Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.2% → 99.4%

Time: 14.9s

Alternatives: 10

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: 61.2% 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 64.2%

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

   1. *-commutative 64.2%

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

   2. log-rec 66.2%

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

   3. distribute-lft-neg-out 66.2%

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

   4. distribute-rgt-neg-in 66.2%

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

   5. sub-neg 66.2%

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

   6. log1p-def 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 88.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \frac{u \cdot \left(u \cdot -16\right)}{8 \cdot \left(u \cdot u\right) - u \cdot 4} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (/ (* u (* u -16.0)) (- (* 8.0 (* u u)) (* u 4.0)))))

C

float code(float s, float u) {
	return s * ((u * (u * -16.0f)) / ((8.0f * (u * u)) - (u * 4.0f)));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((u * (u * (-16.0e0))) / ((8.0e0 * (u * u)) - (u * 4.0e0)))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(u * Float32(u * Float32(-16.0))) / Float32(Float32(Float32(8.0) * Float32(u * u)) - Float32(u * Float32(4.0)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((u * (u * single(-16.0))) / ((single(8.0) * (u * u)) - (u * single(4.0))));
end

Derivation

1. Initial program 64.2%

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

2. Taylor expanded in u around 0 85.8%

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

   1. fma-def 85.8%

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

   2. unpow2 85.8%

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

4. Simplified 85.8%

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

   1. fma-udef 85.8%

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

   2. flip-+ 85.7%

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

   3. swap-sqr 85.7%

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

   4. metadata-eval 85.7%

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

   5. *-commutative 85.7%

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

6. Applied egg-rr 85.7%

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

7. Taylor expanded in u around 0 87.7%

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

   1. unpow2 87.7%

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

   2. *-commutative 87.7%

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

   3. associate-*r* 87.7%

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

9. Simplified 87.7%

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

10. Final simplification 87.7%

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

Alternative 3: 87.8% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ -16 \cdot \frac{u \cdot u}{\frac{u \cdot \left(-4 + u \cdot 8\right)}{s}} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* -16.0 (/ (* u u) (/ (* u (+ -4.0 (* u 8.0))) s))))

C

float code(float s, float u) {
	return -16.0f * ((u * u) / ((u * (-4.0f + (u * 8.0f))) / s));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (-16.0e0) * ((u * u) / ((u * ((-4.0e0) + (u * 8.0e0))) / s))
end function

Julia

function code(s, u)
	return Float32(Float32(-16.0) * Float32(Float32(u * u) / Float32(Float32(u * Float32(Float32(-4.0) + Float32(u * Float32(8.0)))) / s)))
end

MATLAB

function tmp = code(s, u)
	tmp = single(-16.0) * ((u * u) / ((u * (single(-4.0) + (u * single(8.0)))) / s));
end

Derivation

1. Initial program 64.2%

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

2. Taylor expanded in u around 0 85.8%

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

   1. fma-def 85.8%

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

   2. unpow2 85.8%

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

4. Simplified 85.8%

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

   1. fma-udef 85.8%

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

   2. flip-+ 85.7%

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

   3. swap-sqr 85.7%

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

   4. metadata-eval 85.7%

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

   5. *-commutative 85.7%

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

6. Applied egg-rr 85.7%

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

7. Taylor expanded in u around 0 87.7%

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

   1. unpow2 87.7%

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

   2. *-commutative 87.7%

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

   3. associate-*r* 87.7%

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

9. Simplified 87.7%

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

10. Taylor expanded in s around 0 79.4%

    \[\leadsto \color{blue}{-16 \cdot \frac{s \cdot {u}^{2}}{8 \cdot {u}^{2} - 4 \cdot u}} \]
11. Step-by-step derivation

    1. *-commutative 79.4%

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

    2. cancel-sign-sub-inv 79.4%

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

    3. unpow2 79.4%

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

    4. associate-*r* 79.4%

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

    5. *-commutative 79.4%

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

    6. metadata-eval 79.4%

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

    7. fma-udef 79.4%

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

    8. associate-/l* 86.5%

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

    9. unpow2 86.5%

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

    10. fma-udef 86.5%

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

    11. distribute-rgt-out 86.4%

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

12. Simplified 86.4%

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

13. Final simplification 86.4%

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

Alternative 4: 88.6% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \frac{u \cdot -16}{u \cdot \left(-4 + u \cdot 8\right)}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* u (/ (* u -16.0) (* u (+ -4.0 (* u 8.0)))))))

