Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 7.8s

Alternatives: 21

Speedup: 1.1×

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
  :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))))

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

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

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
  :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))))

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

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

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (fma
 (/ (* (exp (/ r (* -3.0 s))) 0.75) s)
 (/ 1.0 (* 18.84955596923828 r))
 (/ (/ 0.125 (* (* PI s) (exp (/ r s)))) r)))

C

float code(float s, float r) {
	return fmaf(((expf((r / (-3.0f * s))) * 0.75f) / s), (1.0f / (18.84955596923828f * r)), ((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)) / s), Float32(Float32(1.0) / Float32(Float32(18.84955596923828) * r)), Float32(Float32(Float32(0.125) / Float32(Float32(Float32(pi) * s) * exp(Float32(r / 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 99.6%

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

4. Evaluated real constant 99.6%

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

Alternative 2: 99.6% accurate, 1.1× speedup

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

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (fma
 (/ (* (exp (/ r (* -3.0 s))) 0.75) s)
 (/ 1.0 (* 18.84955596923828 r))
 (/ 0.125 (* (* (* PI s) (exp (/ r s))) r))))

C

float code(float s, float r) {
	return fmaf(((expf((r / (-3.0f * s))) * 0.75f) / s), (1.0f / (18.84955596923828f * r)), (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)) / s), Float32(Float32(1.0) / Float32(Float32(18.84955596923828) * r)), Float32(Float32(0.125) / Float32(Float32(Float32(Float32(pi) * s) * exp(Float32(r / 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 99.6%

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

4. Evaluated real constant 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{s}, \frac{1}{18.84955596923828 \cdot r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]
5. Step-by-step derivation

6. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.4× speedup

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

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (/
 (* 0.125 (+ (exp (/ r (* -3.0 s))) (exp (/ (- r) s))))
 (* (* s r) PI)))

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp((r / (single(-3.0) * s))) + exp((-r / s)))) / ((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. Step-by-step derivation

3. Applied rewrites 99.6%

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

   \[\leadsto \frac{0.125 \cdot \left(e^{\frac{-0.3333333333333333}{s} \cdot r} + e^{\frac{-r}{s}}\right)}{\left(s \cdot r\right) \cdot \pi} \]
8. Step-by-step derivation

9. Applied rewrites 99.6%

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

Alternative 4: 99.5% accurate, 1.4× speedup

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

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (/
 (* 0.125 (+ (exp (* (/ -0.3333333333333333 s) r)) (exp (/ (- r) s))))
 (* (* s r) PI)))

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp(((single(-0.3333333333333333) / s) * r)) + exp((-r / s)))) / ((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. Step-by-step derivation

3. Applied rewrites 99.6%

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 5: 95.5% accurate, 0.6× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4 \cdot s, \pi, \mathsf{fma}\left(-8 \cdot r, \frac{\pi}{s} \cdot -0.08333333333333333, 2.6666666666666665 \cdot \pi\right) \cdot r\right) \cdot r}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (if (<=
     (+
      (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
      (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
     0.0)
  (/ 0.0 (* r (* s PI)))
  (/
   1.0
   (*
    (fma
     (* 4.0 s)
     PI
     (*
      (fma
       (* -8.0 r)
       (* (/ PI s) -0.08333333333333333)
       (* 2.6666666666666665 PI))
      r))
    r))))

C

float code(float s, float r) {
	float tmp;
	if ((((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))) <= 0.0f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = 1.0f / (fmaf((4.0f * s), ((float) M_PI), (fmaf((-8.0f * r), ((((float) M_PI) / s) * -0.08333333333333333f), (2.6666666666666665f * ((float) M_PI))) * r)) * r);
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (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))) <= Float32(0.0))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(1.0) / Float32(fma(Float32(Float32(4.0) * s), Float32(pi), Float32(fma(Float32(Float32(-8.0) * r), Float32(Float32(Float32(pi) / s) * Float32(-0.08333333333333333)), Float32(Float32(2.6666666666666665) * Float32(pi))) * r)) * r));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   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

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) 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 0

