Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 16.3s

Alternatives: 15

Speedup: N/A×

Specification

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

Math

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

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + (powf(expf(-0.6666666666666666f), ((r / s) / 2.0f)) / 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((exp(Float32(-0.6666666666666666)) ^ Float32(Float32(r / s) / Float32(2.0))) / r)))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around inf 99.6%

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

   1. pow-exp 99.2%

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

   2. sqr-pow 99.2%

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

   3. pow-prod-down 99.2%

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

   4. prod-exp 99.7%

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

   5. metadata-eval 99.7%

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

   6. associate-/l/ 99.7%

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

   7. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/r* 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around inf 99.6%

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

   1. associate-/r* 99.6%

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

   2. div-inv 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + (expf(((r / s) * -0.3333333333333333f)) / 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(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / r)))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around inf 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 15.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + ((1.0f / (1.0f + ((r / s) * 0.3333333333333333f))) / 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(1.0) / Float32(Float32(1.0) + Float32(Float32(r / s) * Float32(0.3333333333333333)))) / r)))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

   1. pow-to-exp 99.3%

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

   2. rem-log-exp 99.6%

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

   3. metadata-eval 99.6%

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

   4. times-frac 99.6%

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

   5. neg-mul-1 99.6%

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

   6. *-commutative 99.6%

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

   7. distribute-frac-neg 99.6%

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

   8. *-commutative 99.6%

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

   9. exp-neg 99.6%

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

   10. add-sqr-sqrt 99.5%

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

   11. sqrt-unprod 99.6%

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

   12. sqr-neg 99.6%

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

   13. sqrt-unprod -0.0%

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

   14. add-sqr-sqrt 7.4%

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

   15. associate-/l/ 7.4%

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

   16. exp-cbrt 7.4%

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

   17. add-sqr-sqrt -0.0%

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

5. Applied egg-rr 98.7%

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

6. Taylor expanded in r around 0 15.8%

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

   1. *-commutative 15.8%

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

8. Simplified 15.8%

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

9. Final simplification 15.8%

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

Alternative 5: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 9.1%

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

5. Taylor expanded in s around 0 9.1%

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

   1. associate-*r/ 9.1%

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

   2. mul-1-neg 9.1%

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

7. Simplified 9.1%

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

8. Final simplification 9.1%

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

Alternative 6: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 9.1%

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

5. Taylor expanded in s around 0 9.1%

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

   1. associate-*r/ 9.1%

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

   2. mul-1-neg 9.1%

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

7. Simplified 9.1%

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

8. Taylor expanded in r around -inf 9.1%

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

   1. associate-*r* 9.1%

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

   2. associate-*r/ 9.1%

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

   3. *-commutative 9.1%

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

   4. times-frac 9.1%

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

   5. sub-neg 9.1%

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

   6. neg-mul-1 9.1%

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

   7. rec-exp 9.1%

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

   8. associate-*r/ 9.1%

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

   9. metadata-eval 9.1%

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

   10. metadata-eval 9.1%

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

10. Simplified 9.1%

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

11. Final simplification 9.1%

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

Alternative 7: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 9.1%

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

5. Taylor expanded in r around inf 9.1%

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

   1. associate-*r/ 9.1%

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

   2. *-commutative 9.1%

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

   3. associate-/r* 9.1%

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

   4. associate-*r/ 9.1%

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

   5. mul-1-neg 9.1%

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

7. Simplified 9.1%

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

8. Taylor expanded in r around inf 9.1%

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

   1. mul-1-neg 9.1%

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

   2. *-commutative 9.1%

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

   3. associate-*r* 9.1%

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

10. Simplified 9.1%

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

11. Final simplification 9.1%

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

Alternative 8: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 9.1%

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

5. Taylor expanded in r around inf 9.1%

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

   1. associate-*r/ 9.1%

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

   2. *-commutative 9.1%

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

   3. associate-/r* 9.1%

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

   4. associate-*r/ 9.1%

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

   5. mul-1-neg 9.1%

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

7. Simplified 9.1%

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

8. Taylor expanded in r around inf 9.1%

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

   1. associate-*r/ 9.1%

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

   2. *-commutative 9.1%

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

   3. associate-*r* 9.1%

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

   4. times-frac 9.1%

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

   5. associate-*r/ 9.1%

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

   6. mul-1-neg 9.1%

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

10. Simplified 9.1%

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

11. Final simplification 9.1%

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

Alternative 9: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 9.1%

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

5. Taylor expanded in r around inf 9.1%

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

   1. associate-*r/ 9.1%

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

   2. *-commutative 9.1%

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

   3. associate-/r* 9.1%

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

   4. associate-*r/ 9.1%

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

   5. mul-1-neg 9.1%

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

7. Simplified 9.1%

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

8. Final simplification 9.1%

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

Alternative 10: 9.1% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (s r) :precision binary32 (* (/ 1.0 PI) (/ 0.25 (* s r))))

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(1.0) / Float32(pi)) * Float32(Float32(0.25) / Float32(s * r)))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 9.1%

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

5. Taylor expanded in s around inf 8.7%

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

   1. associate-/r* 8.7%

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

7. Simplified 8.7%

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

   1. *-un-lft-identity 8.7%

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

   2. *-commutative 8.7%

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

   3. times-frac 8.7%

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

   4. associate-/l/ 8.7%

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

9. Applied egg-rr 8.7%

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

10. Final simplification 8.7%

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

Alternative 11: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 9.1%

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

5. Taylor expanded in s around inf 8.7%

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

   1. frac-2neg 8.7%

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

   2. div-inv 8.7%

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

   3. metadata-eval 8.7%

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

   4. distribute-rgt-neg-in 8.7%

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

   5. distribute-rgt-neg-in 8.7%

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

7. Applied egg-rr 8.7%

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

   1. associate-*r/ 8.7%

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

   2. metadata-eval 8.7%

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

   3. associate-*r* 8.7%

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

9. Simplified 8.7%

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

10. Final simplification 8.7%

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

Alternative 12: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 9.1%

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

5. Taylor expanded in s around inf 8.7%

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

6. Final simplification 8.7%

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

Alternative 13: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 9.1%

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

5. Taylor expanded in s around inf 8.7%

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

   1. associate-/r* 8.7%

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

7. Simplified 8.7%

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

8. Taylor expanded in r around 0 8.7%

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

   1. *-commutative 8.7%

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

   2. associate-*r* 8.7%

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

10. Simplified 8.7%

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

11. Final simplification 8.7%

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

Alternative 14: 9.1% accurate, 4.0× 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 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 9.1%

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

5. Taylor expanded in s around inf 8.7%

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

   1. associate-/r* 8.7%

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

7. Simplified 8.7%

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

8. Final simplification 8.7%

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

Alternative 15: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

4. Taylor expanded in r around 0 9.1%

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

5. Taylor expanded in s around 0 9.1%

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

   1. associate-*r/ 9.1%

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

   2. mul-1-neg 9.1%

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

7. Simplified 9.1%

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

8. Taylor expanded in r around 0 8.7%

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

   1. associate-*r* 8.7%

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

   2. associate-/r* 8.7%

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

10. Simplified 8.7%

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

11. Final simplification 8.7%

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

Reproduce

herbie shell --seed 2023336 
(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))))