Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 60.9% → 99.4%

Time: 11.0s

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: 60.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.3%

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

   1. log-rec 63.4%

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

   2. distribute-rgt-neg-out 63.4%

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

   3. distribute-lft-neg-out 63.4%

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

   4. cancel-sign-sub-inv 63.4%

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

   5. log1p-def 99.4%

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

   6. *-commutative 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 91.0% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.3%

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

   1. log-rec 63.4%

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

   2. sub-neg 63.4%

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

   3. distribute-rgt-neg-out 63.4%

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

   4. +-commutative 63.4%

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

   5. distribute-rgt-neg-out 63.4%

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

   6. neg-mul-1 63.4%

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

   7. metadata-eval 63.4%

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

   8. distribute-lft-in 63.4%

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

   9. metadata-eval 63.4%

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

   10. sub-neg 63.4%

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

   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 63.4%

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

   17. neg-mul-1 63.4%

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

   18. sub-neg 63.4%

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

3. Simplified 63.4%

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

4. Taylor expanded in u around 0 90.8%

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

   1. associate-+r+ 90.8%

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

   2. *-commutative 90.8%

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

   3. cube-mult 90.8%

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

   4. unpow2 90.8%

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

   5. associate-*l* 90.8%

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

   6. *-commutative 90.8%

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

   7. unpow2 90.8%

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

   8. associate-*l* 90.8%

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

   9. distribute-lft-out 90.8%

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

   10. *-commutative 90.8%

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

   11. distribute-lft-out 90.5%

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

   12. unpow2 90.5%

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

   13. associate-*l* 90.5%

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

   14. distribute-lft-out 90.5%

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

6. Simplified 90.5%

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

   1. distribute-rgt-in 90.5%

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

8. Applied egg-rr 90.5%

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

9. Final simplification 90.5%

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

Alternative 3: 91.0% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.3%

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

   1. log-rec 63.4%

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

   2. sub-neg 63.4%

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

   3. distribute-rgt-neg-out 63.4%

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

   4. +-commutative 63.4%

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

   5. distribute-rgt-neg-out 63.4%

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

   6. neg-mul-1 63.4%

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

   7. metadata-eval 63.4%

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

   8. distribute-lft-in 63.4%

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

   9. metadata-eval 63.4%

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

   10. sub-neg 63.4%

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

   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 63.4%

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

   17. neg-mul-1 63.4%

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

   18. sub-neg 63.4%

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

3. Simplified 63.4%

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

4. Taylor expanded in u around 0 90.8%

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

   1. associate-+r+ 90.8%

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

   2. *-commutative 90.8%

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

   3. cube-mult 90.8%

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

   4. unpow2 90.8%

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

   5. associate-*l* 90.8%

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

   6. *-commutative 90.8%

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

   7. unpow2 90.8%

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

   8. associate-*l* 90.8%

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

   9. distribute-lft-out 90.8%

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

   10. *-commutative 90.8%

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

   11. distribute-lft-out 90.5%

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

   12. unpow2 90.5%

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

   13. associate-*l* 90.5%

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

   14. distribute-lft-out 90.5%

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

6. Simplified 90.5%

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

7. Final simplification 90.5%

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

Alternative 4: 87.0% 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 61.3%

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

   1. log-rec 63.4%

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

   2. sub-neg 63.4%

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

   3. distribute-rgt-neg-out 63.4%

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

   4. +-commutative 63.4%

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

   5. distribute-rgt-neg-out 63.4%

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

   6. neg-mul-1 63.4%

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

   7. metadata-eval 63.4%

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

   8. distribute-lft-in 63.4%

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

   9. metadata-eval 63.4%

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

   10. sub-neg 63.4%

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

   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 63.4%

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

   17. neg-mul-1 63.4%

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

   18. sub-neg 63.4%

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

3. Simplified 63.4%

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

4. Taylor expanded in u around 0 85.9%

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

   1. unpow2 85.9%

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

   2. associate-*r* 85.9%

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

   3. *-commutative 85.9%

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

   4. associate-*r* 85.9%

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

   5. distribute-rgt-out 85.7%

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

   6. *-commutative 85.7%

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

6. Simplified 85.7%

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

7. Taylor expanded in s around 0 86.0%

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

   1. +-commutative 86.0%

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

   2. distribute-rgt-in 86.2%

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

   3. *-commutative 86.2%

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

   4. *-commutative 86.2%

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

9. Applied egg-rr 86.2%

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

10. Final simplification 86.2%

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

Alternative 5: 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 61.3%

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

   1. log-rec 63.4%

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

   2. sub-neg 63.4%

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

   3. distribute-rgt-neg-out 63.4%

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

   4. +-commutative 63.4%

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

   5. distribute-rgt-neg-out 63.4%

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

   6. neg-mul-1 63.4%

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

   7. metadata-eval 63.4%

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

   8. distribute-lft-in 63.4%

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

   9. metadata-eval 63.4%

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

   10. sub-neg 63.4%

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

   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 63.4%

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

   17. neg-mul-1 63.4%

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

   18. sub-neg 63.4%

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

3. Simplified 63.4%

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

4. Taylor expanded in u around 0 85.9%