C

float code(float s, float u) {
	return s * (u * ((u * -16.0f) / (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 * ((u * (-16.0e0)) / (u * ((-4.0e0) + (u * 8.0e0)))))
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(Float32(u * Float32(-16.0)) / Float32(u * Float32(Float32(-4.0) + Float32(u * Float32(8.0)))))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * ((u * single(-16.0)) / (u * (single(-4.0) + (u * single(8.0))))));
end

Derivation

1. Initial program 64.2%

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

2. Taylor expanded in u around 0 85.8%

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

   1. fma-def 85.8%

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

   2. unpow2 85.8%

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

4. Simplified 85.8%

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

   1. fma-udef 85.8%

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

   2. flip-+ 85.7%

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

   3. swap-sqr 85.7%

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

   4. metadata-eval 85.7%

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

   5. *-commutative 85.7%

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

6. Applied egg-rr 85.7%

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

7. Taylor expanded in u around 0 87.7%

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

   1. unpow2 87.7%

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

   2. *-commutative 87.7%

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

   3. associate-*r* 87.7%

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

9. Simplified 87.7%

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

    1. expm1-log1p-u 87.7%

       \[\leadsto s \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u \cdot \left(u \cdot -16\right)}{8 \cdot \left(u \cdot u\right) - 4 \cdot u}\right)\right)} \]

    2. expm1-udef 54.6%

       \[\leadsto s \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{u \cdot \left(u \cdot -16\right)}{8 \cdot \left(u \cdot u\right) - 4 \cdot u}\right)} - 1\right)} \]

    3. associate-/l* 54.6%

       \[\leadsto s \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{u}{\frac{8 \cdot \left(u \cdot u\right) - 4 \cdot u}{u \cdot -16}}}\right)} - 1\right) \]

    4. cancel-sign-sub-inv 54.6%

       \[\leadsto s \cdot \left(e^{\mathsf{log1p}\left(\frac{u}{\frac{\color{blue}{8 \cdot \left(u \cdot u\right) + \left(-4\right) \cdot u}}{u \cdot -16}}\right)} - 1\right) \]

    5. associate-*r* 54.6%

       \[\leadsto s \cdot \left(e^{\mathsf{log1p}\left(\frac{u}{\frac{\color{blue}{\left(8 \cdot u\right) \cdot u} + \left(-4\right) \cdot u}{u \cdot -16}}\right)} - 1\right) \]

    6. *-commutative 54.6%

       \[\leadsto s \cdot \left(e^{\mathsf{log1p}\left(\frac{u}{\frac{\color{blue}{\left(u \cdot 8\right)} \cdot u + \left(-4\right) \cdot u}{u \cdot -16}}\right)} - 1\right) \]

    7. fma-def 54.6%

       \[\leadsto s \cdot \left(e^{\mathsf{log1p}\left(\frac{u}{\frac{\color{blue}{\mathsf{fma}\left(u \cdot 8, u, \left(-4\right) \cdot u\right)}}{u \cdot -16}}\right)} - 1\right) \]

    8. metadata-eval 54.6%

       \[\leadsto s \cdot \left(e^{\mathsf{log1p}\left(\frac{u}{\frac{\mathsf{fma}\left(u \cdot 8, u, \color{blue}{-4} \cdot u\right)}{u \cdot -16}}\right)} - 1\right) \]

11. Applied egg-rr 54.6%

    \[\leadsto s \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{u}{\frac{\mathsf{fma}\left(u \cdot 8, u, -4 \cdot u\right)}{u \cdot -16}}\right)} - 1\right)} \]
12. Step-by-step derivation

    1. expm1-def 87.8%

       \[\leadsto s \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u}{\frac{\mathsf{fma}\left(u \cdot 8, u, -4 \cdot u\right)}{u \cdot -16}}\right)\right)} \]

    2. expm1-log1p 87.8%

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

    3. associate-/r/ 87.4%

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

    4. associate-*l/ 87.7%

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

    5. *-rgt-identity 87.7%

       \[\leadsto s \cdot \frac{\color{blue}{\left(u \cdot \left(u \cdot -16\right)\right) \cdot 1}}{\mathsf{fma}\left(u \cdot 8, u, -4 \cdot u\right)} \]

    6. associate-*r/ 87.5%

       \[\leadsto s \cdot \color{blue}{\left(\left(u \cdot \left(u \cdot -16\right)\right) \cdot \frac{1}{\mathsf{fma}\left(u \cdot 8, u, -4 \cdot u\right)}\right)} \]

