Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 11.2s

Alternatives: 13

Speedup: N/A×

Specification

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

Math

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

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: 99.6% 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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{e^{\frac{r}{-s}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  0.125
  (/
   (+ (exp (/ r (- s))) (pow (exp -0.6666666666666666) (/ (/ r s) 2.0)))
   (* r (* s PI)))))

C

float code(float s, float r) {
	return 0.125f * ((expf((r / -s)) + powf(expf(-0.6666666666666666f), ((r / s) / 2.0f))) / (r * (s * ((float) M_PI))));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(exp(Float32(r / Float32(-s))) + (exp(Float32(-0.6666666666666666)) ^ Float32(Float32(r / s) / Float32(2.0)))) / Float32(r * Float32(s * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((r / -s)) + (exp(single(-0.6666666666666666)) ^ ((r / s) / single(2.0)))) / (r * (s * single(pi))));
end

Derivation

1. Initial program 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around inf 99.9%

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

   1. pow-exp 99.8%

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

   2. sqr-pow 99.7%

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

   3. pow-prod-down 99.8%

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

   4. prod-exp 99.9%

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

   5. metadata-eval 99.9%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + {\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.9%

   \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + \color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r \cdot \left(s \cdot \pi\right)} \]
8. Step-by-step derivation

   1. mul-1-neg 99.9%

      \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{-\frac{r}{s}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

   2. exp-neg 99.9%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\frac{1}{e^{\frac{r}{s}}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

9. Applied egg-rr 99.9%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{\frac{1}{e^{\frac{r}{s}}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]
10. Step-by-step derivation

    1. rec-exp 99.9%

       \[\leadsto 0.125 \cdot \frac{\color{blue}{e^{-\frac{r}{s}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

    2. distribute-frac-neg 99.9%

       \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{\frac{-r}{s}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

11. Simplified 99.9%

    \[\leadsto 0.125 \cdot \frac{\color{blue}{e^{\frac{-r}{s}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

12. Final simplification 99.9%

    \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]
13. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around inf 99.9%

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

   1. metadata-eval 99.9%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-1}{3}} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   2. times-frac 99.9%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-1 \cdot r}{3 \cdot s}}}}{r \cdot \left(s \cdot \pi\right)} \]

   3. neg-mul-1 99.9%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\color{blue}{-r}}{3 \cdot s}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. associate-/r* 99.9%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{\frac{-r}{3}}{s}}}}{r \cdot \left(s \cdot \pi\right)} \]

   5. frac-2neg 99.9%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-\frac{-r}{3}}{-s}}}}{r \cdot \left(s \cdot \pi\right)} \]

   6. add-sqr-sqrt -0.0%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{-r} \cdot \sqrt{-r}}}{3}}{-s}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. sqrt-unprod 6.8%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{\left(-r\right) \cdot \left(-r\right)}}}{3}}{-s}}}{r \cdot \left(s \cdot \pi\right)} \]

   8. sqr-neg 6.8%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\sqrt{\color{blue}{r \cdot r}}}{3}}{-s}}}{r \cdot \left(s \cdot \pi\right)} \]

   9. sqrt-unprod 6.8%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{r} \cdot \sqrt{r}}}{3}}{-s}}}{r \cdot \left(s \cdot \pi\right)} \]

   10. add-sqr-sqrt 6.8%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{r}}{3}}{-s}}}{r \cdot \left(s \cdot \pi\right)} \]

   11. add-sqr-sqrt -0.0%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{r}{3}}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}}{r \cdot \left(s \cdot \pi\right)} \]

   12. sqrt-unprod 99.9%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{r}{3}}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}}}{r \cdot \left(s \cdot \pi\right)} \]

   13. sqr-neg 99.9%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{r}{3}}{\sqrt{\color{blue}{s \cdot s}}}}}{r \cdot \left(s \cdot \pi\right)} \]

   14. sqrt-unprod 99.9%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{r}{3}}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}}{r \cdot \left(s \cdot \pi\right)} \]

