Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.5% → 99.4%

Time: 10.6s

Alternatives: 5

Speedup: 21.8×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Initial Program: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{log1p}\left(u \cdot -4\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (- s) (log1p (* u -4.0))))

C

float code(float s, float u) {
	return -s * log1pf((u * -4.0f));
}

Julia

function code(s, u)
	return Float32(Float32(-s) * log1p(Float32(u * Float32(-4.0))))
end

Derivation

1. Initial program 63.5%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Step-by-step derivation

   1. log-rec 66.1%

      \[\leadsto s \cdot \color{blue}{\left(-\log \left(1 - 4 \cdot u\right)\right)} \]

   2. distribute-rgt-neg-out 66.1%

      \[\leadsto \color{blue}{-s \cdot \log \left(1 - 4 \cdot u\right)} \]

   3. distribute-lft-neg-in 66.1%

      \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(1 - 4 \cdot u\right)} \]

   4. cancel-sign-sub-inv 66.1%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + \left(-4\right) \cdot u\right)} \]

   5. log1p-def 99.2%

      \[\leadsto \left(-s\right) \cdot \color{blue}{\mathsf{log1p}\left(\left(-4\right) \cdot u\right)} \]

   6. *-commutative 99.2%

      \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\color{blue}{u \cdot \left(-4\right)}\right) \]

   7. metadata-eval 99.2%

      \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(u \cdot \color{blue}{-4}\right) \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)} \]

4. Final simplification 99.2%

   \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(u \cdot -4\right) \]

Alternative 2: 88.5% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \frac{s}{\frac{0.25}{u} + -0.5} \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (/ s (+ (/ 0.25 u) -0.5)))

C

float code(float s, float u) {
	return s / ((0.25f / u) + -0.5f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s / ((0.25e0 / u) + (-0.5e0))
end function

Julia

function code(s, u)
	return Float32(s / Float32(Float32(Float32(0.25) / u) + Float32(-0.5)))
end

MATLAB

function tmp = code(s, u)
	tmp = s / ((single(0.25) / u) + single(-0.5));
end

Derivation

1. Initial program 63.5%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Taylor expanded in u around 0 85.3%

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u + 8 \cdot {u}^{2}\right)} \]
3. Step-by-step derivation

   1. +-commutative 85.3%

      \[\leadsto s \cdot \color{blue}{\left(8 \cdot {u}^{2} + 4 \cdot u\right)} \]

   2. fma-def 85.3%

      \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(8, {u}^{2}, 4 \cdot u\right)} \]

   3. unpow2 85.3%

      \[\leadsto s \cdot \mathsf{fma}\left(8, \color{blue}{u \cdot u}, 4 \cdot u\right) \]

4. Simplified 85.3%

   \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(8, u \cdot u, 4 \cdot u\right)} \]
5. Step-by-step derivation

   1. fma-udef 85.3%

      \[\leadsto s \cdot \color{blue}{\left(8 \cdot \left(u \cdot u\right) + 4 \cdot u\right)} \]

   2. flip-+ 85.1%

      \[\leadsto s \cdot \color{blue}{\frac{\left(8 \cdot \left(u \cdot u\right)\right) \cdot \left(8 \cdot \left(u \cdot u\right)\right) - \left(4 \cdot u\right) \cdot \left(4 \cdot u\right)}{8 \cdot \left(u \cdot u\right) - 4 \cdot u}} \]

   3. swap-sqr 85.1%

      \[\leadsto s \cdot \frac{\left(8 \cdot \left(u \cdot u\right)\right) \cdot \left(8 \cdot \left(u \cdot u\right)\right) - \color{blue}{\left(4 \cdot 4\right) \cdot \left(u \cdot u\right)}}{8 \cdot \left(u \cdot u\right) - 4 \cdot u} \]

   4. metadata-eval 85.1%

      \[\leadsto s \cdot \frac{\left(8 \cdot \left(u \cdot u\right)\right) \cdot \left(8 \cdot \left(u \cdot u\right)\right) - \color{blue}{16} \cdot \left(u \cdot u\right)}{8 \cdot \left(u \cdot u\right) - 4 \cdot u} \]

   5. *-commutative 85.1%

      \[\leadsto s \cdot \frac{\left(8 \cdot \left(u \cdot u\right)\right) \cdot \left(8 \cdot \left(u \cdot u\right)\right) - \color{blue}{\left(u \cdot u\right) \cdot 16}}{8 \cdot \left(u \cdot u\right) - 4 \cdot u} \]

   6. associate-*l* 85.1%

      \[\leadsto s \cdot \frac{\left(8 \cdot \left(u \cdot u\right)\right) \cdot \left(8 \cdot \left(u \cdot u\right)\right) - \color{blue}{u \cdot \left(u \cdot 16\right)}}{8 \cdot \left(u \cdot u\right) - 4 \cdot u} \]