    7. associate-*l* 87.4%

       \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(\left(u \cdot -16\right) \cdot \frac{1}{\mathsf{fma}\left(u \cdot 8, u, -4 \cdot u\right)}\right)\right)} \]

    8. associate-*r/ 87.4%

       \[\leadsto s \cdot \left(u \cdot \color{blue}{\frac{\left(u \cdot -16\right) \cdot 1}{\mathsf{fma}\left(u \cdot 8, u, -4 \cdot u\right)}}\right) \]

    9. *-commutative 87.4%

       \[\leadsto s \cdot \left(u \cdot \frac{\color{blue}{\left(-16 \cdot u\right)} \cdot 1}{\mathsf{fma}\left(u \cdot 8, u, -4 \cdot u\right)}\right) \]

    10. associate-*r* 87.4%

        \[\leadsto s \cdot \left(u \cdot \frac{\color{blue}{-16 \cdot \left(u \cdot 1\right)}}{\mathsf{fma}\left(u \cdot 8, u, -4 \cdot u\right)}\right) \]

    11. *-rgt-identity 87.4%

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

    12. *-commutative 87.4%

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

    13. fma-udef 87.4%

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

    14. distribute-rgt-out 87.4%

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

13. Simplified 87.4%

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

14. Final simplification 87.4%

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

Alternative 5: 88.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \frac{u \cdot \left(u \cdot -16\right)}{u \cdot \left(-4 + u \cdot 8\right)} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (/ (* u (* u -16.0)) (* u (+ -4.0 (* u 8.0))))))

C

float code(float s, float u) {
	return s * ((u * (u * -16.0f)) / (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 * (u * (-16.0e0))) / (u * ((-4.0e0) + (u * 8.0e0))))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(u * Float32(u * Float32(-16.0))) / Float32(u * Float32(Float32(-4.0) + Float32(u * Float32(8.0))))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((u * (u * single(-16.0))) / (u * (single(-4.0) + (u * single(8.0)))));
end

Derivation

1. Initial program 64.2%

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

2. Taylor expanded in u around 0 85.8%

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

   1. fma-def 85.8%

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

   2. unpow2 85.8%

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

4. Simplified 85.8%

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

   1. fma-udef 85.8%

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

   2. flip-+ 85.7%

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

   3. swap-sqr 85.7%

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

   4. metadata-eval 85.7%

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

   5. *-commutative 85.7%

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

6. Applied egg-rr 85.7%

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

7. Taylor expanded in u around 0 87.7%

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

   1. unpow2 87.7%

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

   2. *-commutative 87.7%

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

   3. associate-*r* 87.7%

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

9. Simplified 87.7%

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

10. Taylor expanded in u around 0 87.7%

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

    1. unpow2 87.7%

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

    2. associate-*r* 87.7%

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

    3. *-commutative 87.7%

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

    4. distribute-rgt-out 87.6%

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

12. Simplified 87.6%

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

13. Final simplification 87.6%

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

Alternative 6: 86.9% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.2%

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

2. Taylor expanded in u around 0 85.8%

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

   1. fma-def 85.8%

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

   2. unpow2 85.8%

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

4. Simplified 85.8%

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

   1. fma-udef 85.8%

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

6. Applied egg-rr 85.8%

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

7. Final simplification 85.8%

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

Alternative 7: 86.7% 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 64.2%

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

   1. flip3-- 63.8%

      \[\leadsto s \cdot \log \left(\frac{1}{\color{blue}{\frac{{1}^{3} - {\left(4 \cdot u\right)}^{3}}{1 \cdot 1 + \left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + 1 \cdot \left(4 \cdot u\right)\right)}}}\right) \]

   2. associate-/r/ 63.8%

      \[\leadsto s \cdot \log \color{blue}{\left(\frac{1}{{1}^{3} - {\left(4 \cdot u\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + 1 \cdot \left(4 \cdot u\right)\right)\right)\right)} \]

   3. log-prod 63.9%

      \[\leadsto s \cdot \color{blue}{\left(\log \left(\frac{1}{{1}^{3} - {\left(4 \cdot u\right)}^{3}}\right) + \log \left(1 \cdot 1 + \left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + 1 \cdot \left(4 \cdot u\right)\right)\right)\right)} \]

