Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 26.1s

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ \mathsf{fma}\left(t_0, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, t_0 \cdot \frac{e^{\frac{-r}{s}}}{r}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* s PI))))
   (fma
    t_0
    (/ (exp (* -0.3333333333333333 (/ r s))) r)
    (* t_0 (/ (exp (/ (- r) s)) r)))))

C

float code(float s, float r) {
	float t_0 = 0.125f / (s * ((float) M_PI));
	return fmaf(t_0, (expf((-0.3333333333333333f * (r / s))) / r), (t_0 * (expf((-r / s)) / r)));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(s * Float32(pi)))
	return fma(t_0, Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) / r), Float32(t_0 * Float32(exp(Float32(Float32(-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 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Taylor expanded in r around inf 99.6%

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

5. Taylor expanded in s around 0 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Taylor expanded in r around inf 99.6%

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

   1. rem-log-exp 99.3%

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

   2. associate-*r/ 99.4%

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

   3. *-commutative 99.4%

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

   4. rem-log-exp 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 11.5% accurate, 1.4× speedup

Math

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

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 7.3%

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

5. Taylor expanded in s around inf 7.0%

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

   1. add-sqr-sqrt 7.0%

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

   2. sqrt-unprod 7.0%

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

   3. sqr-neg 7.0%

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

   4. sqrt-unprod -0.0%

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

   5. add-sqr-sqrt 4.5%

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

   6. distribute-lft-neg-in 4.5%

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

   7. log1p-expm1-u 8.2%

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

   8. *-commutative 8.2%

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

   9. distribute-rgt-neg-in 8.2%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 10.4%

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

   12. sqr-neg 10.4%

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

   13. sqrt-unprod 10.4%

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

   14. add-sqr-sqrt 10.4%

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

   15. associate-*l* 10.4%

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

7. Applied egg-rr 10.4%

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

8. Final simplification 10.4%

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

Alternative 5: 43.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 7.3%

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

5. Taylor expanded in s around inf 7.0%

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

   1. expm1-log1p-u 7.0%

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

   2. expm1-udef 5.9%

      \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)} - 1}} \]

   3. *-commutative 5.9%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{\left(s \cdot \pi\right) \cdot r}\right)} - 1} \]

   4. associate-*l* 5.9%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{s \cdot \left(\pi \cdot r\right)}\right)} - 1} \]

7. Applied egg-rr 5.9%

   \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)} - 1}} \]
8. Step-by-step derivation

   1. expm1-def 7.0%

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

   2. expm1-log1p 7.0%

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

   3. *-commutative 7.0%

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

9. Simplified 7.0%

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

    1. log1p-expm1-u 43.6%

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

11. Applied egg-rr 43.6%

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

12. Final simplification 43.6%

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

Alternative 6: 9.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Taylor expanded in r around 0 7.3%

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

   1. clear-num 7.3%

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

   2. associate-/r/ 7.3%

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

6. Applied egg-rr 7.3%

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

7. Final simplification 7.3%

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

Alternative 7: 9.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 7.3%

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

5. Taylor expanded in s around 0 7.3%

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

   1. associate-*r/ 7.3%

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

   2. times-frac 7.3%

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

   3. associate-*r/ 7.3%

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

   4. mul-1-neg 7.3%

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

7. Simplified 7.3%

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

8. Taylor expanded in r around inf 7.3%

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

   1. mul-1-neg 7.3%

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

   2. *-commutative 7.3%

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

   3. *-commutative 7.3%

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

   4. *-commutative 7.3%

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

   5. associate-*r* 7.3%

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

   6. *-commutative 7.3%

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

10. Simplified 7.3%

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

11. Final simplification 7.3%

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

Alternative 8: 9.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 7.3%

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

5. Taylor expanded in s around 0 7.3%

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

   1. associate-*r/ 7.3%

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

   2. times-frac 7.3%

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

   3. associate-*r/ 7.3%

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

   4. mul-1-neg 7.3%

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

7. Simplified 7.3%

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

8. Taylor expanded in r around inf 7.3%

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

   1. mul-1-neg 7.3%

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

   2. distribute-neg-frac 7.3%

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

10. Simplified 7.3%

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

11. Final simplification 7.3%

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

Alternative 9: 8.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s} \cdot \frac{\frac{2}{r}}{\pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / s) * ((single(2.0) / r) / single(pi));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 7.3%

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

5. Taylor expanded in s around 0 7.3%

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

   1. associate-*r/ 7.3%

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

   2. times-frac 7.3%

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

   3. associate-*r/ 7.3%

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

   4. mul-1-neg 7.3%

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

7. Simplified 7.3%

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

8. Taylor expanded in r around 0 7.0%

   \[\leadsto \frac{0.125}{s} \cdot \color{blue}{\frac{2}{r \cdot \pi}} \]
9. Step-by-step derivation

   1. associate-/r* 7.0%

      \[\leadsto \frac{0.125}{s} \cdot \color{blue}{\frac{\frac{2}{r}}{\pi}} \]

10. Simplified 7.0%

    \[\leadsto \frac{0.125}{s} \cdot \color{blue}{\frac{\frac{2}{r}}{\pi}} \]

11. Final simplification 7.0%

    \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{2}{r}}{\pi} \]

Alternative 10: 8.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 7.3%

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

5. Taylor expanded in s around inf 7.0%

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

6. Final simplification 7.0%

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

Alternative 11: 8.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 7.3%

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

5. Taylor expanded in s around inf 7.0%

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

   1. expm1-log1p-u 7.0%

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

   2. expm1-udef 5.9%

      \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)} - 1}} \]

   3. *-commutative 5.9%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{\left(s \cdot \pi\right) \cdot r}\right)} - 1} \]

   4. associate-*l* 5.9%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{s \cdot \left(\pi \cdot r\right)}\right)} - 1} \]

7. Applied egg-rr 5.9%

   \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)} - 1}} \]
8. Step-by-step derivation

   1. expm1-def 7.0%

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

   2. expm1-log1p 7.0%

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

   3. *-commutative 7.0%

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

9. Simplified 7.0%

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

10. Final simplification 7.0%

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

Alternative 12: 8.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 7.3%

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

5. Taylor expanded in s around inf 7.0%

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

   1. associate-/r* 7.0%

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

7. Simplified 7.0%

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

8. Final simplification 7.0%

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

Reproduce

herbie shell --seed 2023320 
(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))))