Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 4.3s

Alternatives: 14

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

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

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(Float32(-r) / 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

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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 99.6

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

3. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

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 99.6

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

4. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

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 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-*.f32 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 99.6

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

9. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 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 99.6%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi \cdot s}}{r}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

Alternative 5: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 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 99.6%

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

Alternative 6: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 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 99.6%

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

6. Applied rewrites 99.6%

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

Alternative 7: 10.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - \left(\frac{\mathsf{fma}\left(-0.006944444444444444, \frac{r}{s}, 0.041666666666666664\right)}{s} \cdot r - 0.125\right)}{\pi \cdot s}}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (/
   (-
    (* 0.125 (exp (/ (- r) s)))
    (-
     (* (/ (fma -0.006944444444444444 (/ r s) 0.041666666666666664) s) r)
     0.125))
   (* PI s))
  r))

C

float code(float s, float r) {
	return (((0.125f * expf((-r / s))) - (((fmaf(-0.006944444444444444f, (r / s), 0.041666666666666664f) / s) * r) - 0.125f)) / (((float) M_PI) * s)) / r;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(0.125) * exp(Float32(Float32(-r) / s))) - Float32(Float32(Float32(fma(Float32(-0.006944444444444444), Float32(r / s), Float32(0.041666666666666664)) / s) * r) - Float32(0.125))) / Float32(Float32(pi) * s)) / r)
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 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 99.6%

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

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(r \cdot \left(\frac{-1}{144} \cdot \frac{r}{{s}^{2}} + \frac{1}{24} \cdot \frac{1}{s}\right) - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]
6. Step-by-step derivation

   1. lower--.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(r \cdot \left(\frac{-1}{144} \cdot \frac{r}{{s}^{2}} + \frac{1}{24} \cdot \frac{1}{s}\right) - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\left(\frac{-1}{144} \cdot \frac{r}{{s}^{2}} + \frac{1}{24} \cdot \frac{1}{s}\right) \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\left(\frac{-1}{144} \cdot \frac{r}{{s}^{2}} + \frac{1}{24} \cdot \frac{1}{s}\right) \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\left(\frac{r}{{s}^{2}} \cdot \frac{-1}{144} + \frac{1}{24} \cdot \frac{1}{s}\right) \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\mathsf{fma}\left(\frac{r}{{s}^{2}}, \frac{-1}{144}, \frac{1}{24} \cdot \frac{1}{s}\right) \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   6. pow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{-1}{144}, \frac{1}{24} \cdot \frac{1}{s}\right) \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{-1}{144}, \frac{1}{24} \cdot \frac{1}{s}\right) \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{-1}{144}, \frac{1}{24} \cdot \frac{1}{s}\right) \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{-1}{144}, \frac{\frac{1}{24} \cdot 1}{s}\right) \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{-1}{144}, \frac{\frac{1}{24}}{s}\right) \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   11. lower-/.f32 10.3

       \[\leadsto \frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - \left(\mathsf{fma}\left(\frac{r}{s \cdot s}, -0.006944444444444444, \frac{0.041666666666666664}{s}\right) \cdot r - 0.125\right)}{\pi \cdot s}}{r} \]

7. Applied rewrites 10.3%

   \[\leadsto \frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - \left(\mathsf{fma}\left(\frac{r}{s \cdot s}, -0.006944444444444444, \frac{0.041666666666666664}{s}\right) \cdot r - 0.125\right)}{\pi \cdot s}}{r} \]

8. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\frac{\frac{1}{24} + \frac{-1}{144} \cdot \frac{r}{s}}{s} \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]
9. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\frac{\frac{1}{24} + \frac{-1}{144} \cdot \frac{r}{s}}{s} \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\frac{\frac{-1}{144} \cdot \frac{r}{s} + \frac{1}{24}}{s} \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}} - \left(\frac{\mathsf{fma}\left(\frac{-1}{144}, \frac{r}{s}, \frac{1}{24}\right)}{s} \cdot r - \frac{1}{8}\right)}{\pi \cdot s}}{r} \]

   4. lower-/.f32 10.3

      \[\leadsto \frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - \left(\frac{\mathsf{fma}\left(-0.006944444444444444, \frac{r}{s}, 0.041666666666666664\right)}{s} \cdot r - 0.125\right)}{\pi \cdot s}}{r} \]

10. Applied rewrites 10.3%

    \[\leadsto \frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - \left(\frac{\mathsf{fma}\left(-0.006944444444444444, \frac{r}{s}, 0.041666666666666664\right)}{s} \cdot r - 0.125\right)}{\pi \cdot s}}{r} \]
11. Add Preprocessing

Alternative 8: 9.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
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 8.3%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 9.6%

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

8. Taylor expanded in r around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

10. Applied rewrites 9.7%

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

Alternative 9: 9.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 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 99.6%

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

5. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. pow2 N/A

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

   8. lower-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 9.7

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

7. Applied rewrites 9.7%

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

Alternative 10: 9.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 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 99.6%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 9.2%

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

Alternative 11: 8.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
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 8.3%

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

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 8.8

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

7. Applied rewrites 8.8%

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

Alternative 12: 8.8% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in r around 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 99.6%

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

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{\frac{1}{4} + \frac{-1}{6} \cdot \frac{r}{s}}{\pi \cdot s}}{r} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-/.f32 8.8

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

7. Applied rewrites 8.8%

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

Alternative 13: 8.8% 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 99.6%

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

2. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
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 8.3%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 9.6%

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

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 8.8

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

10. Applied rewrites 8.8%

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

Alternative 14: 8.8% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in 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 8.8

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

4. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
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 8.8

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

6. Applied rewrites 8.8%

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

Reproduce

herbie shell --seed 2025101 
(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))))