   4. metadata-eval 63.9%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{\color{blue}{1} - {\left(4 \cdot u\right)}^{3}}\right) + \log \left(1 \cdot 1 + \left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + 1 \cdot \left(4 \cdot u\right)\right)\right)\right) \]

   5. *-commutative 63.9%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {\color{blue}{\left(u \cdot 4\right)}}^{3}}\right) + \log \left(1 \cdot 1 + \left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + 1 \cdot \left(4 \cdot u\right)\right)\right)\right) \]

   6. unpow-prod-down 63.9%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - \color{blue}{{u}^{3} \cdot {4}^{3}}}\right) + \log \left(1 \cdot 1 + \left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + 1 \cdot \left(4 \cdot u\right)\right)\right)\right) \]

   7. metadata-eval 63.9%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot \color{blue}{64}}\right) + \log \left(1 \cdot 1 + \left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + 1 \cdot \left(4 \cdot u\right)\right)\right)\right) \]

   8. metadata-eval 63.9%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \log \left(\color{blue}{1} + \left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + 1 \cdot \left(4 \cdot u\right)\right)\right)\right) \]

   9. log1p-udef 96.0%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \color{blue}{\mathsf{log1p}\left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + 1 \cdot \left(4 \cdot u\right)\right)}\right) \]

   10. *-un-lft-identity 96.0%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + \color{blue}{4 \cdot u}\right)\right) \]

   11. distribute-lft1-in 95.8%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\color{blue}{\left(4 \cdot u + 1\right) \cdot \left(4 \cdot u\right)}\right)\right) \]

   12. fma-def 95.8%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(4, u, 1\right)} \cdot \left(4 \cdot u\right)\right)\right) \]

3. Applied egg-rr 95.8%

   \[\leadsto s \cdot \color{blue}{\left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative 95.8%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right)\right)} \]

   2. log-rec 96.3%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \color{blue}{\left(-\log \left(1 - {u}^{3} \cdot 64\right)\right)}\right) \]

   3. sub-neg 96.3%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \left(-\log \color{blue}{\left(1 + \left(-{u}^{3} \cdot 64\right)\right)}\right)\right) \]

   4. log1p-def 98.9%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \left(-\color{blue}{\mathsf{log1p}\left(-{u}^{3} \cdot 64\right)}\right)\right) \]

   5. unsub-neg 98.9%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) - \mathsf{log1p}\left(-{u}^{3} \cdot 64\right)\right)} \]

   6. *-commutative 98.9%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(4 \cdot u\right) \cdot \mathsf{fma}\left(4, u, 1\right)}\right) - \mathsf{log1p}\left(-{u}^{3} \cdot 64\right)\right) \]

   7. associate-*l* 98.9%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)}\right) - \mathsf{log1p}\left(-{u}^{3} \cdot 64\right)\right) \]

   8. distribute-rgt-neg-in 98.9%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)\right) - \mathsf{log1p}\left(\color{blue}{{u}^{3} \cdot \left(-64\right)}\right)\right) \]

   9. metadata-eval 98.9%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)\right) - \mathsf{log1p}\left({u}^{3} \cdot \color{blue}{-64}\right)\right) \]

5. Simplified 98.9%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)\right) - \mathsf{log1p}\left({u}^{3} \cdot -64\right)\right)} \]

6. Taylor expanded in u around 0 85.3%

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

   1. +-commutative 85.3%

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

   2. *-commutative 85.3%

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

   3. associate-*l* 85.4%

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

   4. *-commutative 85.4%

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

   5. associate-*l* 85.7%

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

   6. distribute-rgt-in 85.8%

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

   7. unpow2 85.8%

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

   8. associate-*r* 85.8%

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

   9. distribute-rgt-out 85.7%

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

   10. *-commutative 85.7%

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

8. Simplified 85.7%

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

9. Final simplification 85.7%

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

Alternative 8: 73.6% 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 64.2%

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

2. Taylor expanded in u around 0 73.2%

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

   1. *-commutative 73.2%

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

4. Simplified 73.2%

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

5. Final simplification 73.2%

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

Alternative 9: 73.8% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.2%

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

2. Taylor expanded in u around 0 73.5%

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

3. Final simplification 73.5%

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

Alternative 10: 16.5% 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 64.2%

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

   1. add-sqr-sqrt 63.7%

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

   2. associate-/r* 61.7%

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

3. Applied egg-rr 18.5%

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

   1. +-inverses 18.5%

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

5. Simplified 18.5%

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

6. Final simplification 18.5%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023187 
(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))))))