Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.6% → 99.4%

Time: 8.9s

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: 61.6% 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 60.6%

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

   1. log-rec 63.2%

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

   2. distribute-rgt-neg-out 63.2%

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

   3. distribute-lft-neg-out 63.2%

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

   4. cancel-sign-sub-inv 63.2%

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

   5. log1p-def 99.5%

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

   6. *-commutative 99.5%

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

   7. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 86.9% 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 60.6%

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

   1. flip3-- 60.2%

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

   2. associate-/r/ 60.0%

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

   3. log-prod 60.1%

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

   4. metadata-eval 60.1%

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

   5. metadata-eval 60.1%

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

   6. log1p-udef 95.6%

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

   7. *-un-lft-identity 95.6%

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

   8. distribute-lft1-in 95.8%

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

   9. fma-def 95.8%

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

3. Applied egg-rr 95.8%

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

   1. +-commutative 95.8%

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

   2. log-rec 96.4%

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

   3. sub-neg 96.4%

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

   4. log1p-def 99.1%

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

   5. unsub-neg 99.1%

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

   6. fma-def 99.1%

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

   7. distribute-rgt1-in 99.1%

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

   8. swap-sqr 99.1%

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

   9. metadata-eval 99.1%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot u + \color{blue}{16} \cdot \left(u \cdot u\right)\right) - \mathsf{log1p}\left(-{\left(4 \cdot u\right)}^{3}\right)\right) \]

   10. unpow2 99.1%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot u + 16 \cdot \color{blue}{{u}^{2}}\right) - \mathsf{log1p}\left(-{\left(4 \cdot u\right)}^{3}\right)\right) \]

   11. *-commutative 99.1%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{u \cdot 4} + 16 \cdot {u}^{2}\right) - \mathsf{log1p}\left(-{\left(4 \cdot u\right)}^{3}\right)\right) \]

   12. *-commutative 99.1%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(u \cdot 4 + \color{blue}{{u}^{2} \cdot 16}\right) - \mathsf{log1p}\left(-{\left(4 \cdot u\right)}^{3}\right)\right) \]

   13. unpow2 99.1%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(u \cdot 4 + \color{blue}{\left(u \cdot u\right)} \cdot 16\right) - \mathsf{log1p}\left(-{\left(4 \cdot u\right)}^{3}\right)\right) \]

   14. associate-*l* 99.1%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(u \cdot 4 + \color{blue}{u \cdot \left(u \cdot 16\right)}\right) - \mathsf{log1p}\left(-{\left(4 \cdot u\right)}^{3}\right)\right) \]

   15. distribute-lft-out 99.1%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{u \cdot \left(4 + u \cdot 16\right)}\right) - \mathsf{log1p}\left(-{\left(4 \cdot u\right)}^{3}\right)\right) \]

   16. cube-prod 99.1%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(u \cdot \left(4 + u \cdot 16\right)\right) - \mathsf{log1p}\left(-\color{blue}{{4}^{3} \cdot {u}^{3}}\right)\right) \]

   17. metadata-eval 99.1%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(u \cdot \left(4 + u \cdot 16\right)\right) - \mathsf{log1p}\left(-\color{blue}{64} \cdot {u}^{3}\right)\right) \]

5. Simplified 99.1%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(u \cdot \left(4 + u \cdot 16\right)\right) - \mathsf{log1p}\left({u}^{3} \cdot -64\right)\right)} \]

6. Taylor expanded in u around 0 87.8%

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

   1. *-commutative 87.8%

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

   2. *-commutative 87.8%

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

   3. +-commutative 87.8%

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

   4. *-commutative 87.8%

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

   5. fma-def 87.8%

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

   6. unpow2 87.8%

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

   7. associate-*r* 87.8%

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

   8. *-commutative 87.8%

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

   9. fma-def 87.8%

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

   10. associate-*r* 87.8%

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

   11. distribute-lft-out 87.7%

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

   12. fma-udef 87.7%

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

   13. associate-*l* 88.0%

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

8. Simplified 88.0%

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

   1. fma-udef 88.0%

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

   2. distribute-lft-in 88.1%

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

   3. *-commutative 88.1%

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

10. Applied egg-rr 88.1%

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

11. Final simplification 88.1%

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

Alternative 3: 86.8% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.6%

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

2. Taylor expanded in u around 0 88.2%

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

   1. *-commutative 88.2%

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

   2. *-commutative 88.2%

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

   3. unpow2 88.2%

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

   4. associate-*l* 88.2%

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

   5. distribute-lft-out 88.0%

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

4. Simplified 88.0%

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

5. Final simplification 88.0%

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

Alternative 4: 73.6% 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 60.6%

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

2. Taylor expanded in u around 0 74.9%

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

   1. *-commutative 74.9%

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

4. Simplified 74.9%

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

5. Final simplification 74.9%

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

Alternative 5: 73.8% 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 60.6%

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

2. Taylor expanded in u around 0 74.9%

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

   1. associate-*r* 75.2%

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

4. Simplified 75.2%

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

5. Final simplification 75.2%

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

Alternative 6: 16.7% 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 60.6%

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

3. Applied egg-rr 17.7%

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

   1. +-inverses 17.7%

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

5. Simplified 17.7%

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

6. Final simplification 17.7%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023322 
(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))))))