Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 18.1s

Alternatives: 18

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

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp(-(r / s)) * single(0.125)) / (r * (s * single(pi)))) + ((single(0.75) * exp(((r / s) / -single(3.0)))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around 0

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} + \frac{0.75 \cdot e^{\frac{\frac{r}{s}}{-3}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp(-(r / s)) * single(0.125)) / (r * (s * single(pi)))) + ((single(0.75) * exp(((r / s) / -single(3.0)))) / (((single(6.0) * single(pi)) * r) * s));
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around 0

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} + \frac{0.75 \cdot e^{\frac{\frac{r}{s}}{-3}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Applied rewrites 99.6%

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

10. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp(-(r / s)) * single(0.125)) / (r * (s * single(pi)))) + ((single(0.75) * exp((single(-0.3333333333333333) * (r / s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around 0

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} + \frac{0.75 \cdot e^{\frac{\frac{r}{s}}{-3}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Applied rewrites 99.6%

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

9. Taylor expanded in s around 0

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

11. Applied rewrites 99.6%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

5. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.125, \frac{1}{e^{0.3333333333333333 \cdot \frac{r}{s}}}, 0.125 \cdot \frac{1}{e^{\frac{r}{s}}}\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   0.125
   (/ 1.0 (exp (* 0.3333333333333333 (/ r s))))
   (* 0.125 (/ 1.0 (exp (/ r s)))))
  (* r (* s PI))))

C

float code(float s, float r) {
	return fmaf(0.125f, (1.0f / expf((0.3333333333333333f * (r / s)))), (0.125f * (1.0f / expf((r / s))))) / (r * (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(fma(Float32(0.125), Float32(Float32(1.0) / exp(Float32(Float32(0.3333333333333333) * Float32(r / s)))), Float32(Float32(0.125) * Float32(Float32(1.0) / exp(Float32(r / s))))) / Float32(r * Float32(s * Float32(pi))))
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

5. Applied rewrites 97.0%

   \[\leadsto \color{blue}{\left(\frac{0.75}{e^{\frac{r}{3 \cdot s}} \cdot 6} + \frac{0.25}{e^{\frac{r}{s}} \cdot 2}\right) \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)}} \]

6. Taylor expanded in s around 0

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

8. Applied rewrites 99.5%

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

Alternative 7: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.125, e^{-\frac{r}{s}}, 0.125 \cdot \frac{1}{e^{0.3333333333333333 \cdot \frac{r}{s}}}\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   0.125
   (exp (- (/ r s)))
   (* 0.125 (/ 1.0 (exp (* 0.3333333333333333 (/ r s))))))
  (* r (* s PI))))

C

float code(float s, float r) {
	return fmaf(0.125f, expf(-(r / s)), (0.125f * (1.0f / expf((0.3333333333333333f * (r / s)))))) / (r * (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(fma(Float32(0.125), exp(Float32(-Float32(r / s))), Float32(Float32(0.125) * Float32(Float32(1.0) / exp(Float32(Float32(0.3333333333333333) * Float32(r / s)))))) / Float32(r * Float32(s * Float32(pi))))
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{-\frac{r}{s}}, \frac{0.25}{2}, \frac{\frac{0.75}{e^{\frac{r}{3 \cdot s}}}}{6}\right)}{s \cdot \pi} \cdot \frac{1}{r}} \]

4. Taylor expanded in s around 0

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

6. Applied rewrites 99.5%

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

Alternative 8: 16.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.75}{6 + 2 \cdot \frac{r}{s}} + \frac{0.25}{e^{\frac{r}{s}} \cdot 2}}{r \cdot s}}{\pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 16.3%

   \[\leadsto \frac{\frac{\frac{0.75}{\color{blue}{6 + 2 \cdot \frac{r}{s}}} + \frac{0.25}{e^{\frac{r}{s}} \cdot 2}}{r \cdot s}}{\pi} \]
9. Add Preprocessing

Alternative 9: 10.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \frac{\frac{\frac{\mathsf{fma}\left(\frac{-0.006944444444444444 + -0.0625}{s}, r, 0.16666666666666666\right)}{\pi} \cdot \left(r \cdot \pi\right) - s \cdot 0.25}{r}}{\left(s \cdot \pi\right) \cdot s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  -1.0
  (/
   (/
    (-
     (*
      (/
       (fma (/ (+ -0.006944444444444444 -0.0625) s) r 0.16666666666666666)
       PI)
      (* r PI))
     (* s 0.25))
    r)
   (* (* s PI) s))))

C

float code(float s, float r) {
	return -1.0f * (((((fmaf(((-0.006944444444444444f + -0.0625f) / s), r, 0.16666666666666666f) / ((float) M_PI)) * (r * ((float) M_PI))) - (s * 0.25f)) / r) / ((s * ((float) M_PI)) * s));
}

Julia

function code(s, r)
	return Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(fma(Float32(Float32(Float32(-0.006944444444444444) + Float32(-0.0625)) / s), r, Float32(0.16666666666666666)) / Float32(pi)) * Float32(r * Float32(pi))) - Float32(s * Float32(0.25))) / r) / Float32(Float32(s * Float32(pi)) * s)))
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -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. Applied rewrites 10.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-0.0625, \frac{r}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Applied rewrites 10.2%

   \[\leadsto -1 \cdot \frac{\frac{\left(\frac{0.16666666666666666}{\pi} + \frac{r}{\pi} \cdot \frac{-0.006944444444444444 + -0.0625}{s}\right) \cdot \left(r \cdot \pi\right) - s \cdot 0.25}{r \cdot \left(s \cdot \pi\right)}}{s} \]