   15. add-sqr-sqrt 99.9%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{r}{3}}{\color{blue}{s}}}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.9%

   \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-\frac{r}{3}}{s}}}}{r \cdot \left(s \cdot \pi\right)} \]

8. Final simplification 99.9%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around inf 99.9%

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

   1. mul-1-neg 99.9%

      \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{-\frac{r}{s}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

   2. exp-neg 99.9%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\frac{1}{e^{\frac{r}{s}}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.8%

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

   1. rec-exp 99.9%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{e^{-\frac{r}{s}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

   2. distribute-frac-neg 99.9%

      \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{\frac{-r}{s}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 4: 11.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (log1p (expm1 (* r (* s PI))))))

C

float code(float s, float r) {
	return 0.25f / log1pf(expm1f((r * (s * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(r * Float32(s * Float32(pi))))))
end

Derivation

1. Initial program 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 8.1%

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

   1. log1p-expm1-u 11.0%

      \[\leadsto \frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)}} \]

7. Applied egg-rr 11.0%

   \[\leadsto \frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)}} \]
8. Add Preprocessing

Alternative 5: 9.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 9.0%

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

   1. +-commutative 9.0%

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

   2. *-commutative 9.0%

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

7. Simplified 9.0%

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

Alternative 6: 9.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{r} \cdot \frac{1}{\pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/ (- (* (/ r (* s PI)) 0.06944444444444445) (/ 0.16666666666666666 PI)) s)
   (* (/ 0.25 r) (/ 1.0 PI)))
  s))

C

float code(float s, float r) {
	return (((((r / (s * ((float) M_PI))) * 0.06944444444444445f) - (0.16666666666666666f / ((float) M_PI))) / s) + ((0.25f / r) * (1.0f / ((float) M_PI)))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(r / Float32(s * Float32(pi))) * Float32(0.06944444444444445)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s) + Float32(Float32(Float32(0.25) / r) * Float32(Float32(1.0) / Float32(pi)))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (((((r / (s * single(pi))) * single(0.06944444444444445)) - (single(0.16666666666666666) / single(pi))) / s) + ((single(0.25) / r) * (single(1.0) / single(pi)))) / s;
end

Derivation

1. Initial program 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 7.6%

   \[\leadsto \color{blue}{\frac{r \cdot \left(0.06944444444444445 \cdot \frac{r}{{s}^{3} \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi}\right) + 0.25 \cdot \frac{1}{s \cdot \pi}}{r}} \]

6. Taylor expanded in s around -inf 9.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.06944444444444445 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
7. Step-by-step derivation

   1. mul-1-neg 9.0%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.06944444444444445 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

   2. mul-1-neg 9.0%

      \[\leadsto -\frac{\color{blue}{\left(-\frac{0.06944444444444445 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}\right)} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   3. *-commutative 9.0%

      \[\leadsto -\frac{\left(-\frac{\color{blue}{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   4. associate-*r/ 9.0%

      \[\leadsto -\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \color{blue}{\frac{0.16666666666666666 \cdot 1}{\pi}}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   5. metadata-eval 9.0%

      \[\leadsto -\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{\color{blue}{0.16666666666666666}}{\pi}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   6. associate-*r/ 9.0%

      \[\leadsto -\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s}\right) - \color{blue}{\frac{0.25 \cdot 1}{r \cdot \pi}}}{s} \]

   7. metadata-eval 9.0%

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

8. Simplified 9.0%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{r \cdot \pi}}{s}} \]
9. Step-by-step derivation

   1. associate-/r* 9.0%

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

   2. div-inv 9.0%

      \[\leadsto -\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s}\right) - \color{blue}{\frac{0.25}{r} \cdot \frac{1}{\pi}}}{s} \]

10. Applied egg-rr 9.0%

    \[\leadsto -\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s}\right) - \color{blue}{\frac{0.25}{r} \cdot \frac{1}{\pi}}}{s} \]