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}{s} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in r around 0

      \[\leadsto \frac{1}{r \cdot \left(4 \cdot \left(s \cdot \pi\right) + r \cdot \left(-8 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\pi}{s} + \frac{5}{36} \cdot \frac{\pi}{s}\right)\right) + \frac{8}{3} \cdot \pi\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 26.2%

      \[\leadsto \frac{1}{r \cdot \mathsf{fma}\left(4, s \cdot \pi, r \cdot \mathsf{fma}\left(-8, r \cdot \mathsf{fma}\left(-0.2222222222222222, \frac{\pi}{s}, 0.1388888888888889 \cdot \frac{\pi}{s}\right), 2.6666666666666665 \cdot \pi\right)\right)} \]

   11. Applied rewrites 26.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(4 \cdot s, \pi, \mathsf{fma}\left(-8 \cdot r, \frac{\pi}{s} \cdot -0.08333333333333333, 2.6666666666666665 \cdot \pi\right) \cdot r\right) \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.5% accurate, 0.6× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(-8 \cdot r, \frac{\pi}{s} \cdot -0.08333333333333333, 2.6666666666666665 \cdot \pi\right), \left(\pi \cdot s\right) \cdot 4\right)}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (if (<=
     (+
      (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
      (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
     0.0)
  (/ 0.0 (* r (* s PI)))
  (/
   1.0
   (*
    r
    (fma
     r
     (fma
      (* -8.0 r)
      (* (/ PI s) -0.08333333333333333)
      (* 2.6666666666666665 PI))
     (* (* PI s) 4.0))))))

C

float code(float s, float r) {
	float tmp;
	if ((((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))) <= 0.0f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = 1.0f / (r * fmaf(r, fmaf((-8.0f * r), ((((float) M_PI) / s) * -0.08333333333333333f), (2.6666666666666665f * ((float) M_PI))), ((((float) M_PI) * s) * 4.0f)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (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))) <= Float32(0.0))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(1.0) / Float32(r * fma(r, fma(Float32(Float32(-8.0) * r), Float32(Float32(Float32(pi) / s) * Float32(-0.08333333333333333)), Float32(Float32(2.6666666666666665) * Float32(pi))), Float32(Float32(Float32(pi) * s) * Float32(4.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   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

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) 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 0

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}{s} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in r around 0

      \[\leadsto \frac{1}{r \cdot \left(4 \cdot \left(s \cdot \pi\right) + r \cdot \left(-8 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\pi}{s} + \frac{5}{36} \cdot \frac{\pi}{s}\right)\right) + \frac{8}{3} \cdot \pi\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 26.2%

      \[\leadsto \frac{1}{r \cdot \mathsf{fma}\left(4, s \cdot \pi, r \cdot \mathsf{fma}\left(-8, r \cdot \mathsf{fma}\left(-0.2222222222222222, \frac{\pi}{s}, 0.1388888888888889 \cdot \frac{\pi}{s}\right), 2.6666666666666665 \cdot \pi\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 26.8%

       \[\leadsto \frac{1}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(-8 \cdot r, \frac{\pi}{s} \cdot -0.08333333333333333, 2.6666666666666665 \cdot \pi\right), \left(\pi \cdot s\right) \cdot 4\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.6% accurate, 0.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot \mathsf{fma}\left(4 \cdot s, \pi, 2.6666666666666665 \cdot \left(\pi \cdot r\right)\right)}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (if (<=
     (+
      (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
      (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
     0.0)
  (/ 0.0 (* r (* s PI)))
  (/ 1.0 (* r (fma (* 4.0 s) PI (* 2.6666666666666665 (* PI r)))))))

C

float code(float s, float r) {
	float tmp;
	if ((((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))) <= 0.0f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = 1.0f / (r * fmaf((4.0f * s), ((float) M_PI), (2.6666666666666665f * (((float) M_PI) * r))));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (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))) <= Float32(0.0))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(1.0) / Float32(r * fma(Float32(Float32(4.0) * s), Float32(pi), Float32(Float32(2.6666666666666665) * Float32(Float32(pi) * r)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   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

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) 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 0

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}{s} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in r around 0

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

   9. Applied rewrites 12.1%

      \[\leadsto \frac{1}{r \cdot \mathsf{fma}\left(2.6666666666666665, r \cdot \pi, 4 \cdot \left(s \cdot \pi\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 12.1%

       \[\leadsto \frac{1}{r \cdot \mathsf{fma}\left(4 \cdot s, \pi, 2.6666666666666665 \cdot \left(\pi \cdot r\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.6% accurate, 0.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot \mathsf{fma}\left(2.6666666666666665, r \cdot \pi, 4 \cdot \left(s \cdot \pi\right)\right)}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (if (<=
     (+
      (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
      (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
     0.0)
  (/ 0.0 (* r (* s PI)))
  (/ 1.0 (* r (fma 2.6666666666666665 (* r PI) (* 4.0 (* s PI)))))))

