Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.3s

Alternatives: 15

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: 99.6% 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

\[\mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{s \cdot r}, 0.053051646798849106, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]

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 r))
 0.053051646798849106
 (/ (/ 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 * r)), 0.053051646798849106f, ((0.125f / ((((float) M_PI) * s) * expf((r / s)))) / r));
}

Julia

function code(s, r)
	return fma(Float32(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) * Float32(0.75)) / Float32(s * r)), Float32(0.053051646798849106), 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%

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

4. Evaluated real constant 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{s \cdot r}, 0.053051646798849106, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]
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

\[\mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{s \cdot r}, 0.053051646798849106, \frac{0.125}{\left(\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}\right) \cdot r}\right) \]

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 r))
 0.053051646798849106
 (/ 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 * r)), 0.053051646798849106f, (0.125f / (((((float) M_PI) * s) * expf((r / s))) * r)));
}

Julia

function code(s, r)
	return fma(Float32(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) * Float32(0.75)) / Float32(s * r)), Float32(0.053051646798849106), 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%

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

4. Evaluated real constant 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{s \cdot r}, 0.053051646798849106, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]
5. Step-by-step derivation

6. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{s \cdot r}, 0.053051646798849106, \frac{0.125}{\left(\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}\right) \cdot r}\right) \]
7. 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 97.8%

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

5. Applied rewrites 99.5%

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

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

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in s around 0

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

8. Applied rewrites 99.5%

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

10. Applied rewrites 99.5%

    \[\leadsto \frac{\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}} + e^{\frac{-r}{s}}}{\pi \cdot s} \cdot 0.125}{r} \]
11. 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

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

FPCore

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

C

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

Julia

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

MATLAB

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

5. Applied rewrites 97.6%

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

7. Applied rewrites 99.5%

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

Alternative 7: 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

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

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) + 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) * (r / 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 97.6%

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

7. Applied rewrites 99.5%

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

Alternative 8: 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

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

FPCore

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

C

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

Julia

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

MATLAB

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

5. Applied rewrites 97.6%

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

Alternative 9: 94.4% accurate, 0.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(4 \cdot r, \pi, 2.6666666666666665 \cdot \frac{\left(r \cdot r\right) \cdot \pi}{s}\right)}\\ \end{array} \]

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)))
     4.999999943633011e-27)
  (/ 0.0 (* r (* s PI)))
  (/
   1.0
   (*
    s
    (fma (* 4.0 r) PI (* 2.6666666666666665 (/ (* (* r 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))) <= 4.999999943633011e-27f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = 1.0f / (s * fmaf((4.0f * r), ((float) M_PI), (2.6666666666666665f * (((r * 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(4.999999943633011e-27))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * fma(Float32(Float32(4.0) * r), Float32(pi), Float32(Float32(2.6666666666666665) * Float32(Float32(Float32(r * r) * 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))) < 4.99999994e-27

   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 9.1%

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

   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 88.5%

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

   if 4.99999994e-27 < (+.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 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 9.1%

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

   6. Applied rewrites 9.1%

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

   7. Taylor expanded in s around inf

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

   9. Applied rewrites 19.8%

      \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(2.6666666666666665, \frac{{r}^{2} \cdot \pi}{s}, 4 \cdot \left(r \cdot \pi\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 19.8%

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

Alternative 10: 94.4% accurate, 0.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot \mathsf{fma}\left(4 \cdot s, \pi, 2.6666666666666665 \cdot \left(\pi \cdot r\right)\right)}\\ \end{array} \]

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)))
     4.999999943633011e-27)
  (/ 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))) <= 4.999999943633011e-27f) {
		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(4.999999943633011e-27))
		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))) < 4.99999994e-27

   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 9.1%

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

   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 88.5%

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

   if 4.99999994e-27 < (+.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 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 9.1%

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

   6. Applied rewrites 9.1%

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

   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 12.5%

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

   11. Applied rewrites 12.5%

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

Alternative 11: 94.4% accurate, 0.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot \mathsf{fma}\left(2.6666666666666665, r \cdot \pi, 4 \cdot \left(s \cdot \pi\right)\right)}\\ \end{array} \]

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)))
     4.999999943633011e-27)
  (/ 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))) <= 4.999999943633011e-27f) {
		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(4.999999943633011e-27))
		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))) < 4.99999994e-27

   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 9.1%

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

   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 88.5%

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

   if 4.99999994e-27 < (+.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 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 9.1%

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

   6. Applied rewrites 9.1%

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

   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 12.5%

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

Alternative 12: 93.6% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s}, 0.25\right)}{\left(\pi \cdot r\right) \cdot s}\\ \end{array} \]

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)))
     1.9999999949504854e-6)
  (/ 0.0 (* r (* s PI)))
  (/ (fma -0.16666666666666666 (/ r s) 0.25) (* (* PI 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))) <= 1.9999999949504854e-6f) {
		tmp = 0.0f / (r * (s * ((float) M_PI)));
	} else {
		tmp = fmaf(-0.16666666666666666f, (r / s), 0.25f) / ((((float) M_PI) * 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(1.9999999949504854e-6))
		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(Float32(pi) * r) * 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))) < 1.99999999e-6

   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 9.1%

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

   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 88.5%

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

   if 1.99999999e-6 < (+.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 97.8%

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

   5. Applied rewrites 99.5%

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

   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 9.1%

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

   10. Applied rewrites 9.1%

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

Alternative 13: 92.8% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 2.000000033724767 \cdot 10^{-16}:\\ \;\;\;\;\frac{0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.07957746833562851}{s}}{r}\\ \end{array} \]

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)))
     2.000000033724767e-16)
  (/ 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))) <= 2.000000033724767e-16f) {
		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(2.000000033724767e-16))
		tmp = Float32(Float32(0.0) / Float32(r * Float32(s * Float32(pi))));
	else
		tmp = Float32(Float32(Float32(0.07957746833562851) / 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(2.000000033724767e-16))
		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))) < 2.00000003e-16

   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 9.1%

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

   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 88.5%

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

   if 2.00000003e-16 < (+.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 9.1%

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

   7. Evaluated real constant 9.1%

      \[\leadsto \frac{0.07957746833562851}{s \cdot r} \]
   8. Step-by-step derivation

   9. Applied rewrites 9.1%

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

Alternative 14: 9.1% 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 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 9.1%

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

7. Evaluated real constant 9.1%

   \[\leadsto \frac{0.07957746833562851}{s \cdot r} \]
8. Step-by-step derivation

9. Applied rewrites 9.1%

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

Alternative 15: 9.1% 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 9.1%

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

7. Evaluated real constant 9.1%

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

Reproduce

herbie shell --seed 2026084 
(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))))