11. Final simplification 9.0%

    \[\leadsto \frac{\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{r} \cdot \frac{1}{\pi}}{s} \]
12. Add Preprocessing

Alternative 7: 9.7% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 7.6%

   \[\leadsto \color{blue}{\frac{r \cdot \left(0.06944444444444445 \cdot \frac{r}{{s}^{3} \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi}\right) + 0.25 \cdot \frac{1}{s \cdot \pi}}{r}} \]

6. Taylor expanded in s around -inf 9.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.06944444444444445 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
7. Step-by-step derivation

   1. mul-1-neg 9.0%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.06944444444444445 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

   2. mul-1-neg 9.0%

      \[\leadsto -\frac{\color{blue}{\left(-\frac{0.06944444444444445 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}\right)} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   3. *-commutative 9.0%

      \[\leadsto -\frac{\left(-\frac{\color{blue}{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   4. associate-*r/ 9.0%

      \[\leadsto -\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \color{blue}{\frac{0.16666666666666666 \cdot 1}{\pi}}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   5. metadata-eval 9.0%

      \[\leadsto -\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{\color{blue}{0.16666666666666666}}{\pi}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   6. associate-*r/ 9.0%

      \[\leadsto -\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s}\right) - \color{blue}{\frac{0.25 \cdot 1}{r \cdot \pi}}}{s} \]

   7. metadata-eval 9.0%

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

8. Simplified 9.0%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{r \cdot \pi}}{s}} \]
9. Step-by-step derivation

   1. associate-*l/ 9.0%

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

   2. associate-/r* 9.0%

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

10. Applied egg-rr 9.0%

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

11. Final simplification 9.0%

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

Alternative 8: 9.7% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 7.6%

   \[\leadsto \color{blue}{\frac{r \cdot \left(0.06944444444444445 \cdot \frac{r}{{s}^{3} \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi}\right) + 0.25 \cdot \frac{1}{s \cdot \pi}}{r}} \]

6. Taylor expanded in s around -inf 9.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.06944444444444445 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
7. Step-by-step derivation

   1. mul-1-neg 9.0%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.06944444444444445 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

   2. mul-1-neg 9.0%

      \[\leadsto -\frac{\color{blue}{\left(-\frac{0.06944444444444445 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}\right)} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   3. *-commutative 9.0%

      \[\leadsto -\frac{\left(-\frac{\color{blue}{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   4. associate-*r/ 9.0%

      \[\leadsto -\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \color{blue}{\frac{0.16666666666666666 \cdot 1}{\pi}}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   5. metadata-eval 9.0%

      \[\leadsto -\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{\color{blue}{0.16666666666666666}}{\pi}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   6. associate-*r/ 9.0%

      \[\leadsto -\frac{\left(-\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s}\right) - \color{blue}{\frac{0.25 \cdot 1}{r \cdot \pi}}}{s} \]

   7. metadata-eval 9.0%

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

8. Simplified 9.0%

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

9. Final simplification 9.0%

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

Alternative 9: 8.7% accurate, 15.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 8.3%

   \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \frac{r}{{s}^{2} \cdot \pi} + 0.25 \cdot \frac{1}{s \cdot \pi}}{r}} \]

6. Taylor expanded in s around -inf 8.4%

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

   1. mul-1-neg 8.4%

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

   2. *-commutative 8.4%

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

   3. associate-*r/ 8.4%

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

   4. metadata-eval 8.4%

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

8. Simplified 8.4%

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

9. Final simplification 8.4%

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

Alternative 10: 8.7% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 8.3%

   \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \frac{r}{{s}^{2} \cdot \pi} + 0.25 \cdot \frac{1}{s \cdot \pi}}{r}} \]

6. Taylor expanded in r around inf 8.3%

   \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi}} \]
7. Step-by-step derivation

