Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 5.6s

Alternatives: 15

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.75
  (/ (/ (exp (* (/ r s) -0.3333333333333333)) (* (* 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 / s) * -0.3333333333333333f)) / ((((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(Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / 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 99.5%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around 0

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

10. Applied rewrites 99.5%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-/r* N/A

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

    4. lower-/.f32 N/A

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

    5. lower-/.f32 99.6

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

    6. lift-*.f32 N/A

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

    7. lift-/.f32 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f32 N/A

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

    10. lift-/.f32 99.6

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

12. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around 0

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

10. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

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(Float32(0.125) * Float32(Float32(exp(Float32(Float32(-r) / s)) + exp(Float32(Float32(r / s) * Float32(-0.3333333333333333)))) / 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 99.5%

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

3. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 4: 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 99.5%

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

3. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 5: 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 99.5%

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

3. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 6: 43.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

5. 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. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 10.9

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

8. Applied rewrites 10.9%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. add-log-exp N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. log-pow-rev N/A

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

   8. lower-log.f32 N/A

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

   9. lower-pow.f32 45.6

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

10. Applied rewrites 45.6%

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

Alternative 7: 41.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.05000000074505806:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.16666666666666666, \frac{r \cdot r}{\pi \cdot s} \cdot 0.06944444444444445\right)}{s} + \frac{0.25}{\pi}}{s}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 0.05000000074505806)
   (/
    (/
     (+
      (/
       (fma
        (/ r PI)
        -0.16666666666666666
        (* (/ (* r r) (* PI s)) 0.06944444444444445))
       s)
      (/ 0.25 PI))
     s)
    r)
   (/ 0.25 (log (pow (exp r) (* PI s))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 0.05000000074505806f) {
		tmp = (((fmaf((r / ((float) M_PI)), -0.16666666666666666f, (((r * r) / (((float) M_PI) * s)) * 0.06944444444444445f)) / s) + (0.25f / ((float) M_PI))) / s) / r;
	} else {
		tmp = 0.25f / logf(powf(expf(r), (((float) M_PI) * s)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(0.05000000074505806))
		tmp = Float32(Float32(Float32(Float32(fma(Float32(r / Float32(pi)), Float32(-0.16666666666666666), Float32(Float32(Float32(r * r) / Float32(Float32(pi) * s)) * Float32(0.06944444444444445))) / s) + Float32(Float32(0.25) / Float32(pi))) / s) / r);
	else
		tmp = Float32(Float32(0.25) / log((exp(r) ^ Float32(Float32(pi) * s))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 0.0500000007

   1. Initial program 99.3%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{r}} \]

   5. Applied rewrites 20.6%

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

   6. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

   8. Applied rewrites 21.9%

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

   if 0.0500000007 < r

   1. Initial program 99.7%

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

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
   4. 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 5.3

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

   5. Applied rewrites 5.3%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. add-log-exp N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 9.0

