Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 6.8s

Alternatives: 15

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, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi}, 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}\right)}{s}} \]

6. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

8. Applied rewrites 99.5%

   \[\leadsto \frac{\frac{e^{\frac{r}{s} \cdot -0.3333333333333333} + e^{\frac{-r}{s}}}{r} \cdot \frac{0.125}{\pi}}{s} \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{s} \cdot \frac{-1}{3}} + e^{\frac{-r}{s}}}{r} \cdot \frac{\frac{1}{8}}{\pi}}{s} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{s} \cdot \frac{-1}{3}} + e^{\frac{-r}{s}}}{r} \cdot \frac{\frac{1}{8}}{\pi}}{s} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{\frac{e^{\frac{r \cdot \frac{-1}{3}}{s}} + e^{\frac{-r}{s}}}{r} \cdot \frac{\frac{1}{8}}{\pi}}{s} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{r \cdot \frac{-1}{3}}{s}} + e^{\frac{-r}{s}}}{r} \cdot \frac{\frac{1}{8}}{\pi}}{s} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{e^{\frac{\frac{-1}{3} \cdot r}{s}} + e^{\frac{-r}{s}}}{r} \cdot \frac{\frac{1}{8}}{\pi}}{s} \]

   6. lower-*.f32 99.5

      \[\leadsto \frac{\frac{e^{\frac{-0.3333333333333333 \cdot r}{s}} + e^{\frac{-r}{s}}}{r} \cdot \frac{0.125}{\pi}}{s} \]

10. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi}, 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}\right)}{s}} \]

6. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

8. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi}, 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}\right)}{s}} \]

6. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/l* N/A

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

   5. associate-/l* N/A

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

   6. associate-/r* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 N/A

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

8. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi}, 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}\right)}{s}} \]

6. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

8. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi}, 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}\right)}{s}} \]

6. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

8. Applied rewrites 99.5%

   \[\leadsto \frac{\frac{e^{\frac{r}{s} \cdot -0.3333333333333333} + e^{\frac{-r}{s}}}{r} \cdot \frac{0.125}{\pi}}{s} \]
9. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{s} \cdot \frac{-1}{3}} + e^{\frac{-r}{s}}}{r} \cdot \frac{\frac{1}{8}}{\pi}}{s} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{s} \cdot \frac{-1}{3}} + e^{\frac{-r}{s}}}{r} \cdot \frac{\frac{1}{8}}{\pi}}{s} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{s} \cdot \frac{-1}{3}} + e^{\frac{-r}{s}}}{r} \cdot \frac{\frac{1}{8}}{\pi}}{s} \]

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/l/ N/A

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

10. Applied rewrites 99.5%

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

Alternative 6: 43.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 8.8

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 8.8

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

6. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-exp N/A

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

   9. lift-*.f32 N/A

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

   10. lower-exp.f32 43.4

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

8. Applied rewrites 43.4%

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

Alternative 7: 9.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-PI.f32 N/A

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

   5. add-log-exp N/A

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

   6. log-pow-rev N/A

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

   7. lower-log.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. pow-exp N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lower-exp.f32 9.8

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 9.8

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

6. Applied rewrites 9.8%

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

Alternative 8: 8.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{2 + -1.3333333333333333 \cdot \frac{r}{s}}{r} \cdot 0.75}{\pi \cdot 6}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (/ (* (/ (+ 2.0 (* -1.3333333333333333 (/ r s))) r) 0.75) (* PI 6.0)) s))

C

float code(float s, float r) {
	return ((((2.0f + (-1.3333333333333333f * (r / s))) / r) * 0.75f) / (((float) M_PI) * 6.0f)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(-1.3333333333333333) * Float32(r / s))) / r) * Float32(0.75)) / Float32(Float32(pi) * Float32(6.0))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((((single(2.0) + (single(-1.3333333333333333) * (r / s))) / r) * single(0.75)) / (single(pi) * single(6.0))) / 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 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%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi}, 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}\right)}{s}} \]

6. Applied rewrites 99.5%

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

7. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{\frac{2 + \frac{-4}{3} \cdot \frac{r}{s}}{r} \cdot \frac{3}{4}}{\pi \cdot 6}}{s} \]
8. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{2 + \frac{-4}{3} \cdot \frac{r}{s}}{r} \cdot \frac{3}{4}}{\pi \cdot 6}}{s} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{\frac{\frac{2 + \frac{-4}{3} \cdot \frac{r}{s}}{r} \cdot \frac{3}{4}}{\pi \cdot 6}}{s} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{2 + \frac{-4}{3} \cdot \frac{r}{s}}{r} \cdot \frac{3}{4}}{\pi \cdot 6}}{s} \]

   4. lower-/.f32 8.9

      \[\leadsto \frac{\frac{\frac{2 + -1.3333333333333333 \cdot \frac{r}{s}}{r} \cdot 0.75}{\pi \cdot 6}}{s} \]

9. Applied rewrites 8.9%

   \[\leadsto \frac{\frac{\frac{2 + -1.3333333333333333 \cdot \frac{r}{s}}{r} \cdot 0.75}{\pi \cdot 6}}{s} \]
10. Add Preprocessing

