Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 3.6s

Alternatives: 10

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{\pi \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ 0.125 (* PI s)) (/ (exp (/ (- r) s)) r))
  (/ (* 0.75 (exp (* -0.3333333333333333 (/ r s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.125f / (((float) M_PI) * s)) * (expf((-r / s)) / r)) + ((0.75f * expf((-0.3333333333333333f * (r / s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(Float32(pi) * s)) * Float32(exp(Float32(Float32(-r) / s)) / r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-0.3333333333333333) * Float32(r / s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / (single(pi) * s)) * (exp((-r / s)) / r)) + ((single(0.75) * exp((single(-0.3333333333333333) * (r / s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. times-frac N/A

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

   8. distribute-frac-neg N/A

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

   9. mul-1-neg N/A

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

   10. lower-*.f32 N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 99.5

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

9. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (exp (/ (- r) s)) (* (* PI s) r)) 0.125)
  (/ (* 0.75 (exp (* -0.3333333333333333 (/ r s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((expf((-r / s)) / ((((float) M_PI) * s) * r)) * 0.125f) + ((0.75f * expf((-0.3333333333333333f * (r / s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(pi) * s) * r)) * Float32(0.125)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-0.3333333333333333) * Float32(r / s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) / ((single(pi) * s) * r)) * single(0.125)) + ((single(0.75) * exp((single(-0.3333333333333333) * (r / s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.5

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

7. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

   \[\leadsto \frac{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi} \cdot 0.125}{\color{blue}{r}} \]
9. Add Preprocessing

Alternative 4: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

Alternative 5: 45.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;\frac{0.25}{s \cdot \log \left({\left(e^{\pi}\right)}^{r}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{\pi} \cdot -1.3333333333333333}{s}, 0.125, \mathsf{fma}\left(\frac{\frac{r \cdot r}{\pi} \cdot 0.5555555555555556}{s \cdot s}, 0.125, \frac{0.25}{\pi}\right)\right)}{s}}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<=
      (+
       (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
       (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
      4.999999841327613e-21)
   (/ 0.25 (* s (log (pow (exp PI) r))))
   (/
    (/
     (fma
      (/ (* (/ r PI) -1.3333333333333333) s)
      0.125
      (fma
       (/ (* (/ (* r r) PI) 0.5555555555555556) (* s s))
       0.125
       (/ 0.25 PI)))
     s)
    r)))

C

float code(float s, float r) {
	float tmp;
	if ((((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))) <= 4.999999841327613e-21f) {
		tmp = 0.25f / (s * logf(powf(expf(((float) M_PI)), r)));
	} else {
		tmp = (fmaf((((r / ((float) M_PI)) * -1.3333333333333333f) / s), 0.125f, fmaf(((((r * r) / ((float) M_PI)) * 0.5555555555555556f) / (s * s)), 0.125f, (0.25f / ((float) M_PI)))) / s) / r;
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (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))) <= Float32(4.999999841327613e-21))
		tmp = Float32(Float32(0.25) / Float32(s * log((exp(Float32(pi)) ^ r))));
	else
		tmp = Float32(Float32(fma(Float32(Float32(Float32(r / Float32(pi)) * Float32(-1.3333333333333333)) / s), Float32(0.125), fma(Float32(Float32(Float32(Float32(r * r) / Float32(pi)) * Float32(0.5555555555555556)) / Float32(s * s)), Float32(0.125), Float32(Float32(0.25) / Float32(pi)))) / s) / r);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 4.99999984e-21

   1. Initial program 99.6%

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

   2. 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 9.3

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

   4. Applied rewrites 9.3%

      \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
   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. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 9.3

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

   6. Applied rewrites 9.3%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-PI.f32 9.3

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

   8. Applied rewrites 9.3%

      \[\leadsto \frac{0.25}{s \cdot \color{blue}{\left(\pi \cdot r\right)}} \]
   9. 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. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. lift-PI.f32 42.9

