Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 12.9s

Alternatives: 11

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf 99.7%

   \[\leadsto \frac{\color{blue}{0.25 \cdot e^{-1 \cdot \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. neg-mul-1 99.7%

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

   2. rec-exp 99.7%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. associate-*r/ 99.7%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32((Float32(exp(1)) ^ Float32(Float32(r / s) * Float32(-0.3333333333333333))) + exp(Float32(r / Float32(-s)))) / Float32(r * Float32(s * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (((single(2.71828182845904523536) ^ ((r / s) * single(-0.3333333333333333))) + exp((r / -s))) / (r * (s * single(pi))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around inf 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-commutative 99.6%

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

   3. *-un-lft-identity 99.6%

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

   4. pow-exp 99.7%

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

   5. e-exp-1 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*r/ 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32((Float32(exp(1)) ^ Float32(r / Float32(-s))) + exp(Float32(Float32(r / s) * Float32(-0.3333333333333333)))) / Float32(r * Float32(s * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (((single(2.71828182845904523536) ^ (r / -s)) + exp(((r / s) * single(-0.3333333333333333)))) / (r * (s * single(pi))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around inf 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. exp-prod 99.7%

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

   3. e-exp-1 99.7%

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

   4. mul-1-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around inf 99.6%

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

   1. mul-1-neg 99.6%

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

   2. distribute-frac-neg2 99.6%

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

   3. expm1-log1p-u 99.6%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. expm1-undefine 99.5%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)} - 1\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.5%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)} - 1\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
8. Step-by-step derivation

   1. log1p-undefine 99.5%

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

   2. rem-exp-log 99.5%

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

   3. associate-+r- 99.6%

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

   4. expm1-undefine 99.6%

      \[\leadsto 0.125 \cdot \frac{\left(1 + \color{blue}{\mathsf{expm1}\left(\frac{r}{-s}\right)}\right) + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   5. rem-exp-log 99.6%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\frac{r}{-s}\right)\right)}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   6. log1p-define 99.5%

      \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{r}{-s}\right)\right)}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. log1p-expm1 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 5: 15.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\frac{r}{s} + 1}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(6 \cdot \left(s \cdot \pi\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (/ 0.125 (+ (/ r s) 1.0)) (* r (* s PI)))
  (* 0.75 (/ (exp (/ (* r -0.3333333333333333) s)) (* r (* 6.0 (* s PI)))))))

C

float code(float s, float r) {
	return ((0.125f / ((r / s) + 1.0f)) / (r * (s * ((float) M_PI)))) + (0.75f * (expf(((r * -0.3333333333333333f) / s)) / (r * (6.0f * (s * ((float) M_PI))))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(Float32(r / s) + Float32(1.0))) / Float32(r * Float32(s * Float32(pi)))) + Float32(Float32(0.75) * Float32(exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / Float32(r * Float32(Float32(6.0) * Float32(s * Float32(pi)))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / ((r / s) + single(1.0))) / (r * (s * single(pi)))) + (single(0.75) * (exp(((r * single(-0.3333333333333333)) / s)) / (r * (single(6.0) * (s * single(pi))))));
end

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-frac-neg 99.7%

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

   4. associate-/l* 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 99.6%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   2. associate-*l/ 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

7. Simplified 99.6%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

8. Taylor expanded in s around 0 99.7%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-\frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 99.7%

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

   2. rec-exp 99.7%

      \[\leadsto \frac{0.125 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   3. associate-*r/ 99.7%

      \[\leadsto \frac{\color{blue}{\frac{0.125 \cdot 1}{e^{\frac{r}{s}}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   4. metadata-eval 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in r around 0 16.6%

    \[\leadsto \frac{\frac{0.125}{\color{blue}{1 + \frac{r}{s}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

12. Final simplification 16.6%

    \[\leadsto \frac{\frac{0.125}{\frac{r}{s} + 1}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(6 \cdot \left(s \cdot \pi\right)\right)} \]
13. Add Preprocessing

Alternative 6: 9.4% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r \cdot \pi} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/ 0.25 (* r PI))
   (/
    (+ (/ (* (/ r PI) -0.06944444444444445) s) (/ 0.16666666666666666 PI))
    s))
  s))

