Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 4.5s

Alternatives: 16

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% 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, 1.0× speedup

Math

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

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(Float32(-r) / 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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 99.6

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

3. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return fmaf(0.75f, (expf((-r / (3.0f * s))) / (((6.0f * ((float) M_PI)) * 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(-r) / Float32(Float32(3.0) * s))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * 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%

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

2. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-*.f32 N/A

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

6. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in 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 99.5%

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

6. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

Alternative 5: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (((single(0.125) * exp((-r / s))) - (single(-0.125) * 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 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 99.5

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

6. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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 99.5%

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

6. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. distribute-lft-out N/A

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

   10. distribute-frac-neg N/A

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

   11. mul-1-neg N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. mul-1-neg N/A

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

   15. distribute-frac-neg N/A

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

8. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. times-frac N/A

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

10. Applied rewrites 99.5%

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

Alternative 7: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(-r) / s)) + exp(Float32(Float32(Float32(-0.3333333333333333) * r) / s)))) / Float32(Float32(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 99.5%

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

6. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. distribute-lft-out N/A

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

   10. distribute-frac-neg N/A

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

   11. mul-1-neg N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. mul-1-neg N/A

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

   15. distribute-frac-neg N/A

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

8. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 99.5

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

10. Applied rewrites 99.5%

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

Alternative 8: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in 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 99.5%

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

6. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. distribute-lft-out N/A

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

   10. distribute-frac-neg N/A

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

   11. mul-1-neg N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. mul-1-neg N/A

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

   15. distribute-frac-neg N/A

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

8. Applied rewrites 99.5%

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

Alternative 9: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f * (expf((-r / s)) + expf(((r / s) * -0.3333333333333333f)))) / ((((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(r / s) * Float32(-0.3333333333333333))))) / Float32(Float32(Float32(pi) * r) * s))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp((-r / s)) + exp(((r / s) * single(-0.3333333333333333))))) / ((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 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 99.5%

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

6. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. distribute-lft-out N/A

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

   10. distribute-frac-neg N/A

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

   11. mul-1-neg N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. mul-1-neg N/A

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

   15. distribute-frac-neg N/A

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

8. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lift-PI.f32 99.5

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

10. Applied rewrites 99.5%

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

Alternative 10: 44.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 9.0

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

8. Applied rewrites 9.0%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lift-PI.f32 44.0

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

10. Applied rewrites 44.0%

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

Alternative 11: 40.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. add-log-exp N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. log-pow N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. pow-unpow N/A

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

   13. pow-to-exp N/A

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

   14. log-pow-rev N/A

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

   15. lift-PI.f32 N/A

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

   16. lift-exp.f32 N/A

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

   17. add-log-exp N/A

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

   18. *-commutative N/A

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

   19. exp-prod N/A

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

8. Applied rewrites 40.5%

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

Alternative 12: 10.4% 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. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. add-log-exp N/A

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

   8. log-pow-rev N/A

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

   9. lower-log.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 10.4

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

6. Applied rewrites 10.4%

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

   1. lift-pow.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow-unpow N/A

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

   4. pow-to-exp N/A

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

   5. log-pow-rev N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. add-log-exp N/A

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

   9. *-commutative N/A

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

   10. lower-exp.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 10.4

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

8. Applied rewrites 10.4%

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

Alternative 13: 9.0% 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. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 9.0

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

6. Applied rewrites 9.0%

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

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lift-PI.f32 9.0

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

8. Applied rewrites 9.0%

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

Alternative 14: 9.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

Alternative 15: 9.0% 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. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 9.0

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

6. Applied rewrites 9.0%

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

Alternative 16: 9.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 9.0

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

8. Applied rewrites 9.0%

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

Reproduce

herbie shell --seed 2025113 
(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))))