Average Error: 0.1 → 0.1
Time: 7.1s
Precision: binary64
Cost: 13376
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
↓
\[\mathsf{fma}\left(\log y, -0.5 - y, y\right) + \left(x - z\right) \]
(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 Float64(fma(log(y), Float64(-0.5 - y), 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[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + y), $MachinePrecision] + N[(x - z), $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\right) + \left(x - z\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. Applied egg-rr0.1

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

  4. Final simplification0.1

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

Alternatives

Alternative 1
Error16.8
Cost7508
\[\begin{array}{l} t_0 := \left(y - y \cdot \log y\right) - z\\ \mathbf{if}\;y \leq 2.676834236016207 \cdot 10^{-204}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.794891806316223 \cdot 10^{-97}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.3988506774425419 \cdot 10^{+48}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.503319563785685 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.8576439601920052 \cdot 10^{+111}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error22.6
Cost7244
\[\begin{array}{l} \mathbf{if}\;y \leq 2.676834236016207 \cdot 10^{-204}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.794891806316223 \cdot 10^{-97}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.6522310505035953 \cdot 10^{+234}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log \left(\frac{1}{y}\right)\right)\\ \end{array} \]
Alternative 3
Error7.1
Cost7244
\[\begin{array}{l} t_0 := \left(y - y \cdot \log y\right) - z\\ t_1 := x + \left(\log y \cdot -0.5 - z\right)\\ \mathbf{if}\;y \leq 1.3988506774425419 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.503319563785685 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.739201804643862 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error22.6
Cost7116
\[\begin{array}{l} \mathbf{if}\;y \leq 2.676834236016207 \cdot 10^{-204}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.794891806316223 \cdot 10^{-97}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.6522310505035953 \cdot 10^{+234}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \]
Alternative 5
Error0.7
Cost7108
\[\begin{array}{l} \mathbf{if}\;y \leq 1.556659961920067 \cdot 10^{-15}:\\ \;\;\;\;x + \left(\log y \cdot -0.5 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(y - y \cdot \log y\right)\right) - z\\ \end{array} \]
Alternative 6
Error0.1
Cost7104
\[\left(x - z\right) + \left(y + \log y \cdot \left(-0.5 - y\right)\right) \]
Alternative 7
Error19.2
Cost6984
\[\begin{array}{l} \mathbf{if}\;x \leq -4.79149069724115 \cdot 10^{+18}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 7.658030261379521 \cdot 10^{+19}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
Alternative 8
Error27.8
Cost6856
\[\begin{array}{l} \mathbf{if}\;z \leq -1.8574584974694644 \cdot 10^{-180}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.787938526000673 \cdot 10^{-186}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
Alternative 9
Error33.5
Cost392
\[\begin{array}{l} \mathbf{if}\;x \leq -2.854718311731688 \cdot 10^{+106}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.254899758870259 \cdot 10^{+90}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 10
Error26.7
Cost192
\[x - z \]
Alternative 11
Error45.1
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022225 
(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))