Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 26.0s

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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(\pi \cdot \left(s \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (/ 0.125 s) PI) (/ (exp (/ (- r) s)) r))
  (* 0.75 (/ (exp (/ (- r) (* s 3.0))) (* r (* PI (* s 6.0)))))))

C

float code(float s, float r) {
	return (((0.125f / s) / ((float) M_PI)) * (expf((-r / s)) / r)) + (0.75f * (expf((-r / (s * 3.0f))) / (r * (((float) M_PI) * (s * 6.0f)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(0.125) / s) / Float32(pi)) * Float32(exp(Float32(Float32(-r) / s)) / r)) + Float32(Float32(0.75) * Float32(exp(Float32(Float32(-r) / Float32(s * Float32(3.0)))) / Float32(r * Float32(Float32(pi) * Float32(s * Float32(6.0)))))))
end

MATLAB

function tmp = code(s, r)
	tmp = (((single(0.125) / s) / single(pi)) * (exp((-r / s)) / r)) + (single(0.75) * (exp((-r / (s * single(3.0)))) / (r * (single(pi) * (s * single(6.0))))));
end

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. *-commutative 99.6%

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

   3. distribute-frac-neg 99.6%

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

   4. associate-/l* 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l* 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.6%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

6. Taylor expanded in s around 0 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \color{blue}{\left(6 \cdot \left(s \cdot \pi\right)\right)}} \]
7. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \color{blue}{\left(\left(s \cdot \pi\right) \cdot 6\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(\color{blue}{\left(\pi \cdot s\right)} \cdot 6\right)} \]

   3. associate-*l* 99.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \color{blue}{\left(\pi \cdot \left(s \cdot 6\right)\right)}} \]

8. Simplified 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \color{blue}{\left(\pi \cdot \left(s \cdot 6\right)\right)}} \]
9. Step-by-step derivation

   1. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(\pi \cdot \left(s \cdot 6\right)\right)} \]

   2. div-inv 99.6%

      \[\leadsto \color{blue}{\left(\frac{0.125}{s} \cdot \frac{1}{\pi}\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(\pi \cdot \left(s \cdot 6\right)\right)} \]

10. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{\left(\frac{0.125}{s} \cdot \frac{1}{\pi}\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(\pi \cdot \left(s \cdot 6\right)\right)} \]
11. Step-by-step derivation

    1. div-inv 99.6%

       \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(\pi \cdot \left(s \cdot 6\right)\right)} \]

12. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(\pi \cdot \left(s \cdot 6\right)\right)} \]

13. Final simplification 99.6%

    \[\leadsto \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(\pi \cdot \left(s \cdot 6\right)\right)} \]
14. Add Preprocessing

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((-r / s)) / r) + (exp((r / (s * single(-3.0)))) / 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 99.3%

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

   1. pow-to-exp 99.4%

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

   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. frac-2neg 99.6%

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

   7. remove-double-neg 99.6%

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

   8. *-commutative 99.6%

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

   9. distribute-rgt-neg-in 99.6%

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

   10. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 12.0% 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 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.4%

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

6. Taylor expanded in s around inf 8.0%

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

   1. log1p-expm1-u 11.0%

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

8. Applied egg-rr 11.0%

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

9. Final simplification 11.0%

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

Alternative 4: 12.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(Float32(pi) * Float32(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 99.3%

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

5. Taylor expanded in r around 0 8.4%

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

6. Taylor expanded in s around inf 8.0%

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

   1. *-commutative 8.0%

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

   2. *-commutative 8.0%

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

   3. associate-*l* 8.0%

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

8. Simplified 8.0%

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

   1. log1p-expm1-u 11.0%

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

10. Applied egg-rr 11.0%

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

11. Final simplification 11.0%

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

Alternative 5: 10.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(Float32(-r) / s)) / r) + Float32(Float32(Float32(1.0) + 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(1.0) + (r * (single(-0.3333333333333333) / 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 99.3%