C

float code(float s, float r) {
	float tmp;
	if ((((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))) <= 0.0f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = 1.0f / (r * fmaf(2.6666666666666665f, (r * ((float) M_PI)), (4.0f * (s * ((float) M_PI)))));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (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))) <= Float32(0.0))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(1.0) / Float32(r * fma(Float32(2.6666666666666665), Float32(r * Float32(pi)), Float32(Float32(4.0) * Float32(s * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   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

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) 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 0

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}{s} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in r around 0

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

   9. Applied rewrites 12.1%

      \[\leadsto \frac{1}{r \cdot \mathsf{fma}\left(2.6666666666666665, r \cdot \pi, 4 \cdot \left(s \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 93.5% accurate, 1.4× speedup

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

Math

\[\mathsf{fma}\left(0.75, \frac{e^{\frac{r}{-3 \cdot s}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (fma
 0.75
 (/ (exp (/ r (* -3.0 s))) (* 6.0 (* r (* s PI))))
 (/ 0.0 (* (* (+ PI PI) s) r))))

C

float code(float s, float r) {
	return fmaf(0.75f, (expf((r / (-3.0f * s))) / (6.0f * (r * (s * ((float) M_PI))))), (0.0f / (((((float) M_PI) + ((float) M_PI)) * s) * r)));
}

Julia

function code(s, r)
	return fma(Float32(0.75), Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(Float32(6.0) * Float32(r * Float32(s * Float32(pi))))), Float32(Float32(0.0) / Float32(Float32(Float32(Float32(pi) + Float32(pi)) * s) * r)))
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

4. Applied rewrites 9.0%

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

5. Taylor expanded in undef-var around zero

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Applied rewrites 93.5%

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

9. Applied rewrites 93.5%

   \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{\frac{r}{-3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

10. Taylor expanded in s around 0

    \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{\frac{r}{-3 \cdot s}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]
11. Step-by-step derivation

12. Applied rewrites 93.5%

    \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{\frac{r}{-3 \cdot s}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]
13. Add Preprocessing

Alternative 10: 93.5% accurate, 1.5× speedup

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

Math

\[\frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{-3 \cdot s}}}{r \cdot \left(s \cdot \pi\right)} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (+
 (/ 0.0 (* (* (* 2.0 PI) s) r))
 (* 0.125 (/ (exp (/ r (* -3.0 s))) (* r (* s PI))))))

C

float code(float s, float r) {
	return (0.0f / (((2.0f * ((float) M_PI)) * s) * r)) + (0.125f * (expf((r / (-3.0f * s))) / (r * (s * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.0) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(0.125) * Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(r * Float32(s * Float32(pi))))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.0) / (((single(2.0) * single(pi)) * s) * r)) + (single(0.125) * (exp((r / (single(-3.0) * s))) / (r * (s * single(pi)))));
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

4. Applied rewrites 9.0%

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

5. Taylor expanded in undef-var around zero

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Applied rewrites 93.5%

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Taylor expanded in s around 0

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
9. Step-by-step derivation

10. Applied rewrites 93.5%

    \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
11. Step-by-step derivation

12. Applied rewrites 93.5%

    \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{-3 \cdot s}}}{r \cdot \left(s \cdot \pi\right)} \]
13. Add Preprocessing

Alternative 11: 93.5% accurate, 1.5× speedup

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

Math

\[\mathsf{fma}\left(\frac{0.75}{18.84955596923828 \cdot s}, \frac{e^{\frac{r}{-3 \cdot s}}}{r}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (fma
 (/ 0.75 (* 18.84955596923828 s))
 (/ (exp (/ r (* -3.0 s))) r)
 (/ 0.0 (* (* (+ PI PI) s) r))))

C

float code(float s, float r) {
	return fmaf((0.75f / (18.84955596923828f * s)), (expf((r / (-3.0f * s))) / r), (0.0f / (((((float) M_PI) + ((float) M_PI)) * s) * r)));
}

