Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.1% → 99.4%

Time: 12.8s

Alternatives: 9

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: 61.1% 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 62.2%

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

   1. log-rec 64.6%

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

   2. distribute-rgt-neg-out 64.6%

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

   3. distribute-lft-neg-out 64.6%

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

   4. cancel-sign-sub-inv 64.6%

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

   5. log1p-def 99.3%

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

   6. *-commutative 99.3%

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

   7. metadata-eval 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 2: 91.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot \left(u \cdot -21.333333333333332\right)\\ s \cdot \left(u \cdot \left(\frac{t\_0 \cdot t\_0 - \left(u \cdot -8\right) \cdot \left(u \cdot -8\right)}{u \cdot -8 - t\_0} - -4\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (* u (* u -21.333333333333332))))
   (*
    s
    (*
     u
     (-
      (/ (- (* t_0 t_0) (* (* u -8.0) (* u -8.0))) (- (* u -8.0) t_0))
      -4.0)))))

C

float code(float s, float u) {
	float t_0 = u * (u * -21.333333333333332f);
	return s * (u * ((((t_0 * t_0) - ((u * -8.0f) * (u * -8.0f))) / ((u * -8.0f) - t_0)) - -4.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: t_0
    t_0 = u * (u * (-21.333333333333332e0))
    code = s * (u * ((((t_0 * t_0) - ((u * (-8.0e0)) * (u * (-8.0e0)))) / ((u * (-8.0e0)) - t_0)) - (-4.0e0)))
end function

Julia

function code(s, u)
	t_0 = Float32(u * Float32(u * Float32(-21.333333333333332)))
	return Float32(s * Float32(u * Float32(Float32(Float32(Float32(t_0 * t_0) - Float32(Float32(u * Float32(-8.0)) * Float32(u * Float32(-8.0)))) / Float32(Float32(u * Float32(-8.0)) - t_0)) - Float32(-4.0))))
end

MATLAB

function tmp = code(s, u)
	t_0 = u * (u * single(-21.333333333333332));
	tmp = s * (u * ((((t_0 * t_0) - ((u * single(-8.0)) * (u * single(-8.0)))) / ((u * single(-8.0)) - t_0)) - single(-4.0)));
end

Derivation

1. Initial program 62.2%

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

   1. log-rec 64.6%

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

   2. sub-neg 64.6%

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

   3. distribute-rgt-neg-out 64.6%

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

   4. +-commutative 64.6%

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

   5. distribute-rgt-neg-out 64.6%

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

   6. neg-mul-1 64.6%

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

   7. metadata-eval 64.6%

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

   8. distribute-lft-in 64.6%

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

   9. metadata-eval 64.6%

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

   10. sub-neg 64.6%

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

   11. log-prod -0.0%

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

   12. distribute-neg-in -0.0%

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

   13. unsub-neg -0.0%

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

   14. unsub-neg -0.0%

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

   15. distribute-neg-in -0.0%

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

   16. log-prod 64.6%

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

   17. sub-neg 64.6%

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

   18. metadata-eval 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in u around 0 90.5%

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

   1. +-commutative 90.5%

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

   2. +-commutative 90.5%

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

   3. associate-+l+ 90.6%

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

   4. *-commutative 90.6%

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

   5. unpow2 90.6%

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

   6. associate-*r* 90.6%

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

   7. unpow3 90.6%

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

   8. unpow2 90.6%

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

   9. associate-*r* 90.6%

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

   10. distribute-rgt-out 90.6%

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

   11. distribute-lft-out 90.3%

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

   12. unpow2 90.3%

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

   13. associate-*r* 90.3%

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

   14. distribute-rgt-out 90.3%

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

   15. *-commutative 90.3%

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

7. Simplified 90.3%

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

   1. distribute-lft-in 90.3%

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

   2. flip-+ 90.3%

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

9. Applied egg-rr 90.3%

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

10. Final simplification 90.3%

    \[\leadsto s \cdot \left(u \cdot \left(\frac{\left(u \cdot \left(u \cdot -21.333333333333332\right)\right) \cdot \left(u \cdot \left(u \cdot -21.333333333333332\right)\right) - \left(u \cdot -8\right) \cdot \left(u \cdot -8\right)}{u \cdot -8 - u \cdot \left(u \cdot -21.333333333333332\right)} - -4\right)\right) \]
11. Add Preprocessing

Alternative 3: 91.1% accurate, 6.8× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return s * (u * (-(-4.0f) - ((u * -8.0f) + (u * (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 * (u * (-21.333333333333332e0))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 62.2%

