Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 11.0s

Precision: binary64

Cost: 7104

Math

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

↓

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

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

↓

(FPCore (x y z) :precision binary64 (- (- x (- (* (+ y 0.5) (log y)) y)) z))

C

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 - (((y + 0.5) * log(y)) - y)) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - ((y + 0.5d0) * log(y))) + y) - z
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - (((y + 0.5d0) * log(y)) - y)) - z
end function

Java

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 - (((y + 0.5) * Math.log(y)) - y)) - z;
}

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

↓

def code(x, y, z):
	return (x - (((y + 0.5) * math.log(y)) - y)) - z

Julia

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(Float64(Float64(y + 0.5) * log(y)) - y)) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - ((y + 0.5) * log(y))) + y) - z;
end

↓

function tmp = code(x, y, z)
	tmp = (x - (((y + 0.5) * log(y)) - y)) - z;
end

Wolfram

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[(N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Target

Original	0.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	7.4
Cost	7768

\[\begin{array}{l} t_0 := \left(x - 0.5 \cdot \log y\right) - z\\ t_1 := \log y \cdot \left(0.5 + y\right)\\ t_2 := x - \left(t_1 - y\right)\\ t_3 := \left(y - z\right) - t_1\\ \mathbf{if}\;x \leq -3.05 \cdot 10^{+113}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{+16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -19:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+66}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+161}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	19.9
Cost	7512

\[\begin{array}{l} t_0 := \left(1 - \log y\right) \cdot y\\ \mathbf{if}\;y \leq 1.7 \cdot 10^{-243}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-192}:\\ \;\;\;\;x - 0.5 \cdot \log y\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-168}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+104}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+184}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	19.3
Cost	7380

\[\begin{array}{l} t_0 := \left(1 - \log y\right) \cdot y\\ \mathbf{if}\;y \leq 2.9 \cdot 10^{-184}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-174}:\\ \;\;\;\;-0.5 \cdot \log y\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+104}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+184}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	19.9
Cost	7380

\[\begin{array}{l} t_0 := \left(1 - \log y\right) \cdot y\\ \mathbf{if}\;y \leq 1.12 \cdot 10^{-243}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-168}:\\ \;\;\;\;x - 0.5 \cdot \log y\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+104}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+184}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	19.6
Cost	7376

\[\begin{array}{l} t_0 := y - \left(0.5 + y\right) \cdot \log y\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-32}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-96}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;x \leq 510:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 6
Error	10.8
Cost	7244

\[\begin{array}{l} t_0 := \left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{if}\;y \leq 2.15 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+140}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+184}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \]

Alternative 7
Error	7.0
Cost	7240

\[\begin{array}{l} \mathbf{if}\;z \leq -3.15 \cdot 10^{+66}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+112}:\\ \;\;\;\;x - \left(\log y \cdot \left(0.5 + y\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \end{array} \]

Alternative 8
Error	0.1
Cost	7104

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

Alternative 9
Error	27.6
Cost	6856

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-151}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-257}:\\ \;\;\;\;-0.5 \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 10
Error	33.4
Cost	392

\[\begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+69}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+133}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 11
Error	26.6
Cost	192

\[x - z \]

Alternative 12
Error	44.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023101 
(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))