?

Average Error: 0.1 → 0.1
Time: 16.8s
Precision: binary64
Cost: 20032

?

\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
↓
\[\left(x + \left(-1 + \left(e^{\mathsf{log1p}\left(y\right)} + \log y \cdot \left(-0.5 - y\right)\right)\right)\right) - z \]
(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))
↓
(FPCore (x y z)
 :precision binary64
 (- (+ x (+ -1.0 (+ (exp (log1p y)) (* (log y) (- -0.5 y))))) 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 (x + (-1.0 + (exp(log1p(y)) + (log(y) * (-0.5 - y))))) - z;
}

public static double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * Math.log(y))) + y) - z;
}
↓
public static double code(double x, double y, double z) {
	return (x + (-1.0 + (Math.exp(Math.log1p(y)) + (Math.log(y) * (-0.5 - y))))) - z;
}

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z
↓
def code(x, y, z):
	return (x + (-1.0 + (math.exp(math.log1p(y)) + (math.log(y) * (-0.5 - y))))) - 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(Float64(x + Float64(-1.0 + Float64(exp(log1p(y)) + Float64(log(y) * Float64(-0.5 - y))))) - 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[(x + N[(-1.0 + N[(N[Exp[N[Log[1 + y], $MachinePrecision]], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(x + \left(-1 + \left(e^{\mathsf{log1p}\left(y\right)} + \log y \cdot \left(-0.5 - y\right)\right)\right)\right) - z

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

    [Start]0.1

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

    associate-+l- [=>]0.1

    \[ \color{blue}{\left(x - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z \]

  3. Applied egg-rr0.1

    \[\leadsto \left(x - \color{blue}{\left(\left(\left(y + 0.5\right) \cdot \log y - e^{\mathsf{log1p}\left(y\right)}\right) + 1\right)}\right) - z \]

  4. Final simplification0.1

    \[\leadsto \left(x + \left(-1 + \left(e^{\mathsf{log1p}\left(y\right)} + \log y \cdot \left(-0.5 - y\right)\right)\right)\right) - z \]

Alternatives

Alternative 1
Error0.1
Cost13376
\[x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right) \]
Alternative 2
Error17.2
Cost7904
\[\begin{array}{l} t_0 := y + \log y \cdot \left(-0.5 - y\right)\\ t_1 := y \cdot \left(1 - \log y\right) - z\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{+155}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -0.0036:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -2.02 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-190}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
Alternative 3
Error16.6
Cost7904
\[\begin{array}{l} t_0 := \left(x + y\right) - y \cdot \log y\\ t_1 := y \cdot \left(1 - \log y\right) - z\\ t_2 := 0.5 \cdot \log y\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-170}:\\ \;\;\;\;\left(x + y\right) - t_2\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-286}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-162}:\\ \;\;\;\;y + \log y \cdot \left(-0.5 - y\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-14}:\\ \;\;\;\;x - t_2\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+143}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error20.2
Cost7645
\[\begin{array}{l} t_0 := \log y \cdot -0.5 - z\\ \mathbf{if}\;y \leq 3.3 \cdot 10^{-179}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-47}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+125} \lor \neg \left(y \leq 8 \cdot 10^{+144}\right) \land y \leq 3.5 \cdot 10^{+190}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
Alternative 5
Error19.5
Cost7381
\[\begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-179}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-105}:\\ \;\;\;\;x - 0.5 \cdot \log y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+125} \lor \neg \left(y \leq 1.65 \cdot 10^{+146}\right) \land y \leq 3.3 \cdot 10^{+190}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
Alternative 6
Error6.9
Cost7244
\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+120}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;\left(x + y\right) + \log y \cdot \left(-0.5 - y\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+140}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
Alternative 7
Error19.1
Cost7117
\[\begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+125} \lor \neg \left(y \leq 1.6 \cdot 10^{+147}\right) \land y \leq 3.2 \cdot 10^{+190}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
Alternative 8
Error6.8
Cost7112
\[\begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+148}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
Alternative 9
Error6.8
Cost7112
\[\begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 2.6 \cdot 10^{+52}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+147}:\\ \;\;\;\;x + t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \]
Alternative 10
Error0.1
Cost7104
\[\left(y - z\right) + \left(x + \log y \cdot \left(-0.5 - y\right)\right) \]
Alternative 11
Error33.8
Cost392
\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+55}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+116}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 12
Error27.2
Cost192
\[x - z \]
Alternative 13
Error45.1
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023020 
(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))