Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.3% → 99.4%

Time: 7.6s

Alternatives: 6

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: 61.3% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.6%

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

   1. *-commutative 60.6%

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

   2. log-rec 63.1%

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

   3. distribute-lft-neg-out 63.1%

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

   4. distribute-rgt-neg-in 63.1%

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

   5. sub-neg 63.1%

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

   6. log1p-def 99.3%

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

   7. *-commutative 99.3%

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

   8. distribute-rgt-neg-in 99.3%

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

   9. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 2: 91.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(u \cdot \left(--4\right) - \left(u \cdot -21.333333333333332\right) \cdot \left(u \cdot u\right)\right) - \left(u \cdot u\right) \cdot -8\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  s
  (-
   (- (* u (- -4.0)) (* (* u -21.333333333333332) (* u u)))
   (* (* u u) -8.0))))

C

float code(float s, float u) {
	return s * (((u * -(-4.0f)) - ((u * -21.333333333333332f) * (u * u))) - ((u * u) * -8.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (((u * -(-4.0e0)) - ((u * (-21.333333333333332e0)) * (u * u))) - ((u * u) * (-8.0e0)))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(u * Float32(-Float32(-4.0))) - Float32(Float32(u * Float32(-21.333333333333332)) * Float32(u * u))) - Float32(Float32(u * u) * Float32(-8.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (((u * -single(-4.0)) - ((u * single(-21.333333333333332)) * (u * u))) - ((u * u) * single(-8.0)));
end

Derivation

1. Initial program 60.6%

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

   1. *-commutative 60.6%

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

   2. log-rec 63.1%

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

   3. distribute-lft-neg-out 63.1%

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

   4. distribute-rgt-neg-in 63.1%

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

   5. sub-neg 63.1%

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

   6. log1p-def 99.3%

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

   7. *-commutative 99.3%

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

   8. distribute-rgt-neg-in 99.3%

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

   9. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in u around 0 91.0%

   \[\leadsto \color{blue}{\left(-4 \cdot u + \left(-8 \cdot {u}^{2} + -21.333333333333332 \cdot {u}^{3}\right)\right)} \cdot \left(-s\right) \]
5. Step-by-step derivation

   1. *-commutative 91.0%

      \[\leadsto \left(\color{blue}{u \cdot -4} + \left(-8 \cdot {u}^{2} + -21.333333333333332 \cdot {u}^{3}\right)\right) \cdot \left(-s\right) \]

   2. fma-def 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, -4, -8 \cdot {u}^{2} + -21.333333333333332 \cdot {u}^{3}\right)} \cdot \left(-s\right) \]

   3. +-commutative 91.0%

      \[\leadsto \mathsf{fma}\left(u, -4, \color{blue}{-21.333333333333332 \cdot {u}^{3} + -8 \cdot {u}^{2}}\right) \cdot \left(-s\right) \]

   4. cube-mult 91.0%

      \[\leadsto \mathsf{fma}\left(u, -4, -21.333333333333332 \cdot \color{blue}{\left(u \cdot \left(u \cdot u\right)\right)} + -8 \cdot {u}^{2}\right) \cdot \left(-s\right) \]

   5. unpow2 91.0%

      \[\leadsto \mathsf{fma}\left(u, -4, -21.333333333333332 \cdot \left(u \cdot \color{blue}{{u}^{2}}\right) + -8 \cdot {u}^{2}\right) \cdot \left(-s\right) \]

   6. associate-*r* 91.0%

      \[\leadsto \mathsf{fma}\left(u, -4, \color{blue}{\left(-21.333333333333332 \cdot u\right) \cdot {u}^{2}} + -8 \cdot {u}^{2}\right) \cdot \left(-s\right) \]

   7. distribute-rgt-out 91.0%

      \[\leadsto \mathsf{fma}\left(u, -4, \color{blue}{{u}^{2} \cdot \left(-21.333333333333332 \cdot u + -8\right)}\right) \cdot \left(-s\right) \]

   8. *-commutative 91.0%

      \[\leadsto \mathsf{fma}\left(u, -4, {u}^{2} \cdot \left(\color{blue}{u \cdot -21.333333333333332} + -8\right)\right) \cdot \left(-s\right) \]

   9. unpow2 91.0%

      \[\leadsto \mathsf{fma}\left(u, -4, \color{blue}{\left(u \cdot u\right)} \cdot \left(u \cdot -21.333333333333332 + -8\right)\right) \cdot \left(-s\right) \]

6. Simplified 91.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, -4, \left(u \cdot u\right) \cdot \left(u \cdot -21.333333333333332 + -8\right)\right)} \cdot \left(-s\right) \]
7. Step-by-step derivation

   1. fma-udef 91.0%

      \[\leadsto \color{blue}{\left(u \cdot -4 + \left(u \cdot u\right) \cdot \left(u \cdot -21.333333333333332 + -8\right)\right)} \cdot \left(-s\right) \]

   2. distribute-rgt-in 91.0%

      \[\leadsto \left(u \cdot -4 + \color{blue}{\left(\left(u \cdot -21.333333333333332\right) \cdot \left(u \cdot u\right) + -8 \cdot \left(u \cdot u\right)\right)}\right) \cdot \left(-s\right) \]