   7. *-commutative 85.1%

      \[\leadsto s \cdot \frac{\left(8 \cdot \left(u \cdot u\right)\right) \cdot \left(8 \cdot \left(u \cdot u\right)\right) - u \cdot \left(u \cdot 16\right)}{8 \cdot \left(u \cdot u\right) - \color{blue}{u \cdot 4}} \]

6. Applied egg-rr 85.1%

   \[\leadsto s \cdot \color{blue}{\frac{\left(8 \cdot \left(u \cdot u\right)\right) \cdot \left(8 \cdot \left(u \cdot u\right)\right) - u \cdot \left(u \cdot 16\right)}{8 \cdot \left(u \cdot u\right) - u \cdot 4}} \]

7. Taylor expanded in s around 0 78.0%

   \[\leadsto \color{blue}{\frac{s \cdot \left(64 \cdot {u}^{4} - 16 \cdot {u}^{2}\right)}{8 \cdot {u}^{2} - 4 \cdot u}} \]
8. Step-by-step derivation

   1. associate-/l* 85.0%

      \[\leadsto \color{blue}{\frac{s}{\frac{8 \cdot {u}^{2} - 4 \cdot u}{64 \cdot {u}^{4} - 16 \cdot {u}^{2}}}} \]

   2. cancel-sign-sub-inv 85.0%

      \[\leadsto \frac{s}{\frac{\color{blue}{8 \cdot {u}^{2} + \left(-4\right) \cdot u}}{64 \cdot {u}^{4} - 16 \cdot {u}^{2}}} \]

   3. unpow2 85.0%

      \[\leadsto \frac{s}{\frac{8 \cdot \color{blue}{\left(u \cdot u\right)} + \left(-4\right) \cdot u}{64 \cdot {u}^{4} - 16 \cdot {u}^{2}}} \]

   4. associate-*r* 85.0%

      \[\leadsto \frac{s}{\frac{\color{blue}{\left(8 \cdot u\right) \cdot u} + \left(-4\right) \cdot u}{64 \cdot {u}^{4} - 16 \cdot {u}^{2}}} \]

   5. distribute-rgt-out 84.9%

      \[\leadsto \frac{s}{\frac{\color{blue}{u \cdot \left(8 \cdot u + \left(-4\right)\right)}}{64 \cdot {u}^{4} - 16 \cdot {u}^{2}}} \]

   6. *-commutative 84.9%

      \[\leadsto \frac{s}{\frac{u \cdot \left(\color{blue}{u \cdot 8} + \left(-4\right)\right)}{64 \cdot {u}^{4} - 16 \cdot {u}^{2}}} \]

   7. metadata-eval 84.9%

      \[\leadsto \frac{s}{\frac{u \cdot \left(u \cdot 8 + \color{blue}{-4}\right)}{64 \cdot {u}^{4} - 16 \cdot {u}^{2}}} \]

   8. unpow2 84.9%

      \[\leadsto \frac{s}{\frac{u \cdot \left(u \cdot 8 + -4\right)}{64 \cdot {u}^{4} - 16 \cdot \color{blue}{\left(u \cdot u\right)}}} \]

   9. *-commutative 84.9%

      \[\leadsto \frac{s}{\frac{u \cdot \left(u \cdot 8 + -4\right)}{\color{blue}{{u}^{4} \cdot 64} - 16 \cdot \left(u \cdot u\right)}} \]

   10. *-commutative 84.9%

       \[\leadsto \frac{s}{\frac{u \cdot \left(u \cdot 8 + -4\right)}{{u}^{4} \cdot 64 - \color{blue}{\left(u \cdot u\right) \cdot 16}}} \]

   11. associate-*r* 84.9%

       \[\leadsto \frac{s}{\frac{u \cdot \left(u \cdot 8 + -4\right)}{{u}^{4} \cdot 64 - \color{blue}{u \cdot \left(u \cdot 16\right)}}} \]

   12. sub-neg 84.9%

       \[\leadsto \frac{s}{\frac{u \cdot \left(u \cdot 8 + -4\right)}{\color{blue}{{u}^{4} \cdot 64 + \left(-u \cdot \left(u \cdot 16\right)\right)}}} \]

   13. +-commutative 84.9%

       \[\leadsto \frac{s}{\frac{u \cdot \left(u \cdot 8 + -4\right)}{\color{blue}{\left(-u \cdot \left(u \cdot 16\right)\right) + {u}^{4} \cdot 64}}} \]

   14. distribute-rgt-neg-in 84.9%

       \[\leadsto \frac{s}{\frac{u \cdot \left(u \cdot 8 + -4\right)}{\color{blue}{u \cdot \left(-u \cdot 16\right)} + {u}^{4} \cdot 64}} \]

   15. fma-def 84.9%

       \[\leadsto \frac{s}{\frac{u \cdot \left(u \cdot 8 + -4\right)}{\color{blue}{\mathsf{fma}\left(u, -u \cdot 16, {u}^{4} \cdot 64\right)}}} \]