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

5. Taylor expanded in r around 0 8.6%

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

   1. *-commutative 8.6%

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

   2. associate-*l/ 8.6%

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

   3. associate-/l* 8.6%

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

7. Simplified 8.6%

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

8. Final simplification 8.6%

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

Alternative 6: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.4%

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

6. Taylor expanded in r around inf 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-commutative 8.4%

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

   3. *-commutative 8.4%

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

   4. times-frac 8.4%

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

   5. associate-/l/ 8.4%

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

   6. associate-*r/ 8.4%

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

   7. neg-mul-1 8.4%

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

8. Simplified 8.4%

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

   1. clear-num 8.4%

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

   2. associate-/r/ 8.4%

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

10. Applied egg-rr 8.4%

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

    1. distribute-frac-neg 8.4%

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

    2. exp-neg 8.4%

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

12. Applied egg-rr 8.4%

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

13. Final simplification 8.4%

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

Alternative 7: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.4%

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

6. Taylor expanded in r around inf 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-commutative 8.4%

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

   3. *-commutative 8.4%

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

   4. times-frac 8.4%

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

   5. associate-/l/ 8.4%

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

   6. associate-*r/ 8.4%

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

   7. neg-mul-1 8.4%

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

8. Simplified 8.4%

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

   1. clear-num 8.4%

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

   2. associate-/r/ 8.4%

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

10. Applied egg-rr 8.4%

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

11. Final simplification 8.4%

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

Alternative 8: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.4%

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

6. Taylor expanded in r around inf 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-commutative 8.4%

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

   3. *-commutative 8.4%

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

   4. times-frac 8.4%

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

   5. associate-/l/ 8.4%

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

   6. associate-*r/ 8.4%

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

   7. neg-mul-1 8.4%

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

8. Simplified 8.4%

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

   1. distribute-frac-neg 8.4%

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

   2. exp-neg 8.4%

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

10. Applied egg-rr 8.4%

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

11. Final simplification 8.4%

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

Alternative 9: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((single(1.0) + 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 99.3%

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

5. Taylor expanded in r around 0 8.4%

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

6. Taylor expanded in r around inf 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-commutative 8.4%

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

   3. *-commutative 8.4%

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

   4. times-frac 8.4%

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

   5. associate-/l/ 8.4%

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

   6. associate-*r/ 8.4%

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

   7. neg-mul-1 8.4%

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

8. Simplified 8.4%

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

9. Taylor expanded in s around 0 8.4%

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

    1. mul-1-neg 8.4%

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

11. Simplified 8.4%

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

12. Final simplification 8.4%

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

Alternative 10: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(-0.125) / r) * ((single(-1.0) - exp((-r / s))) / (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 99.3%

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

5. Taylor expanded in r around 0 8.4%

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

6. Taylor expanded in r around -inf 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-commutative 8.4%

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

   3. times-frac 8.4%

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

   4. sub-neg 8.4%

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

   5. metadata-eval 8.4%

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

   6. +-commutative 8.4%

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

   7. mul-1-neg 8.4%

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

   8. unsub-neg 8.4%

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

   9. associate-*r/ 8.4%

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

   10. neg-mul-1 8.4%

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

   11. *-commutative 8.4%

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

8. Simplified 8.4%

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

9. Final simplification 8.4%

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

Alternative 11: 9.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.4%

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

6. Taylor expanded in r around inf 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-commutative 8.4%

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

   3. *-commutative 8.4%

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

   4. times-frac 8.4%

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

   5. associate-/l/ 8.4%

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

   6. associate-*r/ 8.4%

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

   7. neg-mul-1 8.4%

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

8. Simplified 8.4%

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

9. Taylor expanded in s around 0 8.4%

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

10. Final simplification 8.4%

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

Alternative 12: 9.3% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.125}{s} \cdot \frac{1}{\pi}\right) \cdot \frac{2}{r} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (* (* (/ 0.125 s) (/ 1.0 PI)) (/ 2.0 r)))

C

float code(float s, float r) {
	return ((0.125f / s) * (1.0f / ((float) M_PI))) * (2.0f / r);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / s) * Float32(Float32(1.0) / Float32(pi))) * Float32(Float32(2.0) / r))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / s) * (single(1.0) / single(pi))) * (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 99.3%

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

5. Taylor expanded in r around 0 8.4%

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

6. Taylor expanded in r around inf 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-commutative 8.4%

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

   3. *-commutative 8.4%

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

   4. times-frac 8.4%

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

   5. associate-/l/ 8.4%

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

   6. associate-*r/ 8.4%

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

   7. neg-mul-1 8.4%

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

8. Simplified 8.4%

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

   1. clear-num 8.4%

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

   2. associate-/r/ 8.4%

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

10. Applied egg-rr 8.4%

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

11. Taylor expanded in r around 0 8.0%

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

12. Final simplification 8.0%

    \[\leadsto \left(\frac{0.125}{s} \cdot \frac{1}{\pi}\right) \cdot \frac{2}{r} \]
13. Add Preprocessing

Alternative 13: 9.3% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.4%

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

6. Taylor expanded in s around inf 8.0%

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

   1. clear-num 8.0%

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

   2. associate-/r/ 8.0%

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

   3. *-commutative 8.0%

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

   4. associate-/r* 8.0%

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

   5. *-commutative 8.0%

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

   6. associate-/r* 8.0%

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

8. Applied egg-rr 8.0%

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

9. Final simplification 8.0%

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

Alternative 14: 9.3% 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 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.4%

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

6. Taylor expanded in s around inf 8.0%

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

7. Final simplification 8.0%

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

Alternative 15: 9.3% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 8.4%

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

6. Taylor expanded in s around inf 8.0%

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

   1. *-commutative 8.0%

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

   2. *-commutative 8.0%

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

   3. associate-*l* 8.0%

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

8. Simplified 8.0%

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

9. Final simplification 8.0%

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

Reproduce

herbie shell --seed 2024076 
(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))))