Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 5.7s

Alternatives: 22

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
  :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: 99.5% 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: 99.5% accurate, 1.2× speedup

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

Math

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

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))) (* PI s))
  0.125
  (/ 0.125 (* (* PI s) (exp (/ r s)))))
 r))

C

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

Julia

function code(s, r)
	return Float32(fma(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(Float32(pi) * s)), Float32(0.125), 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 99.5%

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

Alternative 2: 99.5% accurate, 1.3× speedup

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

Math

\[\frac{1}{\frac{r}{\frac{0.125 \cdot \frac{e^{\frac{r}{-3 \cdot s}} + e^{\frac{-r}{s}}}{\pi}}{s}}} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(1.0) / (r / ((single(0.125) * ((exp((r / (single(-3.0) * s))) + exp((-r / s))) / single(pi))) / s));
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

   \[\leadsto \frac{1}{\frac{r}{\frac{0.125 \cdot \frac{e^{\frac{r}{-3 \cdot s}} + e^{\frac{-r}{s}}}{\pi}}{s}}} \]
6. Add Preprocessing

Alternative 3: 99.5% accurate, 1.3× speedup

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

Math

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

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))) 0.125 (/ 0.125 (exp (/ r s))))
  (* PI s))
 r))

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

   \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{e^{\frac{r}{-3 \cdot s}} + e^{\frac{-r}{s}}}{\pi}}{r} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

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

Alternative 5: 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

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

FPCore

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

C

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

Julia

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

MATLAB

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

   \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{e^{\frac{r}{-3 \cdot s}} + e^{\frac{-r}{s}}}{\pi}}{r} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

Alternative 6: 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

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

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) s)) (exp (* -0.3333333333333333 (/ r s))))
   (* PI s)))
 r))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in r around inf

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

Alternative 7: 97.6% accurate, 1.4× speedup

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

   \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{e^{\frac{r}{-3 \cdot s}} + e^{\frac{-r}{s}}}{\pi}}{r} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \frac{e^{\frac{r}{-3 \cdot s}} + e^{\frac{-r}{s}}}{\pi \cdot r} \cdot \frac{0.125}{s} \]
8. Step-by-step derivation

9. Applied rewrites 97.6%

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

Alternative 8: 95.2% accurate, 0.6× speedup

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

Math

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

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
   (fma
    (* (* PI s) 4.0)
    r
    (*
     (*
      (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(((((float) M_PI) * s) * 4.0f), r, ((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) / fma(Float32(Float32(Float32(pi) * s) * Float32(4.0)), r, Float32(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 99.5%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 9.4%

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

   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 87.8%

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

   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 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\frac{r}{\frac{0.125 \cdot \frac{e^{\frac{r}{-3 \cdot s}} + e^{\frac{-r}{s}}}{\pi}}{s}}} \]

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 27.2%

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

   10. Applied rewrites 27.9%

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

Alternative 9: 95.2% accurate, 0.6× speedup

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

Math

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

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)
     (*
      r
      (fma
       2.6666666666666665
       PI
       (* (* -8.0 r) (* (/ PI s) -0.08333333333333333)))))))))

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)), (r * fmaf(2.6666666666666665f, ((float) M_PI), ((-8.0f * r) * ((((float) M_PI) / s) * -0.08333333333333333f))))));
	}
	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(4.0), Float32(s * Float32(pi)), Float32(r * fma(Float32(2.6666666666666665), Float32(pi), Float32(Float32(Float32(-8.0) * r) * Float32(Float32(Float32(pi) / s) * Float32(-0.08333333333333333))))))));
	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 99.5%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 9.4%

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

   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 87.8%

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

   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 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\frac{r}{\frac{0.125 \cdot \frac{e^{\frac{r}{-3 \cdot s}} + e^{\frac{-r}{s}}}{\pi}}{s}}} \]

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 27.2%

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

   10. Applied rewrites 27.8%

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

Alternative 10: 94.5% accurate, 0.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{\pi \cdot r} - \frac{\mathsf{fma}\left(r, \frac{-0.06944444444444445}{\pi \cdot s}, 0.053051646798849106\right)}{s}}{s}\\ \end{array} \]

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)))
     0.0)
  (/ 0.0 (* r (* s PI)))
  (/
   (-
    (/ 0.25 (* PI r))
    (/
     (fma r (/ -0.06944444444444445 (* PI s)) 0.053051646798849106)
     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))) <= 0.0f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = ((0.25f / (((float) M_PI) * r)) - (fmaf(r, (-0.06944444444444445f / (((float) M_PI) * s)), 0.053051646798849106f) / 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(0.0))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(Float32(Float32(0.25) / Float32(Float32(pi) * r)) - Float32(fma(r, Float32(Float32(-0.06944444444444445) / Float32(Float32(pi) * s)), Float32(0.053051646798849106)) / s)) / 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))) < 0.0

