Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 4.9s

Alternatives: 18

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. +-commutative N/A

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

   13. distribute-frac-neg N/A

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

   14. mul-1-neg N/A

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

   15. +-commutative N/A

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

5. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

Alternative 7: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

Alternative 8: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

Alternative 9: 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 99.6%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

6. Applied rewrites 99.5%

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

Alternative 10: 41.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.4399999976158142:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.25, s, -0.16666666666666666 \cdot r\right)}{\pi}, s, \frac{r \cdot r}{\pi} \cdot 0.06944444444444445\right)}{\left(s \cdot s\right) \cdot s}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 0.4399999976158142)
   (/
    (/
     (fma
      (/ (fma 0.25 s (* -0.16666666666666666 r)) PI)
      s
      (* (/ (* r r) PI) 0.06944444444444445))
     (* (* s s) s))
    r)
   (/ 0.25 (log (pow (exp r) (* PI s))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 0.4399999976158142f) {
		tmp = (fmaf((fmaf(0.25f, s, (-0.16666666666666666f * r)) / ((float) M_PI)), s, (((r * r) / ((float) M_PI)) * 0.06944444444444445f)) / ((s * s) * s)) / r;
	} else {
		tmp = 0.25f / logf(powf(expf(r), (((float) M_PI) * s)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(0.4399999976158142))
		tmp = Float32(Float32(fma(Float32(fma(Float32(0.25), s, Float32(Float32(-0.16666666666666666) * r)) / Float32(pi)), s, Float32(Float32(Float32(r * r) / Float32(pi)) * Float32(0.06944444444444445))) / Float32(Float32(s * s) * s)) / r);
	else
		tmp = Float32(Float32(0.25) / log((exp(r) ^ Float32(Float32(pi) * s))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 0.439999998

   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 8.2%

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

   5. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 9.6%

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

   if 0.439999998 < r

   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 8.7

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

   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. *-commutative N/A

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

      8. add-log-exp N/A

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

      9. log-pow-rev N/A

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

      10. lower-log.f32 N/A

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

      11. lower-pow.f32 N/A

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

      12. lower-exp.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lift-*.f32 10.3

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

   6. Applied rewrites 10.3%

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

      1. lift-*.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. pow-exp N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. lift-PI.f32 39.0

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

   8. Applied rewrites 39.0%

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

Alternative 11: 10.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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 8.7

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

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. *-commutative N/A

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

   8. add-log-exp N/A

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

   9. log-pow-rev N/A

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

   10. lower-log.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lower-exp.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-*.f32 10.3

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

6. Applied rewrites 10.3%

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

   1. lift-*.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. pow-exp N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-exp.f32 N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-*.f32 10.3

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

8. Applied rewrites 10.3%

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

Alternative 12: 8.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. times-frac N/A

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

   5. mul-1-neg N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lift-neg.f32 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in r around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. pow2 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. mult-flip-rev N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 8.7

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

8. Applied rewrites 8.7%

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

9. Taylor expanded in s around 0

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

    1. lower-/.f32 N/A

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

    2. +-commutative N/A

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

    3. associate-*r/ N/A

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

    4. associate-*r/ N/A

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

    5. div-add-rev N/A

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

    6. lower-/.f32 N/A

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

    7. lower-fma.f32 N/A

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

    8. lower-*.f32 N/A

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

    9. lift-PI.f32 N/A

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

    10. pow2 N/A

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

    11. lift-*.f32 8.8

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

11. Applied rewrites 8.8%

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

Alternative 13: 8.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \frac{\mathsf{fma}\left(-1.3333333333333333, \frac{r}{s}, 2\right)}{\pi \cdot s}}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (* 0.125 (/ (fma -1.3333333333333333 (/ r s) 2.0) (* PI s))) r))

C

float code(float s, float r) {
	return (0.125f * (fmaf(-1.3333333333333333f, (r / s), 2.0f) / (((float) M_PI) * s))) / r;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(fma(Float32(-1.3333333333333333), Float32(r / s), Float32(2.0)) / Float32(Float32(pi) * s))) / r)
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{8} \cdot \frac{\color{blue}{2 + \left(-1 \cdot \frac{r}{s} + \frac{-1}{3} \cdot \frac{r}{s}\right)}}{\pi \cdot s}}{r} \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{\left(-1 \cdot \frac{r}{s} + \frac{-1}{3} \cdot \frac{r}{s}\right) + \color{blue}{2}}{\pi \cdot s}}{r} \]

   2. distribute-rgt-out N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{\frac{r}{s} \cdot \left(-1 + \frac{-1}{3}\right) + 2}{\pi \cdot s}}{r} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{\frac{r}{s} \cdot \frac{-4}{3} + 2}{\pi \cdot s}}{r} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{\frac{-4}{3} \cdot \frac{r}{s} + 2}{\pi \cdot s}}{r} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{\mathsf{fma}\left(\frac{-4}{3}, \color{blue}{\frac{r}{s}}, 2\right)}{\pi \cdot s}}{r} \]

   6. lift-/.f32 8.7

      \[\leadsto \frac{0.125 \cdot \frac{\mathsf{fma}\left(-1.3333333333333333, \frac{r}{\color{blue}{s}}, 2\right)}{\pi \cdot s}}{r} \]

6. Applied rewrites 8.7%

   \[\leadsto \frac{0.125 \cdot \frac{\color{blue}{\mathsf{fma}\left(-1.3333333333333333, \frac{r}{s}, 2\right)}}{\pi \cdot s}}{r} \]
7. Add Preprocessing

Alternative 14: 8.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 8.8%

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

Alternative 15: 8.7% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / s) * (single(1.0) / 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 8.7

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

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. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.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

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 8.7

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

8. Applied rewrites 8.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. metadata-eval N/A

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

   4. times-frac N/A

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

   5. mult-flip-rev N/A

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

   6. lower-*.f32 N/A

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

   7. mult-flip-rev N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 8.7

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

10. Applied rewrites 8.7%

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

Alternative 16: 8.7% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-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

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/r* N/A

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

   7. mult-flip-rev N/A

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

   8. lower-/.f32 N/A

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

   9. mult-flip-rev N/A

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

   10. associate-/r* N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-/.f32 8.7

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

6. Applied rewrites 8.7%

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

Alternative 17: 8.7% accurate, 6.4× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-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. Add Preprocessing

Alternative 18: 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 99.6%

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

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 8.7

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

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. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.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

Reproduce

herbie shell --seed 2025129 
(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))))