Julia

function code(s, r)
	return fma(Float32(Float32(0.75) / Float32(Float32(18.84955596923828) * s)), Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / r), Float32(Float32(0.0) / Float32(Float32(Float32(Float32(pi) + Float32(pi)) * s) * r)))
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

4. Applied rewrites 9.0%

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

5. Taylor expanded in undef-var around zero

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Applied rewrites 93.5%

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

9. Applied rewrites 93.5%

   \[\leadsto \mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{r}{-3 \cdot s}}}{r}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

10. Evaluated real constant 93.5%

    \[\leadsto \mathsf{fma}\left(\frac{0.75}{18.84955596923828 \cdot s}, \frac{e^{\frac{r}{-3 \cdot s}}}{r}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]
11. Add Preprocessing

Alternative 12: 93.5% accurate, 1.5× speedup

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

Math

\[\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{r}{-3 \cdot s}}}{r}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (fma
 (/ 0.125 (* s PI))
 (/ (exp (/ r (* -3.0 s))) r)
 (/ 0.0 (* (* (+ PI PI) s) r))))

C

float code(float s, float r) {
	return fmaf((0.125f / (s * ((float) M_PI))), (expf((r / (-3.0f * s))) / r), (0.0f / (((((float) M_PI) + ((float) M_PI)) * s) * r)));
}

Julia

function code(s, r)
	return fma(Float32(Float32(0.125) / Float32(s * Float32(pi))), Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / r), Float32(Float32(0.0) / Float32(Float32(Float32(Float32(pi) + Float32(pi)) * s) * r)))
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

4. Applied rewrites 9.0%

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

5. Taylor expanded in undef-var around zero

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Applied rewrites 93.5%

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

9. Applied rewrites 93.5%

   \[\leadsto \mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{r}{-3 \cdot s}}}{r}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

10. Taylor expanded in s around 0

    \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{8}}{s \cdot \pi}, \frac{e^{\frac{r}{-3 \cdot s}}}{r}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]
11. Step-by-step derivation

12. Applied rewrites 93.5%

    \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{r}{-3 \cdot s}}}{r}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]
13. Add Preprocessing

Alternative 13: 93.5% accurate, 1.5× speedup

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

Math

\[\mathsf{fma}\left(0.75, \frac{e^{\frac{r}{-3 \cdot s}}}{\left(18.84955596923828 \cdot s\right) \cdot r}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (fma
 0.75
 (/ (exp (/ r (* -3.0 s))) (* (* 18.84955596923828 s) r))
 (/ 0.0 (* (* (+ PI PI) s) r))))

C

float code(float s, float r) {
	return fmaf(0.75f, (expf((r / (-3.0f * s))) / ((18.84955596923828f * s) * r)), (0.0f / (((((float) M_PI) + ((float) M_PI)) * s) * r)));
}

Julia

function code(s, r)
	return fma(Float32(0.75), Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(Float32(Float32(18.84955596923828) * s) * r)), Float32(Float32(0.0) / Float32(Float32(Float32(Float32(pi) + Float32(pi)) * s) * r)))
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

4. Applied rewrites 9.0%

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

5. Taylor expanded in undef-var around zero

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Applied rewrites 93.5%

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

9. Applied rewrites 93.5%

   \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{\frac{r}{-3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]

10. Evaluated real constant 93.5%

    \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{\frac{r}{-3 \cdot s}}}{\left(18.84955596923828 \cdot s\right) \cdot r}, \frac{0}{\left(\left(\pi + \pi\right) \cdot s\right) \cdot r}\right) \]
11. Add Preprocessing

Alternative 14: 93.5% accurate, 1.6× speedup

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

Math

\[\frac{0}{\left(6.2831854820251465 \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(18.84955596923828 \cdot s\right) \cdot r} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (+
 (/ 0.0 (* (* 6.2831854820251465 s) r))
 (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* 18.84955596923828 s) r))))

C

float code(float s, float r) {
	return (0.0f / ((6.2831854820251465f * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / ((18.84955596923828f * s) * r));
}

Fortran

real(4) function code(s, r)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: r
    code = (0.0e0 / ((6.2831854820251465e0 * s) * r)) + ((0.75e0 * exp((-r / (3.0e0 * s)))) / ((18.84955596923828e0 * s) * r))
end function

