Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.2s

Alternatives: 12

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.125, \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi}, \frac{\frac{e^{\frac{r}{s \cdot -3}}}{r} \cdot 0.75}{\pi \cdot 6}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   0.125
   (/ (exp (* -1.0 (/ r s))) (* r PI))
   (/ (* (/ (exp (/ r (* s -3.0))) r) 0.75) (* PI 6.0)))
  s))

C

float code(float s, float r) {
	return fmaf(0.125f, (expf((-1.0f * (r / s))) / (r * ((float) M_PI))), (((expf((r / (s * -3.0f))) / r) * 0.75f) / (((float) M_PI) * 6.0f))) / s;
}

Julia

function code(s, r)
	return Float32(fma(Float32(0.125), Float32(exp(Float32(Float32(-1.0) * Float32(r / s))) / Float32(r * Float32(pi))), Float32(Float32(Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / r) * Float32(0.75)) / Float32(Float32(pi) * Float32(6.0)))) / s)
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(0.125, \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi}, \frac{\frac{e^{\frac{r}{s \cdot -3}}}{r} \cdot 0.75}{\pi \cdot 6}\right)}{s} \]
7. Add Preprocessing

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 99.5

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

6. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}}{\pi \cdot r} \cdot 0.125}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}}{\pi \cdot r} \cdot 0.125}{s}} \]
6. Add Preprocessing

Alternative 5: 42.7% 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 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 9.1

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

7. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. add-log-exp N/A

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

   4. log-pow-rev N/A

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

   5. lower-log.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-exp.f32 42.7

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

9. Applied rewrites 42.7%

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

Alternative 6: 9.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(fma(Float32(0.125), Float32(exp(Float32(Float32(-1.0) * Float32(r / s))) / Float32(r * Float32(pi))), Float32(Float32(0.125) / Float32(r * 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 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 9.5

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

7. Applied rewrites 9.5%

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

Alternative 7: 9.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-PI.f32 9.0

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

7. Applied rewrites 9.0%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, 0.25 \cdot \frac{1}{\pi}\right)}{r}}{s} \]
8. Add Preprocessing

Alternative 8: 9.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 9.0

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. mult-flip-rev N/A

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

   9. lift-/.f32 9.0

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 9.0

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. mult-flip-rev N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 N/A

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

   19. lower-/.f32 9.0

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

6. Applied rewrites 9.0%

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

Alternative 9: 9.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. mult-flip-rev N/A

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

   4. lift-/.f32 9.0

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 9.0

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

   8. lift-*.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. mult-flip-rev N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. lower-/.f32 9.0

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

6. Applied rewrites 9.0%

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

Alternative 10: 9.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 9.1

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

6. Applied rewrites 9.1%

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

Alternative 11: 9.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 9.1

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

6. Applied rewrites 9.1%

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

Alternative 12: 9.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.1

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

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 2025142 
(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))))