Average Error: 0.1 → 0.1
Time: 9.6s
Precision: binary64
Cost: 13376
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
↓
\[\mathsf{fma}\left(\log y, -0.5 - y, y + \left(x - z\right)\right) \]
(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))
↓
(FPCore (x y z) :precision binary64 (fma (log y) (- -0.5 y) (+ y (- x z))))
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 fma(log(y), (-0.5 - y), (y + (x - z)));
}

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 fma(log(y), Float64(-0.5 - y), Float64(y + Float64(x - z)))
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[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + N[(y + N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

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

  3. Final simplification0.1

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

Alternatives

Alternative 1
Error21.9
Cost7776
\[\begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ t_1 := \log y \cdot -0.5 - z\\ \mathbf{if}\;x \leq -921151009287500400:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -8.174006334402303 \cdot 10^{-260}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.611774457564097 \cdot 10^{-242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.190262272788642 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.5336557511100107 \cdot 10^{-131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.0814862251832016 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.162428272505588 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.847538022615878 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
Alternative 2
Error15.5
Cost7508
\[\begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ t_1 := x + t_0\\ t_2 := t_0 - z\\ \mathbf{if}\;z \leq -1.0159495820178921 \cdot 10^{+160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -9.852267458054785 \cdot 10^{+26}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq -6.7850502988754325 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.1586985361164386 \cdot 10^{-209}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;z \leq 4.270594108366231 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 3
Error14.9
Cost7508
\[\begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ t_1 := x + t_0\\ t_2 := t_0 - z\\ \mathbf{if}\;z \leq -1.0159495820178921 \cdot 10^{+160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -9.852267458054785 \cdot 10^{+26}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq -6.7850502988754325 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.1586985361164386 \cdot 10^{-209}:\\ \;\;\;\;y + \log y \cdot \left(-0.5 - y\right)\\ \mathbf{elif}\;z \leq 4.270594108366231 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 4
Error16.0
Cost7376
\[\begin{array}{l} t_0 := x + y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;z \leq -9.852267458054785 \cdot 10^{+26}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq -6.7850502988754325 \cdot 10^{-171}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.1586985361164386 \cdot 10^{-209}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;z \leq 2.6121471417075755 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
Alternative 5
Error7.1
Cost7244
\[\begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ t_1 := \left(x + \log y \cdot -0.5\right) - z\\ \mathbf{if}\;y \leq 189902241414233.06:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.790846930577521 \cdot 10^{+70}:\\ \;\;\;\;x + t_0\\ \mathbf{elif}\;y \leq 1.6338737371858298 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \]
Alternative 6
Error18.4
Cost7116
\[\begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 7.487266188386265 \cdot 10^{+19}:\\ \;\;\;\;x + \left(1 - z\right)\\ \mathbf{elif}\;y \leq 1.5990021261811702 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.2657220436862592 \cdot 10^{+151}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error0.5
Cost7108
\[\begin{array}{l} \mathbf{if}\;y \leq 0.0001698425387102433:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \]
Alternative 8
Error0.5
Cost7108
\[\begin{array}{l} \mathbf{if}\;y \leq 0.0001698425387102433:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + x\right) - z\right) - y \cdot \log y\\ \end{array} \]
Alternative 9
Error0.1
Cost7104
\[\left(\left(y + x\right) - z\right) + \log y \cdot \left(-0.5 - y\right) \]
Alternative 10
Error32.6
Cost392
\[\begin{array}{l} \mathbf{if}\;z \leq -8.050038012921309 \cdot 10^{+70}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 4.655355521105147 \cdot 10^{+59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
Alternative 11
Error26.1
Cost320
\[x + \left(1 - z\right) \]
Alternative 12
Error26.8
Cost192
\[x - z \]
Alternative 13
Error44.8
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022216 
(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))