Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 11.4s

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.3× speedup

Math

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

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(r / Float32(-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 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in s around 0 99.3%

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

5. Taylor expanded in r around inf 99.6%

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

   1. metadata-eval 99.6%

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

   2. times-frac 99.6%

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

   3. neg-mul-1 99.6%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 6.4%

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

   6. sqr-neg 6.4%

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

   7. sqrt-unprod 6.4%

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

   8. add-sqr-sqrt 6.4%

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

   9. frac-2neg 6.4%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 99.6%

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

   12. sqr-neg 99.6%

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

   13. sqrt-unprod 99.5%

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

   14. add-sqr-sqrt 99.6%

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

   15. *-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. Applied egg-rr 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. Step-by-step derivation

   1. distribute-rgt-neg-in 99.6%

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

   2. metadata-eval 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 1.3× 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 99.3%

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

4. Taylor expanded in s around 0 99.3%

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

5. Taylor expanded in r around inf 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 11.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Taylor expanded in r around 0 8.0%

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

5. Taylor expanded in s around inf 7.7%

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

   1. log1p-expm1-u 10.7%

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

7. Applied egg-rr 10.7%

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

8. Final simplification 10.7%

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

Alternative 4: 11.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Taylor expanded in r around 0 8.0%

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

5. Taylor expanded in s around inf 7.7%

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

   1. associate-*r* 7.7%

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

   2. *-commutative 7.7%

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

7. Simplified 7.7%

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

   1. log1p-expm1-u 10.7%

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

9. Applied egg-rr 10.7%

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

10. Final simplification 10.7%

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

Alternative 5: 10.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Taylor expanded in r around 0 8.0%

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

5. Taylor expanded in s around 0 8.0%

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

   1. associate-*r/ 8.0%

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

   2. associate-/l* 8.0%

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

   3. mul-1-neg 8.0%

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

   4. distribute-frac-neg 8.0%

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

7. Simplified 8.0%

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

8. Final simplification 8.0%

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

Alternative 6: 10.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Taylor expanded in r around 0 8.0%

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

   1. associate-*l/ 8.0%

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

6. Applied egg-rr 8.0%

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

7. Final simplification 8.0%

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

Alternative 7: 10.0% 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 99.3%

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

4. Taylor expanded in r around 0 8.0%

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

5. Taylor expanded in r around inf 8.0%

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

   1. mul-1-neg 8.0%

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

7. Simplified 8.0%

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

8. Final simplification 8.0%

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

Alternative 8: 10.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Taylor expanded in r around 0 8.0%

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

5. Taylor expanded in r around inf 8.0%

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

   1. associate-*r* 8.0%

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

   2. mul-1-neg 8.0%

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

   3. distribute-frac-neg 8.0%

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

   4. *-commutative 8.0%

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

7. Simplified 8.0%

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

8. Final simplification 8.0%

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

Alternative 9: 10.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Taylor expanded in r around 0 8.0%

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

5. Taylor expanded in r around -inf 8.0%

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

   1. associate-*r/ 8.0%

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

   2. times-frac 8.0%

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

   3. sub-neg 8.0%

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

   4. metadata-eval 8.0%

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

   5. +-commutative 8.0%

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

   6. mul-1-neg 8.0%

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

   7. mul-1-neg 8.0%

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

   8. distribute-frac-neg 8.0%

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

   9. unsub-neg 8.0%

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

7. Simplified 8.0%

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

8. Taylor expanded in r around inf 8.0%

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

   1. associate-*r/ 8.0%

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

   2. associate-/l* 8.0%

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

   3. mul-1-neg 8.0%

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

   4. distribute-neg-frac 8.0%

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

10. Simplified 8.0%

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

11. Final simplification 8.0%

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

Alternative 10: 9.4% 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 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.0%

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

5. Taylor expanded in s around inf 7.7%

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

6. Final simplification 7.7%

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

Alternative 11: 9.4% accurate, 4.0× 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 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.0%

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

5. Taylor expanded in s around inf 7.7%

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

   1. associate-*r* 7.7%

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

   2. *-commutative 7.7%

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

7. Simplified 7.7%

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

8. Final simplification 7.7%

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

Alternative 12: 9.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.0%

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

5. Taylor expanded in s around inf 7.7%

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

   1. associate-*r* 7.7%

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

   2. *-commutative 7.7%

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

7. Simplified 7.7%

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

8. Taylor expanded in r around 0 7.7%

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

   1. associate-/r* 7.7%

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

   2. associate-/r* 7.7%

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

10. Simplified 7.7%

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

11. Final simplification 7.7%

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

Reproduce

herbie shell --seed 2023271 
(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))))