Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.6%

Time: 4.1s

Alternatives: 11

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return fma(Float32(exp(Float32(Float32(-r) / Float32(Float32(3.0) * s))) / Float32(Float32(Float32(Float32(pi) * Float32(6.0)) * s) * r)), Float32(0.75), 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. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 99.5

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return fma(Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(Float32(Float32(Float32(pi) * Float32(6.0)) * s) * r)), Float32(0.75), 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. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 99.5

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 99.5

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

9. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp(((r / s) * single(-0.3333333333333333))) + exp((-r / s))) / ((single(pi) * s) * r)) * single(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. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

10. Applied rewrites 99.5%

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

Alternative 6: 42.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.7

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

4. Applied rewrites 8.7%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 8.7

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

6. Applied rewrites 8.7%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 8.7

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

8. Applied rewrites 8.7%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lift-PI.f32 42.5

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

10. Applied rewrites 42.5%

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

Alternative 7: 9.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.7

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

4. Applied rewrites 8.7%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. add-log-exp N/A

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

   8. log-pow-rev N/A

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

   9. lower-log.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 9.8

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

6. Applied rewrites 9.8%

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

Alternative 8: 9.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r} + \left(-\frac{\mathsf{fma}\left(\frac{r \cdot 0.5555555555555556}{s}, -0.125, 0.16666666666666666\right)}{s}\right)}{\pi \cdot s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/ 0.25 r)
   (- (/ (fma (/ (* r 0.5555555555555556) s) -0.125 0.16666666666666666) s)))
  (* PI s)))

C

float code(float s, float r) {
	return ((0.25f / r) + -(fmaf(((r * 0.5555555555555556f) / s), -0.125f, 0.16666666666666666f) / s)) / (((float) M_PI) * s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / r) + Float32(-Float32(fma(Float32(Float32(r * Float32(0.5555555555555556)) / s), Float32(-0.125), Float32(0.16666666666666666)) / s))) / Float32(Float32(pi) * 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. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in s around -inf

   \[\leadsto \frac{-1 \cdot \frac{\frac{1}{6} + \frac{-1}{8} \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s} + \frac{1}{4} \cdot \frac{1}{r}}{\color{blue}{\pi} \cdot s} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r} + -1 \cdot \frac{\frac{1}{6} + \frac{-1}{8} \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}}{\pi \cdot s} \]

    2. lower-+.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r} + -1 \cdot \frac{\frac{1}{6} + \frac{-1}{8} \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}}{\pi \cdot s} \]

    3. associate-*r/ N/A

       \[\leadsto \frac{\frac{\frac{1}{4} \cdot 1}{r} + -1 \cdot \frac{\frac{1}{6} + \frac{-1}{8} \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}}{\pi \cdot s} \]

    4. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{1}{4}}{r} + -1 \cdot \frac{\frac{1}{6} + \frac{-1}{8} \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}}{\pi \cdot s} \]

    5. lower-/.f32 N/A

       \[\leadsto \frac{\frac{\frac{1}{4}}{r} + -1 \cdot \frac{\frac{1}{6} + \frac{-1}{8} \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}}{\pi \cdot s} \]

    6. mul-1-neg N/A

       \[\leadsto \frac{\frac{\frac{1}{4}}{r} + \left(\mathsf{neg}\left(\frac{\frac{1}{6} + \frac{-1}{8} \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right)\right)}{\pi \cdot s} \]

    7. lower-neg.f32 N/A

       \[\leadsto \frac{\frac{\frac{1}{4}}{r} + \left(-\frac{\frac{1}{6} + \frac{-1}{8} \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right)}{\pi \cdot s} \]

    8. lower-/.f32 N/A

       \[\leadsto \frac{\frac{\frac{1}{4}}{r} + \left(-\frac{\frac{1}{6} + \frac{-1}{8} \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right)}{\pi \cdot s} \]

11. Applied rewrites 9.5%

    \[\leadsto \frac{\frac{0.25}{r} + \left(-\frac{\mathsf{fma}\left(\frac{r \cdot 0.5555555555555556}{s}, -0.125, 0.16666666666666666\right)}{s}\right)}{\color{blue}{\pi} \cdot s} \]
12. Add Preprocessing

Alternative 9: 8.7% accurate, 5.7× speedup

Math

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

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(Float32(0.25) / Float32(pi)) / s) / r)
end

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 8.1%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   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 8.6

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

7. Applied rewrites 8.6%

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

8. Taylor expanded in s around inf

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

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 8.7

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

10. Applied rewrites 8.7%

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

Alternative 10: 8.7% 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. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.7

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

4. Applied rewrites 8.7%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 8.7

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

6. Applied rewrites 8.7%

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

Alternative 11: 8.7% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.7

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

4. Applied rewrites 8.7%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 8.7

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

6. Applied rewrites 8.7%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 8.7

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

8. Applied rewrites 8.7%

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

Reproduce

herbie shell --seed 2025120 
(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))))