?

Average Error: 0.1 → 0.1
Time: 13.2s
Precision: binary64
Cost: 19648

?

\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
↓
\[\mathsf{fma}\left(x, \log y, \log t - \left(y + z\right)\right) \]
(FPCore (x y z t) :precision binary64 (+ (- (- (* x (log y)) y) z) (log t)))
↓
(FPCore (x y z t) :precision binary64 (fma x (log y) (- (log t) (+ y z))))
double code(double x, double y, double z, double t) {
	return (((x * log(y)) - y) - z) + log(t);
}
↓
double code(double x, double y, double z, double t) {
	return fma(x, log(y), (log(t) - (y + z)));
}

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * log(y)) - y) - z) + log(t))
end
↓
function code(x, y, z, t)
	return fma(x, log(y), Float64(log(t) - Float64(y + z)))
end

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\mathsf{fma}\left(x, \log y, \log t - \left(y + z\right)\right)

Error?

Derivation?

  1. Initial program 0.1

    \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, \log y, \log t\right) - y\right) - z} \]
    Proof

    [Start]0.1

    \[ \left(\left(x \cdot \log y - y\right) - z\right) + \log t \]

    +-lft-identity [<=]0.1

    \[ \color{blue}{0 + \left(\left(\left(x \cdot \log y - y\right) - z\right) + \log t\right)} \]

    +-commutative [=>]0.1

    \[ 0 + \color{blue}{\left(\log t + \left(\left(x \cdot \log y - y\right) - z\right)\right)} \]

    associate-+r- [=>]0.1

    \[ 0 + \color{blue}{\left(\left(\log t + \left(x \cdot \log y - y\right)\right) - z\right)} \]

    associate-+r- [=>]0.1

    \[ \color{blue}{\left(0 + \left(\log t + \left(x \cdot \log y - y\right)\right)\right) - z} \]

    +-lft-identity [=>]0.1

    \[ \color{blue}{\left(\log t + \left(x \cdot \log y - y\right)\right)} - z \]

    associate-+r- [=>]0.1

    \[ \color{blue}{\left(\left(\log t + x \cdot \log y\right) - y\right)} - z \]

    +-commutative [=>]0.1

    \[ \left(\color{blue}{\left(x \cdot \log y + \log t\right)} - y\right) - z \]

    fma-def [=>]0.1

    \[ \left(\color{blue}{\mathsf{fma}\left(x, \log y, \log t\right)} - y\right) - z \]

  3. Taylor expanded in x around 0 0.1

    \[\leadsto \color{blue}{\left(\log y \cdot x + \log t\right) - \left(y + z\right)} \]

  4. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t - \left(y + z\right)\right)} \]
    Proof

    [Start]0.1

    \[ \left(\log y \cdot x + \log t\right) - \left(y + z\right) \]

    *-commutative [<=]0.1

    \[ \left(\color{blue}{x \cdot \log y} + \log t\right) - \left(y + z\right) \]

    *-rgt-identity [<=]0.1

    \[ \left(\color{blue}{\left(x \cdot \log y\right) \cdot 1} + \log t\right) - \left(y + z\right) \]

    associate-+r- [<=]0.1

    \[ \color{blue}{\left(x \cdot \log y\right) \cdot 1 + \left(\log t - \left(y + z\right)\right)} \]

    *-rgt-identity [=>]0.1

    \[ \color{blue}{x \cdot \log y} + \left(\log t - \left(y + z\right)\right) \]

    fma-def [=>]0.1

    \[ \color{blue}{\mathsf{fma}\left(x, \log y, \log t - \left(y + z\right)\right)} \]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, \log y, \log t - \left(y + z\right)\right) \]

Alternatives

Alternative 1
Error0.1
Cost19648
\[\left(\mathsf{fma}\left(x, \log y, \log t\right) - y\right) - z \]
Alternative 2
Error0.8
Cost13449
\[\begin{array}{l} \mathbf{if}\;x \leq -1320000000 \lor \neg \left(x \leq 2.5 \cdot 10^{-34}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \left(-y\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]
Alternative 3
Error0.1
Cost13376
\[\log t + \left(\left(x \cdot \log y - y\right) - z\right) \]
Alternative 4
Error19.4
Cost7384
\[\begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(-y\right) - z\\ t_3 := \log t - y\\ \mathbf{if}\;z \leq -440:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-269}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-296}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-38}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 310:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 5
Error19.7
Cost7384
\[\begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(-y\right) - z\\ t_3 := \log t - y\\ \mathbf{if}\;z \leq -370:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-269}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-38}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 6
Error16.7
Cost7248
\[\begin{array}{l} t_1 := \left(-y\right) - z\\ t_2 := x \cdot \log y - y\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-46}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;z \leq 1800000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error16.9
Cost7248
\[\begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t_1 - y\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+132}:\\ \;\;\;\;t_1 - z\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-50}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;z \leq 200000000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]
Alternative 8
Error6.9
Cost7248
\[\begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t_1 - y\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+15}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+53}:\\ \;\;\;\;t_1 - z\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+97}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 9
Error0.9
Cost7113
\[\begin{array}{l} \mathbf{if}\;x \leq -1320000000 \lor \neg \left(x \leq 2.5 \cdot 10^{-34}\right):\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]
Alternative 10
Error27.8
Cost6992
\[\begin{array}{l} \mathbf{if}\;y \leq 10^{-260}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-190}:\\ \;\;\;\;\log t\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-165}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-147}:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]
Alternative 11
Error18.8
Cost6857
\[\begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{+182} \lor \neg \left(x \leq 6.2 \cdot 10^{+171}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]
Alternative 12
Error33.5
Cost260
\[\begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{+97}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
Alternative 13
Error26.6
Cost256
\[\left(-y\right) - z \]
Alternative 14
Error44.7
Cost128
\[-y \]

Error

Reproduce?

herbie shell --seed 2023060 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))