C

float code(float s, float r) {
	return ((0.25f / (r * ((float) M_PI))) - (((((r / ((float) M_PI)) * -0.06944444444444445f) / s) + (0.16666666666666666f / ((float) M_PI))) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(r * Float32(pi))) - Float32(Float32(Float32(Float32(Float32(r / Float32(pi)) * Float32(-0.06944444444444445)) / s) + Float32(Float32(0.16666666666666666) / Float32(pi))) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * single(pi))) - (((((r / single(pi)) * single(-0.06944444444444445)) / s) + (single(0.16666666666666666) / single(pi))) / s)) / s;
end

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. times-frac 99.7%

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

   3. fma-define 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.7%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.7%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.7%

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

5. Taylor expanded in s around -inf 12.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. mul-1-neg 12.1%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

7. Simplified 12.1%

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

8. Final simplification 12.1%

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

Alternative 7: 8.5% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \frac{2 + \left(r \cdot \frac{-0.3333333333333333}{s} - \frac{r}{s}\right)}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (/ (+ 2.0 (- (* r (/ -0.3333333333333333 s)) (/ r s))) r)))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((2.0f + ((r * (-0.3333333333333333f / s)) - (r / s))) / r);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(Float32(2.0) + Float32(Float32(r * Float32(Float32(-0.3333333333333333) / s)) - Float32(r / s))) / r))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((single(2.0) + ((r * (single(-0.3333333333333333) / s)) - (r / s))) / r);
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 11.9%

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

6. Taylor expanded in r around 0 11.0%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\frac{1 + -1 \cdot \frac{r}{s}}{r}} + \frac{1 + -0.3333333333333333 \cdot \frac{r}{s}}{r}\right) \]
7. Step-by-step derivation

   1. mul-1-neg 11.0%

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

   2. unsub-neg 11.0%

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

8. Simplified 11.0%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\frac{1 - \frac{r}{s}}{r}} + \frac{1 + -0.3333333333333333 \cdot \frac{r}{s}}{r}\right) \]
9. Step-by-step derivation

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 9.7%

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

   3. pow2 9.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{1 - \frac{r}{s}}{r} + \frac{1 + \sqrt{\color{blue}{{\left(-0.3333333333333333 \cdot \frac{r}{s}\right)}^{2}}}}{r}\right) \]

   4. *-commutative 9.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{1 - \frac{r}{s}}{r} + \frac{1 + \sqrt{{\color{blue}{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}^{2}}}{r}\right) \]

10. Applied egg-rr 9.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{1 - \frac{r}{s}}{r} + \frac{1 + \color{blue}{\sqrt{{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}^{2}}}}{r}\right) \]
11. Step-by-step derivation

    1. unpow2 9.7%

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

    2. rem-sqrt-square 9.7%

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

    3. associate-*l/ 9.7%

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

    4. associate-/l* 9.7%

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

12. Simplified 9.7%

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

13. Taylor expanded in r around 0 9.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\frac{2 + \left(\left|-0.3333333333333333 \cdot \frac{r}{s}\right| + -1 \cdot \frac{r}{s}\right)}{r}} \]
14. Step-by-step derivation

    1. rem-square-sqrt -0.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{2 + \left(\left|\color{blue}{\sqrt{-0.3333333333333333 \cdot \frac{r}{s}} \cdot \sqrt{-0.3333333333333333 \cdot \frac{r}{s}}}\right| + -1 \cdot \frac{r}{s}\right)}{r} \]

    2. fabs-sqr -0.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{2 + \left(\color{blue}{\sqrt{-0.3333333333333333 \cdot \frac{r}{s}} \cdot \sqrt{-0.3333333333333333 \cdot \frac{r}{s}}} + -1 \cdot \frac{r}{s}\right)}{r} \]

    3. rem-square-sqrt 11.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{2 + \left(\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}} + -1 \cdot \frac{r}{s}\right)}{r} \]

    4. neg-mul-1 11.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{2 + \left(-0.3333333333333333 \cdot \frac{r}{s} + \color{blue}{\left(-\frac{r}{s}\right)}\right)}{r} \]

    5. unsub-neg 11.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{2 + \color{blue}{\left(-0.3333333333333333 \cdot \frac{r}{s} - \frac{r}{s}\right)}}{r} \]

