Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 19.1s

Alternatives: 12

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

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + (powf(((float) M_E), (r * (-0.3333333333333333f / s))) / 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(exp(1)) ^ Float32(r * Float32(Float32(-0.3333333333333333) / s))) / r)))
end

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

6. Applied egg-rr 98.7%

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

   1. pow1/3 98.7%

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

   2. pow-exp 99.6%

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

8. Applied egg-rr 99.6%

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

   1. rec-exp 99.5%

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

   2. *-commutative 99.5%

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

   3. metadata-eval 99.5%

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

   4. times-frac 99.5%

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

   5. *-un-lft-identity 99.5%

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

   6. frac-2neg 99.5%

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

   7. distribute-frac-neg 99.5%

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

   8. remove-double-neg 99.5%

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

   9. *-commutative 99.5%

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

   10. distribute-rgt-neg-in 99.5%

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

   11. metadata-eval 99.5%

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

   12. metadata-eval 99.5%

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

   13. div-inv 99.5%

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

   14. un-div-inv 99.5%

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

   15. clear-num 99.6%

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

   16. *-un-lft-identity 99.6%

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

   17. exp-prod 99.7%

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (exp (/ r (- s))) r)
   (/ (/ 1.0 (exp (* (/ 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 / 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(Float32(Float32(1.0) / 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) + ((single(1.0) / exp(((r / s) * single(0.3333333333333333)))) / r));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

6. Applied egg-rr 98.7%

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

   1. pow1/3 98.7%

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

   2. pow-exp 99.6%

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.5%

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

   1. associate-*r/ 99.6%

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

   2. associate-/l* 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 5: 15.7% 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 99.5%

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

3. Simplified 99.1%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in r around 0 16.6%

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

8. Final simplification 16.6%

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

Alternative 6: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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) / r) - Float32(Float32(0.3333333333333333) * Float32(Float32(1.0) / s)))))
end

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in r around 0 10.7%

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

   1. +-commutative 10.7%

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

   2. associate-*r/ 10.7%

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

   3. associate-*l/ 10.7%

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

   4. *-commutative 10.7%

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

9. Simplified 10.7%

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

10. Taylor expanded in r around 0 10.7%

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

11. Final simplification 10.7%

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

Alternative 7: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in r around 0 10.7%

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

   1. +-commutative 10.7%

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

   2. associate-*r/ 10.7%

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

   3. associate-*l/ 10.7%

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

   4. *-commutative 10.7%

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

9. Simplified 10.7%

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

10. Taylor expanded in r around 0 10.7%

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

    1. cancel-sign-sub-inv 10.7%

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

    2. metadata-eval 10.7%

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

    3. associate-*r/ 10.7%

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

    4. metadata-eval 10.7%

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

12. Simplified 10.7%

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

13. Final simplification 10.7%

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

Alternative 8: 9.6% accurate, 2.0× speedup

Math

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

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(0.125) * Float32(Float32(Float32(1.0) + exp(Float32(Float32(-r) / s))) / Float32(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 99.5%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 10.3%

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

6. Taylor expanded in r around inf 10.3%

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

   1. associate-*r/ 10.3%

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

   2. neg-mul-1 10.3%

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

8. Simplified 10.3%

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

9. Final simplification 10.3%

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

Alternative 9: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 10.3%

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

6. Taylor expanded in r around inf 10.3%

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

   1. associate-*r/ 10.3%

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

   2. neg-mul-1 10.3%

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

   3. *-commutative 10.3%

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

   4. *-commutative 10.3%

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

   5. associate-*l* 10.3%

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

8. Simplified 10.3%

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

9. Taylor expanded in r around inf 10.3%

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

    1. associate-*r/ 10.3%

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

    2. associate-*r* 10.3%

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

    3. *-commutative 10.3%

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

    4. times-frac 10.3%

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

    5. associate-*r/ 10.3%

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

    6. mul-1-neg 10.3%

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

11. Simplified 10.3%

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

12. Final simplification 10.3%

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

Alternative 10: 9.1% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in r around 0 10.7%

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

   1. +-commutative 10.7%

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

   2. associate-*r/ 10.7%

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

   3. associate-*l/ 10.7%

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

   4. *-commutative 10.7%

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

9. Simplified 10.7%

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

10. Taylor expanded in r around 0 10.7%

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

11. Taylor expanded in r around 0 10.1%

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

    1. mul-1-neg 10.1%

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

    2. unsub-neg 10.1%

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

13. Simplified 10.1%

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

14. Final simplification 10.1%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\left(\frac{1}{r} - 0.3333333333333333 \cdot \frac{1}{s}\right) + \frac{1 - \frac{r}{s}}{r}\right) \]
15. Add Preprocessing

Alternative 11: 9.1% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 10.3%

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

6. Taylor expanded in s around inf 9.7%

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

   1. associate-*r* 9.7%

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

   2. *-commutative 9.7%

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

   3. *-commutative 9.7%

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

   4. associate-/r* 9.7%

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

   5. div-inv 9.7%

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

   6. *-commutative 9.7%

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

8. Applied egg-rr 9.7%

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

9. Final simplification 9.7%

   \[\leadsto \frac{0.25}{\pi} \cdot \frac{1}{s \cdot r} \]
10. Add Preprocessing

Alternative 12: 9.1% accurate, 33.0× 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 99.5%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 10.3%

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

6. Taylor expanded in s around inf 9.7%

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

7. Final simplification 9.7%

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

Reproduce

herbie shell --seed 2024011 
(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))))