Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 60.7% → 99.3%

Time: 12.7s

Alternatives: 6

Speedup: 21.8×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.7% 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: 99.3% 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 61.2%

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

   1. log-rec 63.6%

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

   2. distribute-rgt-neg-out 63.6%

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

   3. distribute-lft-neg-out 63.6%

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

   4. cancel-sign-sub-inv 63.6%

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

   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. Add Preprocessing

5. Final simplification 99.4%

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

Alternative 2: 91.2% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.2%

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

   1. log-rec 63.6%

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

   2. sub-neg 63.6%

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

   3. distribute-rgt-neg-out 63.6%

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

   4. +-commutative 63.6%

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

   5. distribute-rgt-neg-out 63.6%

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

   6. neg-mul-1 63.6%

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

   7. metadata-eval 63.6%

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

   8. distribute-lft-in 63.6%

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

   9. metadata-eval 63.6%

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

   10. sub-neg 63.6%

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

   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 63.6%

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

   17. sub-neg 63.6%

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

   18. metadata-eval 63.6%

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

3. Simplified 63.6%

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

5. Taylor expanded in u around 0 92.6%

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

   1. +-commutative 92.6%

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

   2. +-commutative 92.6%

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

   3. associate-+l+ 92.7%

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

   4. *-commutative 92.7%

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

   5. unpow2 92.7%

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

   6. associate-*r* 92.7%

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

   7. unpow3 92.7%

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

   8. unpow2 92.7%

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

   9. associate-*r* 92.7%

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

   10. distribute-rgt-out 92.7%

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

   11. distribute-lft-out 92.3%

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

   12. unpow2 92.3%

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

   13. associate-*r* 92.3%

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

   14. distribute-rgt-out 92.3%

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

   15. *-commutative 92.3%

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

7. Simplified 92.3%

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

8. Final simplification 92.3%

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

Alternative 3: 87.1% 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 61.2%

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

   1. log-rec 63.6%

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

   2. sub-neg 63.6%

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

   3. distribute-rgt-neg-out 63.6%

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

   4. +-commutative 63.6%

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

   5. distribute-rgt-neg-out 63.6%

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

   6. neg-mul-1 63.6%

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

   7. metadata-eval 63.6%

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

   8. distribute-lft-in 63.6%

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

   9. metadata-eval 63.6%

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

   10. sub-neg 63.6%

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

   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 63.6%

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

   17. sub-neg 63.6%

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

   18. metadata-eval 63.6%

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

3. Simplified 63.6%

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

5. Taylor expanded in u around 0 88.3%

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

   1. *-commutative 88.3%

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

   2. associate-*l* 88.6%

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

   3. *-commutative 88.6%

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

   4. associate-*l* 88.7%

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

   5. distribute-lft-out 88.8%

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

   6. unpow2 88.8%

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

   7. associate-*l* 88.8%

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

   8. distribute-lft-out 88.6%

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

7. Simplified 88.6%

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

8. Final simplification 88.6%

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

Alternative 4: 74.2% 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 61.2%

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

   1. log-rec 63.6%

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

   2. sub-neg 63.6%

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

   3. distribute-rgt-neg-out 63.6%

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

   4. +-commutative 63.6%

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

   5. distribute-rgt-neg-out 63.6%

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

   6. neg-mul-1 63.6%

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

   7. metadata-eval 63.6%

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

   8. distribute-lft-in 63.6%

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

   9. metadata-eval 63.6%

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

   10. sub-neg 63.6%

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

   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 63.6%

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

   17. sub-neg 63.6%

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

   18. metadata-eval 63.6%

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

3. Simplified 63.6%

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

5. Taylor expanded in u around 0 75.3%

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

6. Final simplification 75.3%

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

Alternative 5: 74.4% 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 61.2%

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

   1. log-rec 63.6%

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

   2. sub-neg 63.6%

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

   3. distribute-rgt-neg-out 63.6%

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

   4. +-commutative 63.6%

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

   5. distribute-rgt-neg-out 63.6%

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

   6. neg-mul-1 63.6%

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

   7. metadata-eval 63.6%

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

   8. distribute-lft-in 63.6%

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

   9. metadata-eval 63.6%

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

   10. sub-neg 63.6%

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

   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 63.6%

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

   17. sub-neg 63.6%

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

   18. metadata-eval 63.6%

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

3. Simplified 63.6%

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

5. Taylor expanded in u around 0 75.3%

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

   1. associate-*r* 75.6%

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

7. Simplified 75.6%

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

8. Final simplification 75.6%

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

Alternative 6: 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 61.2%

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

   1. log-rec 63.6%

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

   2. sub-neg 63.6%

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

   3. distribute-rgt-neg-out 63.6%

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

   4. +-commutative 63.6%

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

   5. distribute-rgt-neg-out 63.6%

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

   6. neg-mul-1 63.6%

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

   7. metadata-eval 63.6%

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

   8. distribute-lft-in 63.6%

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

   9. metadata-eval 63.6%

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

   10. sub-neg 63.6%

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

   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 63.6%

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

   17. sub-neg 63.6%

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

   18. metadata-eval 63.6%

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

3. Simplified 63.6%

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

6. Applied egg-rr 18.8%

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

7. Final simplification 18.8%

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

Reproduce

herbie shell --seed 2024031 
(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))))))