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

   1. unpow2 85.9%

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

   2. associate-*r* 85.9%

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

   3. *-commutative 85.9%

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

   4. associate-*r* 85.9%

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

   5. distribute-rgt-out 85.7%

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

   6. *-commutative 85.7%

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

6. Simplified 85.7%

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

7. Taylor expanded in s around 0 86.0%

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

   1. associate-*r* 85.7%

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

   2. *-commutative 85.7%

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

   3. +-commutative 85.7%

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

   4. distribute-lft-in 85.9%

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

   5. *-commutative 85.9%

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

   6. *-commutative 85.9%

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

   7. associate-*r* 86.2%

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

   8. *-commutative 86.2%

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

   9. associate-*l* 86.2%

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

9. Applied egg-rr 86.2%

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

    1. associate-*l* 86.2%

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

    2. distribute-lft-out 86.3%

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

    3. *-commutative 86.3%

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

11. Applied egg-rr 86.3%

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

12. Final simplification 86.3%

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

Alternative 6: 86.8% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.3%

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

   1. log-rec 63.4%

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

   2. sub-neg 63.4%

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

   3. distribute-rgt-neg-out 63.4%

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

   4. +-commutative 63.4%

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

   5. distribute-rgt-neg-out 63.4%

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

   6. neg-mul-1 63.4%

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

   7. metadata-eval 63.4%

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

   8. distribute-lft-in 63.4%

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

   9. metadata-eval 63.4%

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

   10. sub-neg 63.4%

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

   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 63.4%

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

   17. neg-mul-1 63.4%

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

   18. sub-neg 63.4%

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

3. Simplified 63.4%

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

4. Taylor expanded in u around 0 85.9%

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

   1. unpow2 85.9%

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

   2. associate-*r* 85.9%

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

   3. *-commutative 85.9%

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

   4. associate-*r* 85.9%

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

   5. distribute-rgt-out 85.7%

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

   6. *-commutative 85.7%

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

6. Simplified 85.7%

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

7. Taylor expanded in s around 0 86.0%

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

8. Final simplification 86.0%

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

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 61.3%

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

   1. log-rec 63.4%

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

   2. sub-neg 63.4%

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

   3. distribute-rgt-neg-out 63.4%

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

   4. +-commutative 63.4%

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

   5. distribute-rgt-neg-out 63.4%

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

   6. neg-mul-1 63.4%

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

   7. metadata-eval 63.4%

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

   8. distribute-lft-in 63.4%

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

   9. metadata-eval 63.4%

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

   10. sub-neg 63.4%

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

   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 63.4%

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

   17. neg-mul-1 63.4%

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

   18. sub-neg 63.4%

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

3. Simplified 63.4%

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

4. Taylor expanded in u around 0 73.9%

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

   1. *-commutative 73.9%

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

6. Simplified 73.9%

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

7. Final simplification 73.9%

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

Alternative 8: 74.2% 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 61.3%

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

   1. log-rec 63.4%

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

   2. sub-neg 63.4%

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

   3. distribute-rgt-neg-out 63.4%

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

   4. +-commutative 63.4%

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

   5. distribute-rgt-neg-out 63.4%

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

   6. neg-mul-1 63.4%

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

   7. metadata-eval 63.4%

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

   8. distribute-lft-in 63.4%

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

   9. metadata-eval 63.4%

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

   10. sub-neg 63.4%

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

   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 63.4%

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

   17. neg-mul-1 63.4%

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

   18. sub-neg 63.4%

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

3. Simplified 63.4%

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

4. Taylor expanded in u around 0 73.9%

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

   1. associate-*r* 74.2%

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

   2. *-commutative 74.2%

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

6. Simplified 74.2%

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

7. Final simplification 74.2%

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

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 61.3%

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

   1. log-rec 63.4%

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

   2. sub-neg 63.4%

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

   3. distribute-rgt-neg-out 63.4%

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

   4. +-commutative 63.4%

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

   5. distribute-rgt-neg-out 63.4%

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

   6. neg-mul-1 63.4%

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

   7. metadata-eval 63.4%

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

   8. distribute-lft-in 63.4%

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

   9. metadata-eval 63.4%

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

   10. sub-neg 63.4%

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

   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 63.4%

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

   17. neg-mul-1 63.4%

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

   18. sub-neg 63.4%

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

3. Simplified 63.4%

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

   1. add-sqr-sqrt 4.5%

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

   2. sqrt-unprod 15.0%

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

   3. sqr-neg 15.0%

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

   4. sqrt-unprod 15.0%

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

   5. add-sqr-sqrt 15.0%

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

   6. neg-mul-1 15.0%

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

   7. add-log-exp 14.9%

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

   8. neg-mul-1 14.9%

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

   9. add-sqr-sqrt 14.9%

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

   10. sqrt-unprod 14.9%

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

   11. sqr-neg 14.9%

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

   12. sqrt-unprod 4.5%

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

   13. add-sqr-sqrt 63.4%

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

   14. add-exp-log 63.4%

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

   15. add-sqr-sqrt 61.1%

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

   16. sqrt-unprod 63.4%

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

5. Applied egg-rr 17.5%

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

6. Final simplification 17.5%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023334 
(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))))))