Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 6.6s

Alternatives: 16

Speedup: 1.0×

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

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

FPCore

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

C

float code(float s, float r) {
	return fmaf(0.75f, (expf(((-r / 3.0f) / s)) / (((((float) M_PI) * 6.0f) * s) * r)), ((expf((-r / s)) / ((((float) M_PI) * s) * r)) * 0.125f));
}

Julia

function code(s, r)
	return fma(Float32(0.75), Float32(exp(Float32(Float32(Float32(-r) / Float32(3.0)) / s)) / Float32(Float32(Float32(Float32(pi) * Float32(6.0)) * s) * r)), Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(pi) * s) * r)) * Float32(0.125)))
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.75, \frac{e^{\frac{-r}{3 \cdot s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.6

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

8. Applied rewrites 99.6%

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

Alternative 2: 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 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

6. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

Alternative 5: 10.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / log((exp(Float32(pi)) ^ Float32(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 r around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{\left(\pi + \pi\right) \cdot s}}{r}} \]

5. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 8.8

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

7. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   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

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

   6. associate-*r* N/A

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

   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 10.2

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

9. Applied rewrites 10.2%

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

Alternative 6: 9.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}} \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}\right) \]

   2. lower-neg.f32 N/A

      \[\leadsto -\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

4. Applied rewrites 9.8%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{1}{\pi} \cdot 0.16666666666666666}{s}\right) - \frac{1}{\pi \cdot r} \cdot 0.25}{s}} \]

6. Applied rewrites 9.8%

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

Alternative 7: 9.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)} + r \cdot \left(r \cdot \left(\frac{1}{144} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right) - \left(\frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{r}} \]

4. Applied rewrites 8.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{\pi \cdot s}, 0.25, \left(\left(\frac{1}{\left(\left(s \cdot s\right) \cdot s\right) \cdot \pi} \cdot 0.06944444444444445\right) \cdot r - \frac{1}{\left(s \cdot s\right) \cdot \pi} \cdot 0.16666666666666666\right) \cdot r\right)}{r}} \]

5. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)} + s \cdot \left(\frac{1}{4} \cdot \frac{s}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{\color{blue}{{s}^{3}}} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)} + s \cdot \left(\frac{1}{4} \cdot \frac{s}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{{s}^{\color{blue}{3}}} \]

7. Applied rewrites 9.8%

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

Alternative 8: 9.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}} \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}\right) \]

   2. lower-neg.f32 N/A

      \[\leadsto -\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

4. Applied rewrites 9.8%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{1}{\pi} \cdot 0.16666666666666666}{s}\right) - \frac{1}{\pi \cdot r} \cdot 0.25}{s}} \]

5. Taylor expanded in s around inf

   \[\leadsto -\frac{\left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto -\frac{\left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{4}}{s} \]

   2. metadata-eval N/A

      \[\leadsto -\frac{\left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{r \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{1}{8} + \frac{1}{8}\right)}{s} \]

   3. distribute-rgt-out N/A

      \[\leadsto -\frac{\left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

   4. distribute-lft-out N/A

      \[\leadsto -\frac{\left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{8} \cdot \left(\frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

   5. lower--.f32 N/A

      \[\leadsto -\frac{\left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{8} \cdot \left(\frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

7. Applied rewrites 9.8%

   \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \pi}, -0.06944444444444445, \frac{0.16666666666666666}{\pi \cdot s}\right) - \frac{0.25}{\pi \cdot r}}{s} \]
8. Add Preprocessing

Alternative 9: 9.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}} \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}\right) \]

   2. lower-neg.f32 N/A

      \[\leadsto -\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

4. Applied rewrites 9.8%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{1}{\pi} \cdot 0.16666666666666666}{s}\right) - \frac{1}{\pi \cdot r} \cdot 0.25}{s}} \]

5. Taylor expanded in s around -inf

   \[\leadsto -\frac{-1 \cdot \frac{\frac{5}{72} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto -\frac{-1 \cdot \frac{\frac{5}{72} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{r \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{4}}{s} \]

   2. metadata-eval N/A

      \[\leadsto -\frac{-1 \cdot \frac{\frac{5}{72} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{r \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{1}{8} + \frac{1}{8}\right)}{s} \]

   3. distribute-rgt-out N/A

      \[\leadsto -\frac{-1 \cdot \frac{\frac{5}{72} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

   4. distribute-lft-out N/A

      \[\leadsto -\frac{-1 \cdot \frac{\frac{5}{72} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \left(\frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

   5. lower--.f32 N/A

      \[\leadsto -\frac{-1 \cdot \frac{\frac{5}{72} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \left(\frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

7. Applied rewrites 9.8%

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

Alternative 10: 9.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{\frac{-r}{s}} + 1}{\pi \cdot r} \cdot 0.125}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (* (/ (+ (exp (/ (- r) s)) 1.0) (* PI r)) 0.125) s))