    6. *-commutative 11.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{2 + \left(\color{blue}{\frac{r}{s} \cdot -0.3333333333333333} - \frac{r}{s}\right)}{r} \]

    7. associate-*l/ 11.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{2 + \left(\color{blue}{\frac{r \cdot -0.3333333333333333}{s}} - \frac{r}{s}\right)}{r} \]

    8. associate-*r/ 11.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{2 + \left(\color{blue}{r \cdot \frac{-0.3333333333333333}{s}} - \frac{r}{s}\right)}{r} \]

15. Simplified 11.0%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\frac{2 + \left(r \cdot \frac{-0.3333333333333333}{s} - \frac{r}{s}\right)}{r}} \]
16. Add Preprocessing

Alternative 8: 8.5% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \frac{2 + \frac{r}{s} \cdot -1.3333333333333333}{r} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 11.9%

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

6. Taylor expanded in r around 0 11.0%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\frac{1 + -1 \cdot \frac{r}{s}}{r}} + \frac{1 + -0.3333333333333333 \cdot \frac{r}{s}}{r}\right) \]
7. Step-by-step derivation

   1. mul-1-neg 11.0%

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

   2. unsub-neg 11.0%

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

8. Simplified 11.0%

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

9. Taylor expanded in r around inf 11.0%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{1 - \frac{r}{s}}{r} + \color{blue}{\left(\frac{1}{r} - 0.3333333333333333 \cdot \frac{1}{s}\right)}\right) \]
10. Step-by-step derivation

    1. associate-*r/ 11.0%

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

    2. metadata-eval 11.0%

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

11. Simplified 11.0%

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

12. Taylor expanded in r around 0 11.0%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\frac{2 + -1.3333333333333333 \cdot \frac{r}{s}}{r}} \]
13. Step-by-step derivation

    1. *-commutative 11.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{2 + \color{blue}{\frac{r}{s} \cdot -1.3333333333333333}}{r} \]

14. Simplified 11.0%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\frac{2 + \frac{r}{s} \cdot -1.3333333333333333}{r}} \]
15. Add Preprocessing

Alternative 9: 8.5% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 11.9%

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

6. Taylor expanded in r around 0 11.0%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\frac{1 + -1 \cdot \frac{r}{s}}{r}} + \frac{1 + -0.3333333333333333 \cdot \frac{r}{s}}{r}\right) \]
7. Step-by-step derivation

   1. mul-1-neg 11.0%

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

   2. unsub-neg 11.0%

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

8. Simplified 11.0%

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

9. Taylor expanded in r around inf 11.0%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{1 - \frac{r}{s}}{r} + \color{blue}{\left(\frac{1}{r} - 0.3333333333333333 \cdot \frac{1}{s}\right)}\right) \]
10. Step-by-step derivation

    1. associate-*r/ 11.0%

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

    2. metadata-eval 11.0%

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

11. Simplified 11.0%

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

12. Taylor expanded in r around inf 11.0%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} - 1.3333333333333333 \cdot \frac{1}{s}\right)} \]
13. Step-by-step derivation

    1. associate-*r/ 11.0%

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

    2. metadata-eval 11.0%

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

    3. associate-*r/ 11.0%

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

    4. metadata-eval 11.0%

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

14. Simplified 11.0%

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

Alternative 10: 8.6% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around inf 10.5%

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

   1. div-inv 10.5%

      \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)}} \]

   2. associate-*r* 10.5%

      \[\leadsto 0.25 \cdot \frac{1}{\color{blue}{\left(r \cdot s\right) \cdot \pi}} \]

7. Applied egg-rr 10.5%

   \[\leadsto \color{blue}{0.25 \cdot \frac{1}{\left(r \cdot s\right) \cdot \pi}} \]
8. Step-by-step derivation

   1. associate-*r/ 10.5%

      \[\leadsto \color{blue}{\frac{0.25 \cdot 1}{\left(r \cdot s\right) \cdot \pi}} \]

   2. metadata-eval 10.5%

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

   3. associate-*r* 10.5%

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

   4. associate-/r* 10.5%

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

9. Simplified 10.5%

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

Alternative 11: 8.6% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around inf 10.5%

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

Reproduce

herbie shell --seed 2024152 
(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))))