Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 3.9s

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: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}, 0.25, \frac{0.125}{\pi \cdot s} \cdot \frac{e^{\frac{\frac{-r}{3}}{s}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ (- r) s)) (* (* (* PI 2.0) s) r))
  0.25
  (* (/ 0.125 (* PI s)) (/ (exp (/ (/ (- r) 3.0) s)) r))))

C

float code(float s, float r) {
	return fmaf((expf((-r / s)) / (((((float) M_PI) * 2.0f) * s) * r)), 0.25f, ((0.125f / (((float) M_PI) * s)) * (expf(((-r / 3.0f) / s)) / r)));
}

Julia

function code(s, r)
	return fma(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(Float32(pi) * Float32(2.0)) * s) * r)), Float32(0.25), Float32(Float32(Float32(0.125) / Float32(Float32(pi) * s)) * Float32(exp(Float32(Float32(Float32(-r) / Float32(3.0)) / s)) / r)))
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in s around 0

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}, \frac{1}{4}, \color{blue}{\frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}} \cdot \frac{e^{\frac{\frac{-r}{3}}{s}}}{r}\right) \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 99.6

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}, 0.25, \frac{0.125}{\pi \cdot s} \cdot \frac{e^{\frac{\frac{-r}{3}}{s}}}{r}\right) \]

8. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}, 0.25, \color{blue}{\frac{0.125}{\pi \cdot s}} \cdot \frac{e^{\frac{\frac{-r}{3}}{s}}}{r}\right) \]
9. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}, 0.25, \frac{0.125}{\pi \cdot s} \cdot \frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ (- r) s)) (* (* (* PI 2.0) s) r))
  0.25
  (* (/ 0.125 (* PI s)) (/ (exp (/ (* -0.3333333333333333 r) s)) r))))

C

float code(float s, float r) {
	return fmaf((expf((-r / s)) / (((((float) M_PI) * 2.0f) * s) * r)), 0.25f, ((0.125f / (((float) M_PI) * s)) * (expf(((-0.3333333333333333f * r) / s)) / r)));
}

Julia

function code(s, r)
	return fma(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(Float32(pi) * Float32(2.0)) * s) * r)), Float32(0.25), Float32(Float32(Float32(0.125) / Float32(Float32(pi) * s)) * Float32(exp(Float32(Float32(Float32(-0.3333333333333333) * r) / s)) / r)))
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in s around 0

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}, \frac{1}{4}, \color{blue}{\frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}} \cdot \frac{e^{\frac{\frac{-r}{3}}{s}}}{r}\right) \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 99.6

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}, 0.25, \frac{0.125}{\pi \cdot s} \cdot \frac{e^{\frac{\frac{-r}{3}}{s}}}{r}\right) \]

8. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}, 0.25, \color{blue}{\frac{0.125}{\pi \cdot s}} \cdot \frac{e^{\frac{\frac{-r}{3}}{s}}}{r}\right) \]

9. Taylor expanded in r around 0

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}, \frac{1}{4}, \frac{\frac{1}{8}}{\pi \cdot s} \cdot \frac{e^{\frac{\color{blue}{\frac{-1}{3} \cdot r}}{s}}}{r}\right) \]
10. Step-by-step derivation

    1. lower-*.f32 99.6

       \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}, 0.25, \frac{0.125}{\pi \cdot s} \cdot \frac{e^{\frac{-0.3333333333333333 \cdot \color{blue}{r}}{s}}}{r}\right) \]

11. Applied rewrites 99.6%

    \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}, 0.25, \frac{0.125}{\pi \cdot s} \cdot \frac{e^{\frac{\color{blue}{-0.3333333333333333 \cdot r}}{s}}}{r}\right) \]
12. Add Preprocessing

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in r around 0

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

   1. lower-*.f32 99.6

      \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{\frac{-0.3333333333333333 \cdot \color{blue}{r}}{s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125\right) \]

9. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(fma(Float32(0.125), exp(Float32(Float32(-r) / s)), Float32(Float32(0.125) * exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))))) / Float32(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 r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-pow.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-/r* N/A

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

   14. lower-/.f32 N/A

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

6. Applied rewrites 99.3%

   \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(0.125, e^{\frac{-r}{s}}, 0.125 \cdot {\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}\right)}{\pi}}{s}}{r} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. pow-exp N/A

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

   5. lower-exp.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 99.6

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(0.125, e^{\frac{-r}{s}}, 0.125 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{\pi}}{s}}{r} \]

8. Applied rewrites 99.6%

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

Alternative 5: 99.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (((single(0.125) * (exp((-r / s)) + (exp(single(-0.3333333333333333)) ^ (r / s)))) / 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 r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-pow.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-/r* N/A

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

   14. lower-/.f32 N/A

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

6. Applied rewrites 99.3%

   \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(0.125, e^{\frac{-r}{s}}, 0.125 \cdot {\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}\right)}{\pi}}{s}}{r} \]
7. Step-by-step derivation

8. Applied rewrites 99.3%

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

Alternative 6: 99.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) * (exp((-r / s)) + (exp(single(-0.3333333333333333)) ^ (r / s)))) / 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 r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-pow.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-/r* N/A

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

   14. lower-/.f32 N/A

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

6. Applied rewrites 99.3%

   \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(0.125, e^{\frac{-r}{s}}, 0.125 \cdot {\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}\right)}{\pi}}{s}}{r} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

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

8. Applied rewrites 99.3%

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

Alternative 7: 43.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]
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 8.8%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 8.8

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

7. Applied rewrites 8.8%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lift-PI.f32 43.5

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

9. Applied rewrites 43.5%

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

Alternative 8: 10.1% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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 8.8

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

4. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
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. lower-*.f32 10.1

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

6. Applied rewrites 10.1%

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

Alternative 9: 8.8% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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 8.8%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 8.8

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

7. Applied rewrites 8.8%

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

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f32 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 8.8

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

9. Applied rewrites 8.8%

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

Alternative 10: 8.8% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / ((r * s) * single(pi));
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 8.8

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

4. Applied rewrites 8.8%

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

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

   9. lift-PI.f32 8.8

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

6. Applied rewrites 8.8%

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

Reproduce

herbie shell --seed 2025101 
(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))))