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

   1. log-rec 64.6%

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

   2. sub-neg 64.6%

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

   3. distribute-rgt-neg-out 64.6%

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

   4. +-commutative 64.6%

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

   5. distribute-rgt-neg-out 64.6%

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

   6. neg-mul-1 64.6%

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

   7. metadata-eval 64.6%

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

   8. distribute-lft-in 64.6%

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

   9. metadata-eval 64.6%

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

   10. sub-neg 64.6%

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

   11. log-prod -0.0%

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

   12. distribute-neg-in -0.0%

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

   13. unsub-neg -0.0%

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

   14. unsub-neg -0.0%

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

   15. distribute-neg-in -0.0%

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

   16. log-prod 64.6%

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

   17. sub-neg 64.6%

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

   18. metadata-eval 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in u around 0 90.5%

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

   1. +-commutative 90.5%

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

   2. +-commutative 90.5%

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

   3. associate-+l+ 90.6%

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

   4. *-commutative 90.6%

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

   5. unpow2 90.6%

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

   6. associate-*r* 90.6%

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

   7. unpow3 90.6%

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

   8. unpow2 90.6%

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

   9. associate-*r* 90.6%

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

   10. distribute-rgt-out 90.6%

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

   11. distribute-lft-out 90.3%

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

   12. unpow2 90.3%

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

   13. associate-*r* 90.3%

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

   14. distribute-rgt-out 90.3%

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

   15. *-commutative 90.3%

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

7. Simplified 90.3%

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

   1. +-commutative 90.3%

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

   2. distribute-rgt-in 90.3%

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

9. Applied egg-rr 90.3%

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

10. Final simplification 90.3%

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

Alternative 4: 91.1% accurate, 7.8× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return s * (u * ((u * (-(-8.0f) - (u * -21.333333333333332f))) - -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) - (u * (-21.333333333333332e0)))) - (-4.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 62.2%

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

   1. log-rec 64.6%

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

   2. sub-neg 64.6%

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

   3. distribute-rgt-neg-out 64.6%

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

   4. +-commutative 64.6%

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

   5. distribute-rgt-neg-out 64.6%

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

   6. neg-mul-1 64.6%

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

   7. metadata-eval 64.6%

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

   8. distribute-lft-in 64.6%

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

   9. metadata-eval 64.6%

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

   10. sub-neg 64.6%

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

   11. log-prod -0.0%

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

   12. distribute-neg-in -0.0%

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

   13. unsub-neg -0.0%

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

   14. unsub-neg -0.0%

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

   15. distribute-neg-in -0.0%

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

   16. log-prod 64.6%

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

   17. sub-neg 64.6%

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

   18. metadata-eval 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in u around 0 90.5%

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

   1. +-commutative 90.5%

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

   2. +-commutative 90.5%

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

   3. associate-+l+ 90.6%

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

   4. *-commutative 90.6%

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

   5. unpow2 90.6%

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

   6. associate-*r* 90.6%

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

   7. unpow3 90.6%

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

   8. unpow2 90.6%

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

   9. associate-*r* 90.6%

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

   10. distribute-rgt-out 90.6%

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

   11. distribute-lft-out 90.3%

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

   12. unpow2 90.3%

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

   13. associate-*r* 90.3%

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

   14. distribute-rgt-out 90.3%

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

   15. *-commutative 90.3%

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

7. Simplified 90.3%

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

8. Final simplification 90.3%

   \[\leadsto s \cdot \left(u \cdot \left(u \cdot \left(\left(--8\right) - u \cdot -21.333333333333332\right) - -4\right)\right) \]
9. Add Preprocessing

Alternative 5: 87.1% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.2%

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

   1. log-rec 64.6%

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

   2. sub-neg 64.6%

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

   3. distribute-rgt-neg-out 64.6%

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

   4. +-commutative 64.6%

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

   5. distribute-rgt-neg-out 64.6%

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

   6. neg-mul-1 64.6%

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

   7. metadata-eval 64.6%

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

   8. distribute-lft-in 64.6%

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

   9. metadata-eval 64.6%

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

   10. sub-neg 64.6%

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

   11. log-prod -0.0%

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

   12. distribute-neg-in -0.0%

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

   13. unsub-neg -0.0%

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

   14. unsub-neg -0.0%

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

   15. distribute-neg-in -0.0%

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

   16. log-prod 64.6%

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

   17. sub-neg 64.6%

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

   18. metadata-eval 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in u around 0 86.1%

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

   1. *-commutative 86.1%

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

   2. associate-*l* 86.3%

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

   3. *-commutative 86.3%

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

   4. associate-*l* 86.5%

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

   5. distribute-lft-out 86.6%

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

   6. unpow2 86.6%

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

   7. associate-*l* 86.6%

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

   8. distribute-lft-out 86.4%

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

7. Simplified 86.4%

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

   1. +-commutative 86.4%

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

   2. distribute-rgt-in 86.6%

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

9. Applied egg-rr 86.6%

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

10. Final simplification 86.6%

    \[\leadsto s \cdot \left(u \cdot \left(u \cdot 8\right) + u \cdot 4\right) \]
11. Add Preprocessing

