?

Average Error: 0.1 → 0.1
Time: 1.5min
Precision: binary64
Cost: 13376

?

\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
↓
\[x - \mathsf{fma}\left(\log y, 0.5 + y, z - y\right) \]
(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))
↓
(FPCore (x y z) :precision binary64 (- x (fma (log y) (+ 0.5 y) (- z y))))
double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + y) - z;
}
↓
double code(double x, double y, double z) {
	return x - fma(log(y), (0.5 + y), (z - y));
}

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end
↓
function code(x, y, z)
	return Float64(x - fma(log(y), Float64(0.5 + y), Float64(z - y)))
end

code[x_, y_, z_] := N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]
↓
code[x_, y_, z_] := N[(x - N[(N[Log[y], $MachinePrecision] * N[(0.5 + y), $MachinePrecision] + N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
x - \mathsf{fma}\left(\log y, 0.5 + y, z - y\right)

Error?

Target

Original0.1
Target0.1
Herbie0.1
\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y \]

Derivation?

  1. Initial program 0.1

    \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

  2. Simplified0.1

    \[\leadsto \color{blue}{x - \mathsf{fma}\left(\log y, 0.5 + y, z - y\right)} \]
    Proof

Alternatives

Alternative 1
Error20.8
Cost7904
\[\begin{array}{l} t_0 := y - \left(0.5 + y\right) \cdot \log y\\ \mathbf{if}\;x \leq -25:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-232}:\\ \;\;\;\;y - z\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-275}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-306}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-202}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-170}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 230:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
Alternative 2
Error15.9
Cost7508
\[\begin{array}{l} t_0 := x - \frac{\log y}{2}\\ \mathbf{if}\;y \leq 1.8 \cdot 10^{-252}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-92}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+31}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \end{array} \]
Alternative 3
Error20.0
Cost7380
\[\begin{array}{l} t_0 := x - \frac{\log y}{2}\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{-252}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-100}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+166}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \]
Alternative 4
Error6.9
Cost7240
\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+55}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+80}:\\ \;\;\;\;\left(y + x\right) - \left(0.5 + y\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
Alternative 5
Error0.1
Cost7104
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
Alternative 6
Error19.4
Cost6984
\[\begin{array}{l} \mathbf{if}\;z \leq -2800000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 100:\\ \;\;\;\;x - \frac{\log y}{2}\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
Alternative 7
Error6.9
Cost6980
\[\begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+28}:\\ \;\;\;\;\left(x - \frac{\log y}{2}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \end{array} \]
Alternative 8
Error33.4
Cost392
\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+17}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 9
Error27.1
Cost192
\[x - z \]
Alternative 10
Error45.0
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023033 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))