C

float code(float s, float r) {
	return (((expf((-r / s)) + 1.0f) / (((float) M_PI) * r)) * 0.125f) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) + Float32(1.0)) / Float32(Float32(pi) * r)) * Float32(0.125)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (((exp((-r / s)) + single(1.0)) / (single(pi) * r)) * single(0.125)) / 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. Taylor expanded in s around inf

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

7. Applied rewrites 9.3%

   \[\leadsto \frac{0.125 \cdot \left(\frac{e^{\frac{-r}{s}}}{\pi \cdot r} + \frac{1}{\pi \cdot r}\right)}{s} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 9.3

      \[\leadsto \frac{\left(\frac{e^{\frac{-r}{s}}}{\pi \cdot r} + \frac{1}{\pi \cdot r}\right) \cdot 0.125}{s} \]

9. Applied rewrites 9.3%

   \[\leadsto \frac{\frac{e^{\frac{-r}{s}} + 1}{\pi \cdot r} \cdot 0.125}{s} \]
10. Add Preprocessing

Alternative 11: 9.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{\frac{e^{\frac{-r}{s}} + 1}{\pi \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* 0.125 (/ (/ (+ (exp (/ (- r) s)) 1.0) (* PI r)) s)))

C

float code(float s, float r) {
	return 0.125f * (((expf((-r / s)) + 1.0f) / (((float) M_PI) * r)) / s);
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) + Float32(1.0)) / Float32(Float32(pi) * r)) / s))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (((exp((-r / s)) + single(1.0)) / (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. Taylor expanded in s around inf

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

7. Applied rewrites 9.3%

   \[\leadsto \frac{0.125 \cdot \left(\frac{e^{\frac{-r}{s}}}{\pi \cdot r} + \frac{1}{\pi \cdot r}\right)}{s} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 9.3

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

9. Applied rewrites 9.3%

   \[\leadsto 0.125 \cdot \color{blue}{\frac{\frac{e^{\frac{-r}{s}} + 1}{\pi \cdot r}}{s}} \]
10. Add Preprocessing

Alternative 12: 8.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.5}{\pi + \pi}}{s}}{r} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ (/ (/ 0.5 (+ PI PI)) s) r))

C

float code(float s, float r) {
	return ((0.5f / (((float) M_PI) + ((float) M_PI))) / s) / r;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.5) / Float32(Float32(pi) + Float32(pi))) / s) / r)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.5) / (single(pi) + 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 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{\left(\pi + \pi\right) \cdot s}}{r}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{\frac{1}{2}}{\pi + \pi}}{s}}{r} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{\frac{1}{2}}{\pi + \pi}}{s}}{r} \]

   5. lower-/.f32 8.8

      \[\leadsto \frac{\frac{\frac{0.5}{\pi + \pi}}{s}}{r} \]

6. Applied rewrites 8.8%

   \[\leadsto \frac{\frac{\frac{0.5}{\pi + \pi}}{s}}{r} \]
7. Add Preprocessing

Alternative 13: 8.8% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}} \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}\right) \]

   2. lower-neg.f32 N/A

      \[\leadsto -\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

4. Applied rewrites 9.8%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{1}{\pi} \cdot 0.16666666666666666}{s}\right) - \frac{1}{\pi \cdot r} \cdot 0.25}{s}} \]

5. Taylor expanded in r around 0

   \[\leadsto -\frac{\frac{r \cdot \left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{s} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto -\frac{\frac{r \cdot \left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{s} \]

7. Applied rewrites 9.8%

   \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \pi}, -0.06944444444444445, \frac{0.16666666666666666}{\pi \cdot s}\right) \cdot r - \frac{0.25}{\pi}}{r}}{s} \]

8. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lift-PI.f32 8.8

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

10. Applied rewrites 8.8%

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

Alternative 14: 8.8% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 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. Taylor expanded in s around inf

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

   1. distribute-lft-out N/A

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

   2. distribute-rgt-out N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 8.8

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

7. Applied rewrites 8.8%

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

Alternative 15: 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 r around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{\left(\pi + \pi\right) \cdot s}}{r}} \]

5. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 8.8

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

7. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   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

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   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 8.8

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

9. Applied rewrites 8.8%

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

    1. lift-/.f32 N/A

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

    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

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

    4. associate-/r* N/A

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

    5. lower-/.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. *-commutative N/A

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

    8. lower-/.f32 N/A

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

    9. *-commutative N/A

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

    10. lift-*.f32 N/A

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

    11. lift-PI.f32 8.8

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

11. Applied rewrites 8.8%

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

Alternative 16: 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 r around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{\left(\pi + \pi\right) \cdot s}}{r}} \]

5. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 8.8

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

7. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   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

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   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 8.8

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

9. Applied rewrites 8.8%

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

Reproduce

herbie shell --seed 2025124 
(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))))