Alternative 6: 86.9% 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 62.2%

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

   1. log-rec 64.6%

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

   2. sub-neg 64.6%

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

   3. distribute-rgt-neg-out 64.6%

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

   4. +-commutative 64.6%

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

   5. distribute-rgt-neg-out 64.6%

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

   6. neg-mul-1 64.6%

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

   7. metadata-eval 64.6%

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

   8. distribute-lft-in 64.6%

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

   9. metadata-eval 64.6%

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

   10. sub-neg 64.6%

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

   11. log-prod -0.0%

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

   12. distribute-neg-in -0.0%

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

   13. unsub-neg -0.0%

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

   14. unsub-neg -0.0%

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

   15. distribute-neg-in -0.0%

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

   16. log-prod 64.6%

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

   17. sub-neg 64.6%

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

   18. metadata-eval 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in u around 0 86.1%

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

   1. *-commutative 86.1%

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

   2. associate-*l* 86.3%

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

   3. *-commutative 86.3%

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

   4. associate-*l* 86.5%

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

   5. distribute-lft-out 86.6%

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

   6. unpow2 86.6%

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

   7. associate-*l* 86.6%

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

   8. distribute-lft-out 86.4%

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

7. Simplified 86.4%

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

8. Final simplification 86.4%

   \[\leadsto s \cdot \left(u \cdot \left(u \cdot 8 + 4\right)\right) \]
9. Add Preprocessing

Alternative 7: 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 62.2%

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

   1. log-rec 64.6%

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

   2. sub-neg 64.6%

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

   3. distribute-rgt-neg-out 64.6%

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

   4. +-commutative 64.6%

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

   5. distribute-rgt-neg-out 64.6%

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

   6. neg-mul-1 64.6%

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

   7. metadata-eval 64.6%

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

   8. distribute-lft-in 64.6%

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

   9. metadata-eval 64.6%

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

   10. sub-neg 64.6%

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

   11. log-prod -0.0%

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

   12. distribute-neg-in -0.0%

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

   13. unsub-neg -0.0%

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

   14. unsub-neg -0.0%

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

   15. distribute-neg-in -0.0%

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

   16. log-prod 64.6%

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

   17. sub-neg 64.6%

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

   18. metadata-eval 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in u around 0 73.4%

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

6. Final simplification 73.4%

   \[\leadsto 4 \cdot \left(s \cdot u\right) \]
7. Add Preprocessing

Alternative 8: 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 62.2%

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

   1. log-rec 64.6%

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

   2. sub-neg 64.6%

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

   3. distribute-rgt-neg-out 64.6%

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

   4. +-commutative 64.6%

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

   5. distribute-rgt-neg-out 64.6%

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

   6. neg-mul-1 64.6%

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

   7. metadata-eval 64.6%

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

   8. distribute-lft-in 64.6%

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

   9. metadata-eval 64.6%

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

   10. sub-neg 64.6%

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

   11. log-prod -0.0%

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

   12. distribute-neg-in -0.0%

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

   13. unsub-neg -0.0%

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

   14. unsub-neg -0.0%

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

   15. distribute-neg-in -0.0%

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

   16. log-prod 64.6%

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

   17. sub-neg 64.6%

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

   18. metadata-eval 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in u around 0 73.4%

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

   1. associate-*r* 73.5%

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

7. Simplified 73.5%

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

8. Final simplification 73.5%

   \[\leadsto u \cdot \left(s \cdot 4\right) \]
9. Add Preprocessing

Alternative 9: 16.6% 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 62.2%

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

   1. log-rec 64.6%

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

   2. sub-neg 64.6%

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

   3. distribute-rgt-neg-out 64.6%

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

   4. +-commutative 64.6%

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

   5. distribute-rgt-neg-out 64.6%

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

   6. neg-mul-1 64.6%

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

   7. metadata-eval 64.6%

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

   8. distribute-lft-in 64.6%

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

   9. metadata-eval 64.6%

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

   10. sub-neg 64.6%

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

   11. log-prod -0.0%

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

   12. distribute-neg-in -0.0%

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

   13. unsub-neg -0.0%

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

   14. unsub-neg -0.0%

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

   15. distribute-neg-in -0.0%

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

   16. log-prod 64.6%

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

   17. sub-neg 64.6%

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

   18. metadata-eval 64.6%

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

3. Simplified 64.6%

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

6. Applied egg-rr 16.5%

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

7. Final simplification 16.5%

   \[\leadsto s \cdot 0 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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))))))