8. Applied rewrites 10.2%

   \[\leadsto -1 \cdot \frac{\frac{\frac{\mathsf{fma}\left(\frac{-0.006944444444444444 + -0.0625}{s}, r, 0.16666666666666666\right)}{\pi} \cdot \left(r \cdot \pi\right) - s \cdot 0.25}{r}}{\color{blue}{\left(s \cdot \pi\right) \cdot s}} \]
9. Add Preprocessing

Alternative 10: 10.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r \cdot \pi} - \frac{\frac{0.16666666666666666}{\pi} + \frac{r}{\pi} \cdot \frac{-0.006944444444444444 + -0.0625}{s}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/ 0.25 (* r PI))
   (/
    (+
     (/ 0.16666666666666666 PI)
     (* (/ r PI) (/ (+ -0.006944444444444444 -0.0625) s)))
    s))
  s))

C

float code(float s, float r) {
	return ((0.25f / (r * ((float) M_PI))) - (((0.16666666666666666f / ((float) M_PI)) + ((r / ((float) M_PI)) * ((-0.006944444444444444f + -0.0625f) / s))) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(r * Float32(pi))) - Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(pi)) + Float32(Float32(r / Float32(pi)) * Float32(Float32(Float32(-0.006944444444444444) + Float32(-0.0625)) / s))) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * single(pi))) - (((single(0.16666666666666666) / single(pi)) + ((r / single(pi)) * ((single(-0.006944444444444444) + single(-0.0625)) / 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 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. Applied rewrites 10.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-0.0625, \frac{r}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Applied rewrites 10.2%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} - \frac{\frac{0.16666666666666666}{\pi} + \frac{r}{\pi} \cdot \frac{-0.006944444444444444 + -0.0625}{s}}{s}}{s}} \]
7. Add Preprocessing

Alternative 11: 10.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \frac{\frac{r \cdot \left(0.16666666666666666 + -0.06944444444444445 \cdot \frac{r}{s}\right) - s \cdot 0.25}{r \cdot \left(s \cdot \pi\right)}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  -1.0
  (/
   (/
    (-
     (* r (+ 0.16666666666666666 (* -0.06944444444444445 (/ r s))))
     (* s 0.25))
    (* r (* s PI)))
   s)))

C

float code(float s, float r) {
	return -1.0f * ((((r * (0.16666666666666666f + (-0.06944444444444445f * (r / s)))) - (s * 0.25f)) / (r * (s * ((float) M_PI)))) / s);
}

Julia

function code(s, r)
	return Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(r * Float32(Float32(0.16666666666666666) + Float32(Float32(-0.06944444444444445) * Float32(r / s)))) - Float32(s * Float32(0.25))) / Float32(r * Float32(s * Float32(pi)))) / s))
end

MATLAB

function tmp = code(s, r)
	tmp = single(-1.0) * ((((r * (single(0.16666666666666666) + (single(-0.06944444444444445) * (r / s)))) - (s * single(0.25))) / (r * (s * single(pi)))) / s);
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -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. Applied rewrites 10.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-0.0625, \frac{r}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Applied rewrites 10.2%

   \[\leadsto -1 \cdot \frac{\frac{\left(\frac{0.16666666666666666}{\pi} + \frac{r}{\pi} \cdot \frac{-0.006944444444444444 + -0.0625}{s}\right) \cdot \left(r \cdot \pi\right) - s \cdot 0.25}{r \cdot \left(s \cdot \pi\right)}}{s} \]

7. Taylor expanded in r around 0

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

9. Applied rewrites 10.2%

   \[\leadsto -1 \cdot \frac{\frac{r \cdot \left(0.16666666666666666 + -0.06944444444444445 \cdot \frac{r}{s}\right) - s \cdot 0.25}{r \cdot \left(s \cdot \pi\right)}}{s} \]
10. Add Preprocessing

Alternative 12: 9.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(-1.0) * ((((single(0.16666666666666666) * (r / (s * single(pi)))) - (single(0.25) * (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 -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. Applied rewrites 10.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-0.0625, \frac{r}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

5. Taylor expanded in r around 0

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

7. Applied rewrites 9.1%

   \[\leadsto -1 \cdot \frac{\frac{0.16666666666666666 \cdot \frac{r}{s \cdot \pi} - 0.25 \cdot \frac{1}{\pi}}{r}}{s} \]
8. Add Preprocessing

Alternative 13: 9.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around inf

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

6. Applied rewrites 9.2%

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

7. Taylor expanded in s around 0

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

9. Applied rewrites 9.1%

   \[\leadsto \frac{\frac{0.25 \cdot \frac{s}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}}{s} \]
10. Add Preprocessing

Alternative 14: 9.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around inf

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

6. Applied rewrites 9.2%

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

8. Applied rewrites 9.2%

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

Alternative 15: 9.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / (s * single(pi))) * (single(1.0) / r);
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{-\frac{r}{s}}, \frac{0.25}{2}, \frac{\frac{0.75}{e^{\frac{r}{3 \cdot s}}}}{6}\right)}{s \cdot \pi} \cdot \frac{1}{r}} \]

4. Taylor expanded in s around inf

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

6. Applied rewrites 9.1%

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

Alternative 16: 9.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

4. Applied rewrites 9.1%

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

6. Applied rewrites 9.1%

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

Alternative 17: 9.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

4. Applied rewrites 9.1%

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

6. Applied rewrites 9.1%

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

Alternative 18: 9.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

4. Applied rewrites 9.1%

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

Reproduce

herbie shell --seed 2025153 
(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))))