   3. associate-+r+ 91.0%

      \[\leadsto \color{blue}{\left(\left(u \cdot -4 + \left(u \cdot -21.333333333333332\right) \cdot \left(u \cdot u\right)\right) + -8 \cdot \left(u \cdot u\right)\right)} \cdot \left(-s\right) \]

8. Applied egg-rr 91.0%

   \[\leadsto \color{blue}{\left(\left(u \cdot -4 + \left(u \cdot -21.333333333333332\right) \cdot \left(u \cdot u\right)\right) + -8 \cdot \left(u \cdot u\right)\right)} \cdot \left(-s\right) \]

9. Final simplification 91.0%

   \[\leadsto s \cdot \left(\left(u \cdot \left(--4\right) - \left(u \cdot -21.333333333333332\right) \cdot \left(u \cdot u\right)\right) - \left(u \cdot u\right) \cdot -8\right) \]

Alternative 3: 87.0% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot 4 + \left(u \cdot u\right) \cdot 8\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (+ (* u 4.0) (* (* u u) 8.0))))

C

float code(float s, float u) {
	return s * ((u * 4.0f) + ((u * u) * 8.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((u * 4.0e0) + ((u * u) * 8.0e0))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(u * Float32(4.0)) + Float32(Float32(u * u) * Float32(8.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((u * single(4.0)) + ((u * u) * single(8.0)));
end

Derivation

1. Initial program 60.6%

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

   1. *-commutative 60.6%

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

   2. log-rec 63.1%

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

   3. distribute-lft-neg-out 63.1%

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

   4. distribute-rgt-neg-in 63.1%

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

   5. sub-neg 63.1%

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

   6. log1p-def 99.3%

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

   7. *-commutative 99.3%

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

   8. distribute-rgt-neg-in 99.3%

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

   9. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in u around 0 86.8%

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

   1. *-commutative 86.8%

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

   2. associate-*l* 87.1%

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

   3. *-commutative 87.1%

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

   4. associate-*l* 87.2%

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

   5. distribute-lft-out 87.3%

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

   6. *-commutative 87.3%

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

   7. *-commutative 87.3%

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

   8. unpow2 87.3%

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

6. Simplified 87.3%

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

7. Final simplification 87.3%

   \[\leadsto s \cdot \left(u \cdot 4 + \left(u \cdot u\right) \cdot 8\right) \]

Alternative 4: 86.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(4 + u \cdot 8\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (* u (+ 4.0 (* u 8.0)))))

C

float code(float s, float u) {
	return s * (u * (4.0f + (u * 8.0f)));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (u * (4.0e0 + (u * 8.0e0)))
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(Float32(4.0) + Float32(u * Float32(8.0)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * (single(4.0) + (u * single(8.0))));
end

Derivation

1. Initial program 60.6%

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

   1. *-commutative 60.6%

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

   2. log-rec 63.1%

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

   3. distribute-lft-neg-out 63.1%

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

   4. distribute-rgt-neg-in 63.1%

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

   5. sub-neg 63.1%

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

   6. log1p-def 99.3%

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

   7. *-commutative 99.3%

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

   8. distribute-rgt-neg-in 99.3%

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

   9. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in u around 0 86.8%

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

   1. *-commutative 86.8%

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

   2. associate-*l* 87.1%

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

   3. *-commutative 87.1%

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

   4. associate-*l* 87.2%

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

   5. distribute-lft-out 87.3%

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

   6. *-commutative 87.3%

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

   7. *-commutative 87.3%

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

   8. unpow2 87.3%

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

6. Simplified 87.3%

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

   1. +-commutative 87.3%

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

   2. associate-*r* 87.3%

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

   3. distribute-rgt-out 87.2%

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

8. Applied egg-rr 87.2%

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

9. Final simplification 87.2%

   \[\leadsto s \cdot \left(u \cdot \left(4 + u \cdot 8\right)\right) \]

Alternative 5: 73.8% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.6%

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

   1. *-commutative 60.6%

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

   2. log-rec 63.1%

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

   3. distribute-lft-neg-out 63.1%

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

   4. distribute-rgt-neg-in 63.1%

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

   5. sub-neg 63.1%

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

   6. log1p-def 99.3%

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

   7. *-commutative 99.3%

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

   8. distribute-rgt-neg-in 99.3%

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

   9. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in u around 0 74.3%

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

5. Final simplification 74.3%

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

Alternative 6: 74.0% 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 60.6%

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

   1. *-commutative 60.6%

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

   2. log-rec 63.1%

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

   3. distribute-lft-neg-out 63.1%

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

   4. distribute-rgt-neg-in 63.1%

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

   5. sub-neg 63.1%

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

   6. log1p-def 99.3%

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

   7. *-commutative 99.3%

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

   8. distribute-rgt-neg-in 99.3%

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

   9. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in u around 0 74.3%

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

   1. add-log-exp 21.5%

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

   2. associate-*r* 21.5%

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

   3. exp-prod 20.9%

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

6. Applied egg-rr 20.9%

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

   1. log-pow 27.6%

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

   2. rem-log-exp 74.6%

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

8. Simplified 74.6%

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

9. Final simplification 74.6%

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

Reproduce

herbie shell --seed 2023194 
(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))))))