Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.5% → 99.4%

Time: 10.7s

Alternatives: 8

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

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

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(s * Float32(-log1p(Float32(u * Float32(-4.0)))))
end

Derivation

1. Initial program 59.4%

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

   1. log-rec 61.8%

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

   2. cancel-sign-sub-inv 61.8%

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

   3. log1p-def 99.3%

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

   4. *-commutative 99.3%

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

   5. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 2: 91.1% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

   1. log-rec 61.8%

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

3. Simplified 61.8%

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

4. Taylor expanded in u around 0 90.6%

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

   1. associate-+r+ 90.6%

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

   2. +-commutative 90.6%

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

   3. *-commutative 90.6%

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

   4. unpow3 90.6%

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

   5. unpow2 90.6%

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

   6. associate-*r* 90.6%

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

   7. unpow2 90.6%

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

   8. associate-*r* 90.6%

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

   9. distribute-rgt-out 90.6%

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

   10. distribute-lft-out 90.4%

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

   11. unpow2 90.4%

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

   12. associate-*r* 90.4%

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

   13. *-commutative 90.4%

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

   14. distribute-rgt-out 90.4%

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

6. Simplified 90.4%

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

   1. distribute-rgt-in 90.4%

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

   2. *-commutative 90.4%

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

8. Applied egg-rr 90.4%

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

9. Final simplification 90.4%

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

Alternative 3: 91.1% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

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) + (u * (-21.333333333333332e0))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

