Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 7.0s

Alternatives: 13

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

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: 99.5% accurate, 1.1× speedup

Math

\[\mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{r \cdot 18.84955596923828}, \frac{1}{s}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]

FPCore

(FPCore (s r)
  :precision binary32
  (fma
 (/ (* (exp (/ r (* -3.0 s))) 0.75) (* r 18.84955596923828))
 (/ 1.0 s)
 (/ (/ 0.125 (* (* PI s) (exp (/ r s)))) r)))

C

float code(float s, float r) {
	return fmaf(((expf((r / (-3.0f * s))) * 0.75f) / (r * 18.84955596923828f)), (1.0f / s), ((0.125f / ((((float) M_PI) * s) * expf((r / s)))) / r));
}

Julia

function code(s, r)
	return fma(Float32(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) * Float32(0.75)) / Float32(r * Float32(18.84955596923828))), Float32(Float32(1.0) / s), Float32(Float32(Float32(0.125) / Float32(Float32(Float32(pi) * s) * exp(Float32(r / s)))) / r))
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.3%

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

4. Evaluated real constant 99.3%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{r \cdot \color{blue}{18.84955596923828}}, \frac{1}{s}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 4: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((single(-1.0) * (r / s))) + exp((single(-0.3333333333333333) * (r / s)))) / (r * (s * single(pi))));
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
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 99.4%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 99.2%

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

   6. lift-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-out N/A

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

   9. *-commutative N/A

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

6. Applied rewrites 99.2%

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

7. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 99.5%

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

9. Applied rewrites 99.5%

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

Alternative 5: 99.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
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 99.4%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 99.4%

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

Alternative 6: 42.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0%

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 9.0%

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 9.0%

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

6. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-exp.f32 42.9%

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

8. Applied rewrites 42.9%

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

Alternative 7: 41.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0%

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 9.0%

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

6. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-PI.f32 N/A

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

   5. add-log-exp N/A

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

   6. log-pow-rev N/A

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

   7. log-pow-rev N/A

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

   8. lower-log.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. pow-exp N/A

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

   12. lift-*.f32 N/A

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

   13. lower-exp.f32 41.7%

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

8. Applied rewrites 41.7%

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

Alternative 8: 9.5% accurate, 1.7× speedup

Math

\[\frac{\mathsf{fma}\left(0.75, \frac{0.16666666666666666}{\pi \cdot s}, \frac{0.125}{e^{\frac{r}{s}} \cdot \left(\pi \cdot s\right)}\right)}{r} \]

FPCore

(FPCore (s r)
  :precision binary32
  (/
 (fma
  0.75
  (/ 0.16666666666666666 (* PI s))
  (/ 0.125 (* (exp (/ r s)) (* PI s))))
 r))

C

float code(float s, float r) {
	return fmaf(0.75f, (0.16666666666666666f / (((float) M_PI) * s)), (0.125f / (expf((r / s)) * (((float) M_PI) * s)))) / r;
}

Julia

function code(s, r)
	return Float32(fma(Float32(0.75), Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * s)), Float32(Float32(0.125) / Float32(exp(Float32(r / s)) * Float32(Float32(pi) * s)))) / r)
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

4. Applied rewrites 9.5%

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

6. Applied rewrites 9.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.75, \frac{0.16666666666666666}{\pi \cdot s}, \frac{0.125}{e^{\frac{r}{s}} \cdot \left(\pi \cdot s\right)}\right)}{r}} \]
7. Add Preprocessing

Alternative 9: 9.4% accurate, 2.0× speedup

Math

\[\mathsf{fma}\left(\frac{0.75}{r \cdot \left(6 \cdot \pi\right)}, \frac{1}{s}, \frac{\frac{0.125}{\mathsf{fma}\left(r, \pi, s \cdot \pi\right)}}{r}\right) \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in s around inf

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{3}{4}}}{r \cdot \left(6 \cdot \pi\right)}, \frac{1}{s}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]
5. Step-by-step derivation

6. Applied rewrites 9.5%

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

7. Taylor expanded in r around 0

   \[\leadsto \mathsf{fma}\left(\frac{0.75}{r \cdot \left(6 \cdot \pi\right)}, \frac{1}{s}, \frac{\frac{0.125}{\color{blue}{r \cdot \pi + s \cdot \pi}}}{r}\right) \]
8. Step-by-step derivation