Julia

function code(s, r)
	return Float32(Float32(Float32(0.0) / Float32(Float32(Float32(6.2831854820251465) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(18.84955596923828) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.0) / ((single(6.2831854820251465) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / ((single(18.84955596923828) * 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

   \[\leadsto \frac{\frac{1}{4}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

4. Applied rewrites 9.0%

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

5. Taylor expanded in undef-var around zero

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Applied rewrites 93.5%

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Evaluated real constant 93.5%

   \[\leadsto \frac{0}{\left(6.2831854820251465 \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

9. Evaluated real constant 93.5%

   \[\leadsto \frac{0}{\left(6.2831854820251465 \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(18.84955596923828 \cdot s\right) \cdot r} \]
10. Add Preprocessing

Alternative 15: 93.5% accurate, 1.6× speedup

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

Math

\[\frac{0}{\left(6.2831854820251465 \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (+
 (/ 0.0 (* (* 6.2831854820251465 s) r))
 (* 0.125 (/ (exp (* -0.3333333333333333 (/ r s))) (* r (* s PI))))))

C

float code(float s, float r) {
	return (0.0f / ((6.2831854820251465f * s) * r)) + (0.125f * (expf((-0.3333333333333333f * (r / s))) / (r * (s * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.0) / Float32(Float32(Float32(6.2831854820251465) * s) * r)) + Float32(Float32(0.125) * Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) / Float32(r * Float32(s * Float32(pi))))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.0) / ((single(6.2831854820251465) * s) * r)) + (single(0.125) * (exp((single(-0.3333333333333333) * (r / s))) / (r * (s * single(pi)))));
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

4. Applied rewrites 9.0%

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

5. Taylor expanded in undef-var around zero

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

7. Applied rewrites 93.5%

   \[\leadsto \frac{0}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Evaluated real constant 93.5%

   \[\leadsto \frac{0}{\left(6.2831854820251465 \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

9. Taylor expanded in s around 0

   \[\leadsto \frac{0}{\left(6.2831854820251465 \cdot s\right) \cdot r} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
10. Step-by-step derivation

11. Applied rewrites 93.5%

    \[\leadsto \frac{0}{\left(6.2831854820251465 \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
12. Add Preprocessing

Alternative 16: 93.5% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0.004000000189989805:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{0.25}{r}, 0.31830987334251404, \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (if (<=
     (+
      (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
      (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
     0.004000000189989805)
  (/ 0.0 (* r (* s PI)))
  (/
   (fma
    (/ 0.25 r)
    0.31830987334251404
    (/ -0.16666666666666666 (* s PI)))
   s)))

C

float code(float s, float r) {
	float tmp;
	if ((((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))) <= 0.004000000189989805f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = fmaf((0.25f / r), 0.31830987334251404f, (-0.16666666666666666f / (s * ((float) M_PI)))) / s;
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (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))) <= Float32(0.004000000189989805))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(fma(Float32(Float32(0.25) / r), Float32(0.31830987334251404), Float32(Float32(-0.16666666666666666) / Float32(s * Float32(pi)))) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.00400000019

   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

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]

   if 0.00400000019 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) 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

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

   4. Applied rewrites 8.8%

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

   6. Applied rewrites 8.8%

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

   7. Evaluated real constant 8.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{0.25}{r}, 0.31830987334251404, -\frac{0.16666666666666666}{\pi \cdot s}\right)}{s} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{0.25}{r}, 0.31830987334251404, \frac{\frac{-1}{6}}{s \cdot \pi}\right)}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 8.8%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{0.25}{r}, 0.31830987334251404, \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 93.5% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0.004000000189989805:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s}, 0.25\right)}{\left(s \cdot r\right) \cdot \pi}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (if (<=
     (+
      (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
      (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
     0.004000000189989805)
  (/ 0.0 (* r (* s PI)))
  (/ (fma -0.16666666666666666 (/ r s) 0.25) (* (* s r) PI))))

C

float code(float s, float r) {
	float tmp;
	if ((((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))) <= 0.004000000189989805f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = fmaf(-0.16666666666666666f, (r / s), 0.25f) / ((s * r) * ((float) M_PI));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (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))) <= Float32(0.004000000189989805))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(fma(Float32(-0.16666666666666666), Float32(r / s), Float32(0.25)) / Float32(Float32(s * r) * Float32(pi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.00400000019