Alternative 9: 8.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{1.5 + -1 \cdot \frac{r}{s}}{r}}{\pi \cdot 6}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (/ (/ (+ 1.5 (* -1.0 (/ r s))) r) (* PI 6.0)) s))

C

float code(float s, float r) {
	return (((1.5f + (-1.0f * (r / s))) / r) / (((float) M_PI) * 6.0f)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(1.5) + Float32(Float32(-1.0) * Float32(r / s))) / r) / Float32(Float32(pi) * Float32(6.0))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (((single(1.5) + (single(-1.0) * (r / s))) / r) / (single(pi) * single(6.0))) / 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 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%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi}, 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}\right)}{s}} \]

6. Applied rewrites 99.5%

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

7. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{\frac{\frac{3}{2} + -1 \cdot \frac{r}{s}}{r}}{\pi \cdot 6}}{s} \]
8. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{\frac{3}{2} + -1 \cdot \frac{r}{s}}{r}}{\pi \cdot 6}}{s} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{\frac{\frac{\frac{3}{2} + -1 \cdot \frac{r}{s}}{r}}{\pi \cdot 6}}{s} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{\frac{3}{2} + -1 \cdot \frac{r}{s}}{r}}{\pi \cdot 6}}{s} \]

   4. lower-/.f32 8.9

      \[\leadsto \frac{\frac{\frac{1.5 + -1 \cdot \frac{r}{s}}{r}}{\pi \cdot 6}}{s} \]

9. Applied rewrites 8.9%

   \[\leadsto \frac{\frac{\frac{1.5 + -1 \cdot \frac{r}{s}}{r}}{\pi \cdot 6}}{s} \]
10. Add Preprocessing

Alternative 10: 8.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1.5 \cdot \frac{1}{r} - \frac{1}{s}}{\pi \cdot 6}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (/ (- (* 1.5 (/ 1.0 r)) (/ 1.0 s)) (* PI 6.0)) s))

C

float code(float s, float r) {
	return (((1.5f * (1.0f / r)) - (1.0f / s)) / (((float) M_PI) * 6.0f)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(1.5) * Float32(Float32(1.0) / r)) - Float32(Float32(1.0) / s)) / Float32(Float32(pi) * Float32(6.0))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (((single(1.5) * (single(1.0) / r)) - (single(1.0) / s)) / (single(pi) * single(6.0))) / 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 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%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi}, 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}\right)}{s}} \]

6. Applied rewrites 99.5%

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

7. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{\frac{3}{2} \cdot \frac{1}{r} - \frac{1}{s}}{\pi \cdot 6}}{s} \]
8. Step-by-step derivation

   1. lower--.f32 N/A

      \[\leadsto \frac{\frac{\frac{3}{2} \cdot \frac{1}{r} - \frac{1}{s}}{\pi \cdot 6}}{s} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{3}{2} \cdot \frac{1}{r} - \frac{1}{s}}{\pi \cdot 6}}{s} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{3}{2} \cdot \frac{1}{r} - \frac{1}{s}}{\pi \cdot 6}}{s} \]

   4. lower-/.f32 8.9

      \[\leadsto \frac{\frac{1.5 \cdot \frac{1}{r} - \frac{1}{s}}{\pi \cdot 6}}{s} \]

9. Applied rewrites 8.9%

   \[\leadsto \frac{\frac{1.5 \cdot \frac{1}{r} - \frac{1}{s}}{\pi \cdot 6}}{s} \]
10. Add Preprocessing

Alternative 11: 8.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 + -0.16666666666666666 \cdot \frac{r}{s}}{\pi \cdot s} \cdot \frac{1}{r} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Applied rewrites 9.3%

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

6. Applied rewrites 9.3%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. lower-*.f32 N/A

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

8. Applied rewrites 9.3%

   \[\leadsto \color{blue}{\frac{\frac{0.75}{6} + \frac{0.125}{e^{\frac{r}{s}}}}{\pi \cdot s} \cdot \frac{1}{r}} \]

9. Taylor expanded in r around 0

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

    1. lower-+.f32 N/A

       \[\leadsto \frac{\frac{1}{4} + \color{blue}{\frac{-1}{6} \cdot \frac{r}{s}}}{\pi \cdot s} \cdot \frac{1}{r} \]

    2. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} + \frac{-1}{6} \cdot \color{blue}{\frac{r}{s}}}{\pi \cdot s} \cdot \frac{1}{r} \]

    3. lower-/.f32 8.9

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

11. Applied rewrites 8.9%

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

Alternative 12: 8.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s \cdot r} \cdot \frac{1}{\pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / (s * r)) * (single(1.0) / 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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 8.8

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

6. Applied rewrites 8.8%

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

   1. lift-/.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 8.8

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

8. Applied rewrites 8.8%

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

Alternative 13: 8.8% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / (s * r)) / 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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 8.8

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

6. Applied rewrites 8.8%

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

Alternative 14: 8.8% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / ((s * r) * 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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 8.8

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

6. Applied rewrites 8.8%

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

Alternative 15: 8.8% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

Reproduce

herbie shell --seed 2025150 
(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))))