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

   10. Applied rewrites 42.9%

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

   if 4.99999984e-21 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 99.6%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-exp.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f32 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in r around inf

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

      1. lower-/.f32 N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot \frac{-1 \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{3} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{1}{8} \cdot \frac{\frac{1}{18} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{1}{2} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{{s}^{2}} + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s}}{r} \]
   8. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{\frac{\frac{1}{8} \cdot \frac{-1 \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{3} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{1}{8} \cdot \frac{\frac{1}{18} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{1}{2} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{{s}^{2}} + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s}}{r} \]

   9. Applied rewrites 10.3%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{\pi} \cdot -1.3333333333333333}{s}, 0.125, \mathsf{fma}\left(\frac{\frac{r \cdot r}{\pi} \cdot 0.5555555555555556}{s \cdot s}, 0.125, \frac{0.25}{\pi}\right)\right)}{s}}{r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 42.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / Float32(s * log((exp(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 99.6%

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

2. 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 9.3

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

4. Applied rewrites 9.3%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
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. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 9.3

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

6. Applied rewrites 9.3%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 9.3

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

8. Applied rewrites 9.3%

   \[\leadsto \frac{0.25}{s \cdot \color{blue}{\left(\pi \cdot r\right)}} \]
9. 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. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lift-PI.f32 42.9

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

10. Applied rewrites 42.9%

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

Alternative 7: 13.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 280:\\ \;\;\;\;-\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 280.0)
   (-
    (/
     (-
      (-
       (/
        (-
         (- (/ (* (/ r PI) -0.06944444444444445) s))
         (/ 0.16666666666666666 PI))
        s))
      (/ 0.25 (* PI r)))
     s))
   (/ 0.25 (log (pow (exp PI) (* s r))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 280.0f) {
		tmp = -((-((-(((r / ((float) M_PI)) * -0.06944444444444445f) / s) - (0.16666666666666666f / ((float) M_PI))) / s) - (0.25f / (((float) M_PI) * r))) / s);
	} else {
		tmp = 0.25f / logf(powf(expf(((float) M_PI)), (s * r)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(280.0))
		tmp = Float32(-Float32(Float32(Float32(-Float32(Float32(Float32(-Float32(Float32(Float32(r / Float32(pi)) * Float32(-0.06944444444444445)) / s)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s)) - Float32(Float32(0.25) / Float32(Float32(pi) * r))) / s));
	else
		tmp = Float32(Float32(0.25) / log((exp(Float32(pi)) ^ Float32(s * r))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if (r <= single(280.0))
		tmp = -((-((-(((r / single(pi)) * single(-0.06944444444444445)) / s) - (single(0.16666666666666666) / single(pi))) / s) - (single(0.25) / (single(pi) * r))) / s);
	else
		tmp = single(0.25) / log((exp(single(pi)) ^ (s * r)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if r < 280

   1. Initial program 99.6%

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

   2. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

   4. Applied rewrites 10.4%

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

   if 280 < r

   1. Initial program 99.6%

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

   2. 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 9.3

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

   4. Applied rewrites 9.3%

      \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
   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. add-log-exp N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 9.7

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

   6. Applied rewrites 9.7%

      \[\leadsto \frac{0.25}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 10.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = -((-((-(((r / single(pi)) * single(-0.06944444444444445)) / s) - (single(0.16666666666666666) / single(pi))) / s) - (single(0.25) / (single(pi) * r))) / s);
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-/.f32 N/A

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

4. Applied rewrites 10.4%

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

Alternative 9: 9.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. 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. lower-/.f32 N/A

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

4. Applied rewrites 9.5%

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

Alternative 10: 9.3% 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 99.6%

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

2. 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 9.3

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

4. Applied rewrites 9.3%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
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. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 9.3

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

6. Applied rewrites 9.3%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 9.3

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

8. Applied rewrites 9.3%

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

Reproduce

herbie shell --seed 2025127 
(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))))