   1. log-rec 61.8%

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

3. Simplified 61.8%

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

4. Taylor expanded in u around 0 90.6%

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

   1. associate-+r+ 90.6%

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

   2. +-commutative 90.6%

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

   3. *-commutative 90.6%

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

   4. unpow3 90.6%

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

   5. unpow2 90.6%

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

   6. associate-*r* 90.6%

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

   7. unpow2 90.6%

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

   8. associate-*r* 90.6%

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

   9. distribute-rgt-out 90.6%

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

   10. distribute-lft-out 90.4%

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

   11. unpow2 90.4%

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

   12. associate-*r* 90.4%

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

   13. *-commutative 90.4%

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

   14. distribute-rgt-out 90.4%

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

6. Simplified 90.4%

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

7. Final simplification 90.4%

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

Alternative 4: 87.0% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

   1. log-rec 61.8%

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

3. Simplified 61.8%

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

4. Taylor expanded in u around 0 86.2%

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

   1. associate-*r* 86.4%

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

   2. *-commutative 86.4%

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

   3. associate-*r* 86.4%

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

   4. fma-def 86.7%

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

   5. associate-*r* 86.9%

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

6. Applied egg-rr 86.9%

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

   1. *-commutative 86.9%

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

   2. associate-*r* 86.9%

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

   3. fma-def 86.5%

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

   4. unpow2 86.5%

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

   5. associate-*r* 86.5%

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

   6. distribute-rgt-in 86.6%

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

   7. distribute-lft-in 86.4%

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

   8. *-commutative 86.4%

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

   9. associate-*r* 86.3%

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

   10. +-commutative 86.3%

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

   11. fma-def 86.3%

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

8. Applied egg-rr 86.3%

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

   1. fma-def 86.3%

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

   2. distribute-rgt-in 86.5%

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

10. Applied egg-rr 86.5%

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

11. Final simplification 86.5%

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

Alternative 5: 86.8% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

   1. log-rec 61.8%

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

3. Simplified 61.8%

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

4. Taylor expanded in u around 0 86.2%

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

   1. +-commutative 86.2%

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

   2. *-commutative 86.2%

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

   3. associate-*l* 86.3%

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

   4. *-commutative 86.3%

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

   5. associate-*l* 86.5%

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

   6. *-commutative 86.5%

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

   7. distribute-lft-out 86.6%

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

   8. unpow2 86.6%

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

   9. associate-*l* 86.6%

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

   10. *-commutative 86.6%

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

   11. distribute-lft-out 86.4%

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

6. Simplified 86.4%

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

7. Final simplification 86.4%

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

Alternative 6: 73.9% 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 59.4%

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

   1. log-rec 61.8%

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

3. Simplified 61.8%

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

4. Taylor expanded in u around 0 73.5%

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

   1. *-commutative 73.5%

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

6. Simplified 73.5%

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

7. Final simplification 73.5%

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

Alternative 7: 74.1% 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 59.4%

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

   1. log-rec 61.8%

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

3. Simplified 61.8%

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

4. Taylor expanded in u around 0 73.5%

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

   1. associate-*r* 73.7%

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

   2. *-commutative 73.7%

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

6. Simplified 73.7%

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

7. Final simplification 73.7%

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

Alternative 8: 17.1% 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 59.4%

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

   1. log-rec 61.8%

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

3. Simplified 61.8%

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

   1. add-sqr-sqrt 59.1%

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

   2. log-prod 59.4%

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

   3. pow1/2 59.4%

      \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \color{blue}{\left({\left(1 - 4 \cdot u\right)}^{0.5}\right)}\right)\right) \]

   4. add-exp-log 59.4%

      \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \left({\color{blue}{\left(e^{\log \left(1 - 4 \cdot u\right)}\right)}}^{0.5}\right)\right)\right) \]

   5. add-sqr-sqrt 4.5%

      \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \left({\left(e^{\color{blue}{\sqrt{\log \left(1 - 4 \cdot u\right)} \cdot \sqrt{\log \left(1 - 4 \cdot u\right)}}}\right)}^{0.5}\right)\right)\right) \]

   6. sqrt-unprod 19.4%

      \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \left({\left(e^{\color{blue}{\sqrt{\log \left(1 - 4 \cdot u\right) \cdot \log \left(1 - 4 \cdot u\right)}}}\right)}^{0.5}\right)\right)\right) \]

   7. sqr-neg 19.4%

      \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \left({\left(e^{\sqrt{\color{blue}{\left(-\log \left(1 - 4 \cdot u\right)\right) \cdot \left(-\log \left(1 - 4 \cdot u\right)\right)}}}\right)}^{0.5}\right)\right)\right) \]

   8. sqrt-unprod 19.4%

      \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \left({\left(e^{\color{blue}{\sqrt{-\log \left(1 - 4 \cdot u\right)} \cdot \sqrt{-\log \left(1 - 4 \cdot u\right)}}}\right)}^{0.5}\right)\right)\right) \]

   9. add-sqr-sqrt 19.4%

      \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \left({\left(e^{\color{blue}{-\log \left(1 - 4 \cdot u\right)}}\right)}^{0.5}\right)\right)\right) \]

   10. neg-mul-1 19.4%

       \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \left({\left(e^{\color{blue}{-1 \cdot \log \left(1 - 4 \cdot u\right)}}\right)}^{0.5}\right)\right)\right) \]

   11. *-commutative 19.4%

       \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \left({\left(e^{\color{blue}{\log \left(1 - 4 \cdot u\right) \cdot -1}}\right)}^{0.5}\right)\right)\right) \]

   12. pow-to-exp 19.4%

       \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \left({\color{blue}{\left({\left(1 - 4 \cdot u\right)}^{-1}\right)}}^{0.5}\right)\right)\right) \]

   13. pow-pow 17.7%

       \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \color{blue}{\left({\left(1 - 4 \cdot u\right)}^{\left(-1 \cdot 0.5\right)}\right)}\right)\right) \]

   14. metadata-eval 17.7%

       \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \left({\left(1 - 4 \cdot u\right)}^{\color{blue}{-0.5}}\right)\right)\right) \]

   15. metadata-eval 17.7%

       \[\leadsto s \cdot \left(-\left(\log \left(\sqrt{1 - 4 \cdot u}\right) + \log \left({\left(1 - 4 \cdot u\right)}^{\color{blue}{\left(0.5 + -1\right)}}\right)\right)\right) \]

5. Applied egg-rr 15.2%

   \[\leadsto s \cdot \left(-\color{blue}{0}\right) \]

6. Final simplification 15.2%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023310 
(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))))))