N/A

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

10. Applied rewrites 8.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-1.3333333333333333, \frac{r}{s}, 2\right)} \cdot \frac{0.125}{\left(s \cdot r\right) \cdot \pi} \]
11. Add Preprocessing

Alternative 9: 8.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Taylor expanded in r around 0

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

   1. metadata-eval N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 N/A

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

   6. lift-/.f32 8.9

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

6. Applied rewrites 8.9%

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

Alternative 10: 8.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / r) - (single(0.16666666666666666) / 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

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

   1. div-sub N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. associate-/r* N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-/r* N/A

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

   10. associate-/l/ N/A

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

   11. *-commutative N/A

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

   12. sub-div N/A

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

4. Applied rewrites 8.9%

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

Alternative 11: 8.9% accurate, 6.0× speedup

Math

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

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(0.25) / s) / Float32(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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.9

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

4. Applied rewrites 8.9%

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

   1. lift-/.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. times-frac N/A

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

   9. associate-/r* N/A

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

   10. frac-times N/A

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

   11. lower-/.f32 N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lift-PI.f32 8.9

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

6. Applied rewrites 8.9%

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

Alternative 12: 8.9% accurate, 6.4× speedup

Math

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

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(0.25) / Float32(s * Float32(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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.9

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

4. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. add-log-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)} \]

   7. log-pow-rev N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{s}\right)} \]

   8. log-pow-rev N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{s}\right)}^{r}\right)} \]

   9. pow-unpow N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(r \cdot s\right)}\right)} \]

   11. pow-unpow N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right)}^{s}\right)} \]

   12. log-pow-rev N/A

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

   13. log-pow-rev N/A

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

   14. add-log-exp N/A

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

   15. lower-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 N/A

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

   18. lift-PI.f32 8.9

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

6. Applied rewrites 8.9%

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

Reproduce

herbie shell --seed 2025130 
(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))))