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

   7. Applied rewrites 9.0%

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

      1. lift-*.f32 N/A

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

      2. lift-pow.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. exp-prod N/A

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

      11. lower-pow.f32 N/A

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

      12. lower-exp.f32 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f32 N/A

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

      15. lift-PI.f32 64.0

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

   9. Applied rewrites 64.0%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.05000000074505806:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.16666666666666666, \frac{r \cdot r}{\pi \cdot s} \cdot 0.06944444444444445\right)}{s} + \frac{0.25}{\pi}}{s}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.05000000074505806:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi}, 0.16666666666666666, \left(r \cdot \frac{r}{\pi \cdot s}\right) \cdot -0.06944444444444445\right)}{s} - \frac{0.25}{\pi}}{-s}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 0.05000000074505806)
   (/
    (/
     (-
      (/
       (fma
        (/ r PI)
        0.16666666666666666
        (* (* r (/ r (* PI s))) -0.06944444444444445))
       s)
      (/ 0.25 PI))
     (- s))
    r)
   (/ 0.25 (log (pow (exp r) (* PI s))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 0.05000000074505806f) {
		tmp = (((fmaf((r / ((float) M_PI)), 0.16666666666666666f, ((r * (r / (((float) M_PI) * s))) * -0.06944444444444445f)) / s) - (0.25f / ((float) M_PI))) / -s) / r;
	} else {
		tmp = 0.25f / logf(powf(expf(r), (((float) M_PI) * s)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(0.05000000074505806))
		tmp = Float32(Float32(Float32(Float32(fma(Float32(r / Float32(pi)), Float32(0.16666666666666666), Float32(Float32(r * Float32(r / Float32(Float32(pi) * s))) * Float32(-0.06944444444444445))) / s) - Float32(Float32(0.25) / Float32(pi))) / Float32(-s)) / r);
	else
		tmp = Float32(Float32(0.25) / log((exp(r) ^ Float32(Float32(pi) * s))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 0.0500000007

   1. Initial program 99.3%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{r}} \]

   5. Applied rewrites 20.6%

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

   6. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

   8. Applied rewrites 21.9%

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

   9. Taylor expanded in s around inf

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

       1. metadata-eval N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. lower-/.f32 N/A

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

   11. Applied rewrites 21.9%

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

   if 0.0500000007 < r

   1. Initial program 99.7%

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

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
   4. 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 5.3

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

   5. Applied rewrites 5.3%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. add-log-exp N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 9.0

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

   7. Applied rewrites 9.0%

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

      1. lift-*.f32 N/A

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

      2. lift-pow.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. exp-prod N/A

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

      11. lower-pow.f32 N/A

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

      12. lower-exp.f32 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f32 N/A

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

      15. lift-PI.f32 64.0

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

   9. Applied rewrites 64.0%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.05000000074505806:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi}, 0.16666666666666666, \left(r \cdot \frac{r}{\pi \cdot s}\right) \cdot -0.06944444444444445\right)}{s} - \frac{0.25}{\pi}}{-s}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 10.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{r}} \]

5. Applied rewrites 10.6%

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

6. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-/.f32 N/A

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

8. Applied rewrites 11.9%

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

9. Taylor expanded in s around inf

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

    1. metadata-eval N/A

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

    2. fp-cancel-sign-sub-inv N/A

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

    3. lower-/.f32 N/A

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

11. Applied rewrites 11.9%

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

12. Final simplification 11.9%

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

Alternative 10: 10.1% 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 99.5%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{r}} \]

5. Applied rewrites 10.6%

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

6. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

8. Applied rewrites 11.8%

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

Alternative 11: 10.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s} - \frac{0.25}{\pi \cdot r}}{-s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-/.f32 N/A

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

5. Applied rewrites 11.8%

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

6. Final simplification 11.8%

   \[\leadsto \frac{\frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s} - \frac{0.25}{\pi \cdot r}}{-s} \]
7. Add Preprocessing

Alternative 12: 10.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{r}{\pi \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s} \end{array} \]

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(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 99.5%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{r}} \]

5. Applied rewrites 10.6%

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

6. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-/.f32 N/A

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

8. Applied rewrites 11.8%

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

9. Final simplification 11.8%

   \[\leadsto \frac{\frac{\frac{r}{\pi \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s} \]
10. Add Preprocessing

Alternative 13: 10.1% 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 99.5%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
4. 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 10.9

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

5. Applied rewrites 10.9%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
6. 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 11.2

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

7. Applied rewrites 11.2%

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

   1. lift-*.f32 N/A

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

   2. lift-pow.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

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 11.2

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

9. Applied rewrites 11.2%

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

Alternative 14: 9.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{r}} \]

5. Applied rewrites 10.6%

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

6. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 10.9

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

8. Applied rewrites 10.9%

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

Alternative 15: 9.1% 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 99.5%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
4. 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 10.9

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

5. Applied rewrites 10.9%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
6. 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 10.9

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

7. Applied rewrites 10.9%

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

Reproduce

herbie shell --seed 2025064 
(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))))