   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

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

   4. Applied rewrites 9.4%

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

   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 87.8%

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

   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 99.5%

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

   2. Taylor expanded in s around -inf

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\pi} + \frac{-1}{144} \cdot \frac{r}{\pi}}{s} - \frac{1}{6} \cdot \frac{1}{\pi}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \pi}}{s} \]
   3. Step-by-step derivation

   4. Applied rewrites 10.6%

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-0.0625, \frac{r}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   6. Applied rewrites 10.6%

      \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{0.16666666666666666}{\pi} + \frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}}{s}}{s} \]

   7. Evaluated real constant 10.6%

      \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{0.053051646798849106 + \frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}}{s}}{s} \]
   8. Step-by-step derivation

   9. Applied rewrites 10.6%

      \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{\mathsf{fma}\left(r, \frac{-0.06944444444444445}{\pi \cdot s}, 0.053051646798849106\right)}{s}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 94.1% accurate, 0.7× speedup

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

Math

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

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
   (* s (* r (fma 2.6666666666666665 (/ (* r PI) s) (* 4.0 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 / (s * (r * fmaf(2.6666666666666665f, ((r * ((float) M_PI)) / s), (4.0f * ((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(s * Float32(r * fma(Float32(2.6666666666666665), Float32(Float32(r * Float32(pi)) / s), Float32(Float32(4.0) * 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 99.5%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 9.4%

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

   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 87.8%

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

   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 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\frac{r}{\frac{0.125 \cdot \frac{e^{\frac{r}{-3 \cdot s}} + e^{\frac{-r}{s}}}{\pi}}{s}}} \]

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 20.6%

      \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, -0.3333333333333333 \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \pi\right)\right)} \]

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 20.6%

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

Alternative 12: 94.1% 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 99.5%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 9.4%

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

   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 87.8%

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

   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 99.5%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\frac{r}{\frac{0.125 \cdot \frac{e^{\frac{r}{-3 \cdot s}} + e^{\frac{-r}{s}}}{\pi}}{s}}} \]

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 12.9%

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

Alternative 13: 93.4% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, 0.07957746833562851\right)}{s}}{r}\\ \end{array} \]

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)))
     1.9999999494757503e-5)
  (/ 0.0 (* r (* s PI)))
  (/
   (/ (fma -0.16666666666666666 (/ 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))) <= 1.9999999494757503e-5f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = (fmaf(-0.16666666666666666f, (r / (s * ((float) M_PI))), 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(1.9999999494757503e-5))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(fma(Float32(-0.16666666666666666), Float32(r / Float32(s * Float32(pi))), Float32(0.07957746833562851)) / s) / 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))) < 1.99999995e-5

   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

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

   4. Applied rewrites 9.4%

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

   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 87.8%

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

   if 1.99999995e-5 < (+.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 99.5%

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

   2. Taylor expanded in r around 0

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

   4. Applied rewrites 9.6%

      \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{{s}^{2} \cdot \pi}, 0.25 \cdot \frac{1}{s \cdot \pi}\right)}{r} \]

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 9.6%

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

   8. Evaluated real constant 9.6%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, 0.07957746833562851\right)}{s}}{r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 93.4% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{\pi \cdot r} - \frac{0.053051646798849106}{s}}{s}\\ \end{array} \]

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)))
     1.9999999494757503e-5)
  (/ 0.0 (* r (* s PI)))
  (/ (- (/ 0.25 (* PI r)) (/ 0.053051646798849106 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))) <= 1.9999999494757503e-5f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = ((0.25f / (((float) M_PI) * r)) - (0.053051646798849106f / 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(1.9999999494757503e-5))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(Float32(Float32(0.25) / Float32(Float32(pi) * r)) - Float32(Float32(0.053051646798849106) / 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(1.9999999494757503e-5))
		tmp = single(0.0) / (r * (s * single(pi)));
	else
		tmp = ((single(0.25) / (single(pi) * r)) - (single(0.053051646798849106) / 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))) < 1.99999995e-5

   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

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

   4. Applied rewrites 9.4%

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

   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 87.8%

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

   if 1.99999995e-5 < (+.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 99.5%

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

   2. Taylor expanded in s around -inf

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\pi} + \frac{-1}{144} \cdot \frac{r}{\pi}}{s} - \frac{1}{6} \cdot \frac{1}{\pi}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \pi}}{s} \]
   3. Step-by-step derivation