   1. associate-*r/ 8.3%

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

   2. metadata-eval 8.3%

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

   3. associate-/r* 8.3%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi}} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   4. *-commutative 8.3%

      \[\leadsto \frac{\frac{0.25}{r}}{\color{blue}{\pi \cdot s}} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   5. associate-/r* 8.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{0.25}{r}}{\pi}}{s}} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   6. associate-/r* 8.3%

      \[\leadsto \frac{\color{blue}{\frac{0.25}{r \cdot \pi}}}{s} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   7. metadata-eval 8.3%

      \[\leadsto \frac{\frac{\color{blue}{0.25 \cdot 1}}{r \cdot \pi}}{s} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   8. associate-*r/ 8.3%

      \[\leadsto \frac{\color{blue}{0.25 \cdot \frac{1}{r \cdot \pi}}}{s} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   9. unpow2 8.3%

      \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\color{blue}{\left(s \cdot s\right)} \cdot \pi} \]

   10. associate-*l* 8.3%

       \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\color{blue}{s \cdot \left(s \cdot \pi\right)}} \]

   11. associate-/l/ 8.3%

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

   12. associate-/l* 8.3%

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

   13. div-sub 8.3%

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

8. Simplified 8.3%

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

Alternative 11: 8.8% accurate, 33.0× speedup

Math

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

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) / Float32(r * s)) / Float32(pi))
end

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 8.3%

   \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \frac{r}{{s}^{2} \cdot \pi} + 0.25 \cdot \frac{1}{s \cdot \pi}}{r}} \]

6. Taylor expanded in r around 0 8.1%

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

   1. frac-2neg 8.1%

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

   2. div-inv 8.1%

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

   3. distribute-neg-frac 8.1%

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

   4. metadata-eval 8.1%

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

8. Applied egg-rr 8.1%

   \[\leadsto \color{blue}{\frac{-0.25}{s \cdot \pi} \cdot \frac{1}{-r}} \]
9. Step-by-step derivation

   1. *-commutative 8.1%

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

   2. frac-2neg 8.1%

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

   3. metadata-eval 8.1%

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

   4. frac-times 8.1%

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

   5. metadata-eval 8.1%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 4.5%

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

   8. sqr-neg 4.5%

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

   9. sqrt-unprod 4.5%

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

   10. add-sqr-sqrt 4.5%

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

   11. add-sqr-sqrt -0.0%

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

   12. sqrt-unprod 8.1%

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

   13. sqr-neg 8.1%

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

   14. sqrt-unprod 8.1%

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

   15. add-sqr-sqrt 8.1%

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

   16. associate-*r* 8.1%

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

   17. associate-/r* 8.1%

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

10. Applied egg-rr 8.1%

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

Alternative 12: 8.8% 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 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in r around 0 8.3%

   \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \frac{r}{{s}^{2} \cdot \pi} + 0.25 \cdot \frac{1}{s \cdot \pi}}{r}} \]

6. Taylor expanded in r around 0 8.1%

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

   1. frac-2neg 8.1%

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

   2. div-inv 8.1%

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

   3. distribute-neg-frac 8.1%

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

   4. metadata-eval 8.1%

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

8. Applied egg-rr 8.1%

   \[\leadsto \color{blue}{\frac{-0.25}{s \cdot \pi} \cdot \frac{1}{-r}} \]
9. Step-by-step derivation

   1. frac-2neg 8.1%

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

   2. metadata-eval 8.1%

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

   3. frac-times 8.1%

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

   4. metadata-eval 8.1%

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

   5. add-sqr-sqrt -0.0%

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

   6. sqrt-unprod 4.5%

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

   7. sqr-neg 4.5%

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

   8. sqrt-unprod 4.5%

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

   9. add-sqr-sqrt 4.5%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 8.1%

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

   12. sqr-neg 8.1%

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

   13. sqrt-unprod 8.1%

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

   14. add-sqr-sqrt 8.1%

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

   15. *-commutative 8.1%

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

   16. associate-/r* 8.1%

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

10. Applied egg-rr 8.1%

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

Alternative 13: 8.8% 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 99.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 8.1%

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

Reproduce

herbie shell --seed 2024177 
(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))))