   16. distribute-rgt-neg-in 84.9%

       \[\leadsto \frac{s}{\frac{u \cdot \left(u \cdot 8 + -4\right)}{\mathsf{fma}\left(u, \color{blue}{u \cdot \left(-16\right)}, {u}^{4} \cdot 64\right)}} \]

   17. metadata-eval 84.9%

       \[\leadsto \frac{s}{\frac{u \cdot \left(u \cdot 8 + -4\right)}{\mathsf{fma}\left(u, u \cdot \color{blue}{-16}, {u}^{4} \cdot 64\right)}} \]

   18. *-commutative 84.9%

       \[\leadsto \frac{s}{\frac{u \cdot \left(u \cdot 8 + -4\right)}{\mathsf{fma}\left(u, u \cdot -16, \color{blue}{64 \cdot {u}^{4}}\right)}} \]

9. Simplified 84.9%

   \[\leadsto \color{blue}{\frac{s}{\frac{u \cdot \left(u \cdot 8 + -4\right)}{\mathsf{fma}\left(u, u \cdot -16, 64 \cdot {u}^{4}\right)}}} \]

10. Taylor expanded in u around 0 87.2%

    \[\leadsto \frac{s}{\color{blue}{0.25 \cdot \frac{1}{u} - 0.5}} \]
11. Step-by-step derivation

    1. sub-neg 87.2%

       \[\leadsto \frac{s}{\color{blue}{0.25 \cdot \frac{1}{u} + \left(-0.5\right)}} \]

    2. associate-*r/ 87.2%

       \[\leadsto \frac{s}{\color{blue}{\frac{0.25 \cdot 1}{u}} + \left(-0.5\right)} \]

    3. metadata-eval 87.2%

       \[\leadsto \frac{s}{\frac{\color{blue}{0.25}}{u} + \left(-0.5\right)} \]

    4. metadata-eval 87.2%

       \[\leadsto \frac{s}{\frac{0.25}{u} + \color{blue}{-0.5}} \]

12. Simplified 87.2%

    \[\leadsto \frac{s}{\color{blue}{\frac{0.25}{u} + -0.5}} \]

13. Final simplification 87.2%

    \[\leadsto \frac{s}{\frac{0.25}{u} + -0.5} \]

Alternative 3: 73.3% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(s \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* 4.0 (* s u)))

C

float code(float s, float u) {
	return 4.0f * (s * u);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = 4.0e0 * (s * u)
end function

Julia

function code(s, u)
	return Float32(Float32(4.0) * Float32(s * u))
end

MATLAB

function tmp = code(s, u)
	tmp = single(4.0) * (s * u);
end

Derivation

1. Initial program 63.5%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Taylor expanded in u around 0 72.2%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
3. Step-by-step derivation

   1. *-commutative 72.2%

      \[\leadsto 4 \cdot \color{blue}{\left(u \cdot s\right)} \]

4. Simplified 72.2%

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot s\right)} \]

5. Final simplification 72.2%

   \[\leadsto 4 \cdot \left(s \cdot u\right) \]

Alternative 4: 73.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot 4\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* u (* s 4.0)))

C

float code(float s, float u) {
	return u * (s * 4.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * (s * 4.0e0)
end function

Julia

function code(s, u)
	return Float32(u * Float32(s * Float32(4.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = u * (s * single(4.0));
end

Derivation

1. Initial program 63.5%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Taylor expanded in u around 0 72.2%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
3. Step-by-step derivation

   1. associate-*r* 72.5%

      \[\leadsto \color{blue}{\left(4 \cdot s\right) \cdot u} \]

4. Simplified 72.5%

   \[\leadsto \color{blue}{\left(4 \cdot s\right) \cdot u} \]

5. Final simplification 72.5%

   \[\leadsto u \cdot \left(s \cdot 4\right) \]

Alternative 5: 16.4% accurate, 36.3× speedup

Math

\[\begin{array}{l} \\ s \cdot 0 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s 0.0))

C

float code(float s, float u) {
	return s * 0.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * 0.0e0
end function

Julia

function code(s, u)
	return Float32(s * Float32(0.0))
end

MATLAB

function tmp = code(s, u)
	tmp = s * single(0.0);
end

Derivation

1. Initial program 63.5%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

3. Applied egg-rr 17.0%

   \[\leadsto s \cdot \color{blue}{\left(0.5 \cdot \mathsf{log1p}\left(4 \cdot u\right) - 0.5 \cdot \mathsf{log1p}\left(4 \cdot u\right)\right)} \]
4. Step-by-step derivation

   1. +-inverses 17.0%

      \[\leadsto s \cdot \color{blue}{0} \]

5. Simplified 17.0%

   \[\leadsto s \cdot \color{blue}{0} \]

6. Final simplification 17.0%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))