   4. Applied rewrites 10.6%

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-0.0625, \frac{r}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   6. Applied rewrites 10.6%

      \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{0.16666666666666666}{\pi} + \frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}}{s}}{s} \]

   7. Evaluated real constant 10.6%

      \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{0.053051646798849106 + \frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}}{s}}{s} \]

   8. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{14240943}{268435456}}{s}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.6%

       \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{0.053051646798849106}{s}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 93.3% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.07957746833562851}{r} + \frac{-0.16666666666666666}{\pi \cdot s}}{s}\\ \end{array} \]

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)))
     1.9999999494757503e-5)
  (/ 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))) <= 1.9999999494757503e-5f) {
		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(1.9999999494757503e-5))
		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(1.9999999494757503e-5))
		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))) < 1.99999995e-5

   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

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

   4. Applied rewrites 9.4%

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

   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 87.8%

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

   if 1.99999995e-5 < (+.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 99.5%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 9.6%

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

   6. Applied rewrites 9.6%

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

   7. Evaluated real constant 9.6%

      \[\leadsto \frac{\mathsf{fma}\left(0.07957746833562851, \frac{1}{r}, \frac{-0.16666666666666666}{\pi \cdot s}\right)}{s} \]

   9. Applied rewrites 9.6%

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

Alternative 16: 92.4% accurate, 0.8× speedup

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

Math

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

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) (/ 1.0 (* 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) * (1.0f / (((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(1.0) / 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(1.0) / (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 99.5%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 9.4%

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

   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 87.8%

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

   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 99.5%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 9.4%

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

   6. Applied rewrites 9.4%

      \[\leadsto \frac{0.25}{r} \cdot \frac{1}{\pi \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 92.4% accurate, 0.8× speedup

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

Math

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

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)))
  (/ (/ -0.25 (* s 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 = (-0.25f / (s * ((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(Float32(-0.25) / Float32(s * Float32(pi))) / Float32(-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(-0.25) / (s * single(pi))) / -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 99.5%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 9.4%

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

   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 87.8%

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

   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 99.5%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in s around inf

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

   6. Applied rewrites 9.4%

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

Alternative 18: 92.4% accurate, 0.8× speedup

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

Math

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

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)))
     0.0)
  (/ 0.0 (* r (* s PI)))
  (/ 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))) <= 0.0f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = 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(0.0))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(0.25) / Float32(Float32(s * r) * Float32(pi)));
	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) / ((s * r) * single(pi));
	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 99.5%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 9.4%

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

   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 87.8%

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

   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 99.5%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 9.4%

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

   6. Applied rewrites 9.4%

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

Alternative 19: 92.4% accurate, 0.8× speedup

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

Math

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

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)))
  (/ (/ 0.07957746833562851 r) 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 = (0.07957746833562851f / r) / 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(0.07957746833562851) / r) / 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(0.07957746833562851) / r) / 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 99.5%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 9.4%

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

   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 87.8%

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

   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 99.5%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 9.6%

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

   6. Applied rewrites 9.6%

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

   7. Evaluated real constant 9.6%

      \[\leadsto \frac{\mathsf{fma}\left(0.07957746833562851, \frac{1}{r}, \frac{-0.16666666666666666}{\pi \cdot s}\right)}{s} \]

   8. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{10680707}{134217728}}{r}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.4%

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

Alternative 20: 9.4% accurate, 8.1× speedup

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

4. Applied rewrites 9.6%

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

6. Applied rewrites 9.6%

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

7. Evaluated real constant 9.6%

   \[\leadsto \frac{\mathsf{fma}\left(0.07957746833562851, \frac{1}{r}, \frac{-0.16666666666666666}{\pi \cdot s}\right)}{s} \]

8. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{\frac{10680707}{134217728}}{r}}{s} \]
9. Step-by-step derivation

10. Applied rewrites 9.4%

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

Alternative 21: 9.4% accurate, 8.1× speedup

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

Math

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

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(Float32(0.07957746833562851) / s) / r)
end

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in r around 0

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

4. Applied rewrites 9.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{{s}^{2} \cdot \pi}, 0.25 \cdot \frac{1}{s \cdot \pi}\right)}{r} \]

5. Taylor expanded in s around inf

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

7. Applied rewrites 9.6%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 9.4%

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

11. Evaluated real constant 9.4%

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

Alternative 22: 9.4% 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 99.5%

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

3. Applied rewrites 99.5%

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

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 9.4%

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

7. Evaluated real constant 9.4%

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

Reproduce

herbie shell --seed 2026070 
(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))))