   1. lower-fma.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.4%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{r \cdot \left(6 \cdot \pi\right)}, \frac{1}{s}, \frac{\frac{0.125}{\mathsf{fma}\left(r, \pi, s \cdot \pi\right)}}{r}\right) \]

9. Applied rewrites 9.4%

   \[\leadsto \mathsf{fma}\left(\frac{0.75}{r \cdot \left(6 \cdot \pi\right)}, \frac{1}{s}, \frac{\frac{0.125}{\color{blue}{\mathsf{fma}\left(r, \pi, s \cdot \pi\right)}}}{r}\right) \]
10. Add Preprocessing

Alternative 10: 9.0% accurate, 2.6× speedup

Math

\[\frac{1}{s} \cdot \left(\frac{2 + -1.3333333333333333 \cdot \frac{r}{s}}{\pi \cdot r} \cdot 0.125\right) \]

FPCore

(FPCore (s r)
  :precision binary32
  (*
 (/ 1.0 s)
 (* (/ (+ 2.0 (* -1.3333333333333333 (/ r s))) (* PI r)) 0.125)))

C

float code(float s, float r) {
	return (1.0f / s) * (((2.0f + (-1.3333333333333333f * (r / s))) / (((float) M_PI) * r)) * 0.125f);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(1.0) / s) * Float32(Float32(Float32(Float32(2.0) + Float32(Float32(-1.3333333333333333) * Float32(r / s))) / Float32(Float32(pi) * r)) * Float32(0.125)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / s) * (((single(2.0) + (single(-1.3333333333333333) * (r / s))) / (single(pi) * r)) * single(0.125));
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
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 99.4%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 99.2%

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

   6. lift-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-out N/A

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

   9. *-commutative N/A

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

6. Applied rewrites 99.2%

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

7. Taylor expanded in r around 0

   \[\leadsto \frac{1}{s} \cdot \left(\frac{2 + \frac{-4}{3} \cdot \frac{r}{s}}{\pi \cdot r} \cdot 0.125\right) \]
8. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 9.0%

      \[\leadsto \frac{1}{s} \cdot \left(\frac{2 + -1.3333333333333333 \cdot \frac{r}{s}}{\pi \cdot r} \cdot 0.125\right) \]

9. Applied rewrites 9.0%

   \[\leadsto \frac{1}{s} \cdot \left(\frac{2 + -1.3333333333333333 \cdot \frac{r}{s}}{\pi \cdot r} \cdot 0.125\right) \]
10. Add Preprocessing

Alternative 11: 9.0% accurate, 2.7× speedup

Math

\[\frac{0.25}{\left(6.2831854820251465 \cdot s\right) \cdot r} + \frac{0.75}{\pi \cdot \left(6 \cdot \left(s \cdot r\right)\right)} \]

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (/ 0.25 (* (* 6.2831854820251465 s) r))
 (/ 0.75 (* PI (* 6.0 (* s r))))))

C

float code(float s, float r) {
	return (0.25f / ((6.2831854820251465f * s) * r)) + (0.75f / (((float) M_PI) * (6.0f * (s * r))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / Float32(Float32(Float32(6.2831854820251465) * s) * r)) + Float32(Float32(0.75) / Float32(Float32(pi) * Float32(Float32(6.0) * Float32(s * r)))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / ((single(6.2831854820251465) * s) * r)) + (single(0.75) / (single(pi) * (single(6.0) * (s * r))));
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

4. Applied rewrites 9.5%

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

5. Taylor expanded in s around inf

   \[\leadsto \frac{\color{blue}{\frac{1}{4}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Applied rewrites 9.0%

   \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 9.0%

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

9. Applied rewrites 9.0%

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

10. Evaluated real constant 9.0%

    \[\leadsto \frac{0.25}{\left(\color{blue}{6.2831854820251465} \cdot s\right) \cdot r} + \frac{0.75}{\pi \cdot \left(6 \cdot \left(s \cdot r\right)\right)} \]
11. Add Preprocessing

Alternative 12: 9.0% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0%

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 9.0%

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

6. Applied rewrites 9.0%

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

Alternative 13: 9.0% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0%

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

4. Applied rewrites 9.0%

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

Reproduce

herbie shell --seed 2025212 
(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))))