   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

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]

   if 0.00400000019 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 99.6%

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

   3. Applied rewrites 96.4%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 8.8%

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

   10. Applied rewrites 8.8%

       \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s}, 0.25\right)}{\left(s \cdot r\right) \cdot \pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 93.5% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0.004000000189989805:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.07957746833562851}{r} - \frac{0.16666666666666666}{\pi \cdot s}}{s}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (if (<=
     (+
      (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
      (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
     0.004000000189989805)
  (/ 0.0 (* r (* s PI)))
  (/
   (- (/ 0.07957746833562851 r) (/ 0.16666666666666666 (* PI s)))
   s)))

C

float code(float s, float r) {
	float tmp;
	if ((((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))) <= 0.004000000189989805f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = ((0.07957746833562851f / r) - (0.16666666666666666f / (((float) M_PI) * s))) / s;
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (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))) <= Float32(0.004000000189989805))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(Float32(Float32(0.07957746833562851) / r) - Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * s))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if ((((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))) <= single(0.004000000189989805))
		tmp = single(0.0) / (r * (s * single(pi)));
	else
		tmp = ((single(0.07957746833562851) / r) - (single(0.16666666666666666) / (single(pi) * s))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.00400000019

   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

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]

   if 0.00400000019 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) 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

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

   4. Applied rewrites 8.8%

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

   6. Applied rewrites 8.8%

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

   7. Evaluated real constant 8.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{0.25}{r}, 0.31830987334251404, -\frac{0.16666666666666666}{\pi \cdot s}\right)}{s} \]
   8. Step-by-step derivation

   9. Applied rewrites 8.8%

      \[\leadsto \frac{\frac{0.07957746833562851}{r} - \frac{0.16666666666666666}{\pi \cdot s}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 93.1% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{r}}{\pi \cdot s}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (if (<=
     (+
      (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
      (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
     0.0)
  (/ 0.0 (* r (* s PI)))
  (/ (/ 0.25 r) (* PI s))))

C

float code(float s, float r) {
	float tmp;
	if ((((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))) <= 0.0f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = (0.25f / r) / (((float) M_PI) * s);
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (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))) <= Float32(0.0))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(Float32(0.25) / r) / Float32(Float32(pi) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if ((((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))) <= single(0.0))
		tmp = single(0.0) / (r * (s * single(pi)));
	else
		tmp = (single(0.25) / r) / (single(pi) * s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   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

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) 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

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

   4. Applied rewrites 8.8%

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

   6. Applied rewrites 8.8%

      \[\leadsto \frac{\frac{0.25}{r}}{\pi \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 93.1% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.07957746833562851}{s \cdot r}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (if (<=
     (+
      (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
      (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
     0.0)
  (/ 0.0 (* r (* s PI)))
  (/ 0.07957746833562851 (* s r))))

C

float code(float s, float r) {
	float tmp;
	if ((((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))) <= 0.0f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = 0.07957746833562851f / (s * r);
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (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))) <= Float32(0.0))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(0.07957746833562851) / Float32(s * r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if ((((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))) <= single(0.0))
		tmp = single(0.0) / (r * (s * single(pi)));
	else
		tmp = single(0.07957746833562851) / (s * r);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   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

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0}{r \cdot \left(s \cdot \pi\right)} \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) 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. Step-by-step derivation

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in s around inf

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

   6. Applied rewrites 8.8%

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

   7. Evaluated real constant 8.8%

      \[\leadsto \frac{0.07957746833562851}{s \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 8.8% accurate, 8.9× speedup

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

Math

\[\frac{0.07957746833562851}{s \cdot r} \]

FPCore

(FPCore (s r)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (< 1e-6 r) (< r 1000000.0)))
  (/ 0.07957746833562851 (* s r)))

C

float code(float s, float r) {
	return 0.07957746833562851f / (s * r);
}

Fortran

real(4) function code(s, r)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: r
    code = 0.07957746833562851e0 / (s * r)
end function

Julia

function code(s, r)
	return Float32(Float32(0.07957746833562851) / Float32(s * r))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.07957746833562851) / (s * r);
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around inf

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

6. Applied rewrites 8.8%

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

7. Evaluated real constant 8.8%

   \[\leadsto \frac{0.07957746833562851}{s \cdot r} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2026086 
(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))))