sqrt sqr

Average Error: 50.48% → 0%

Time: 3.0s

Precision: binary64

Cost: 6720

Math

\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]

↓

\[1 - \frac{\left|x\right|}{x} \]

FPCore

(FPCore (x) :precision binary64 (- (/ x x) (* (/ 1.0 x) (sqrt (* x x)))))

↓

(FPCore (x) :precision binary64 (- 1.0 (/ (fabs x) x)))

C

double code(double x) {
	return (x / x) - ((1.0 / x) * sqrt((x * x)));
}

↓

double code(double x) {
	return 1.0 - (fabs(x) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / x) - ((1.0d0 / x) * sqrt((x * x)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (abs(x) / x)
end function

Java

public static double code(double x) {
	return (x / x) - ((1.0 / x) * Math.sqrt((x * x)));
}

↓

public static double code(double x) {
	return 1.0 - (Math.abs(x) / x);
}

Python

def code(x):
	return (x / x) - ((1.0 / x) * math.sqrt((x * x)))

↓

def code(x):
	return 1.0 - (math.fabs(x) / x)

Julia

function code(x)
	return Float64(Float64(x / x) - Float64(Float64(1.0 / x) * sqrt(Float64(x * x))))
end

↓

function code(x)
	return Float64(1.0 - Float64(abs(x) / x))
end

MATLAB

function tmp = code(x)
	tmp = (x / x) - ((1.0 / x) * sqrt((x * x)));
end

↓

function tmp = code(x)
	tmp = 1.0 - (abs(x) / x);
end

Wolfram

code[x_] := N[(N[(x / x), $MachinePrecision] - N[(N[(1.0 / x), $MachinePrecision] * N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(1.0 - N[(N[Abs[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Target

Original	50.48%
Target	0%
Herbie	0%

\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Derivation

1. Initial program 50.48

   \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]

2. Simplified 0

   \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   Proof

   [Start] 50.48	\[ \frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
   cancel-sign-sub-inv [=>] 50.48	\[ \color{blue}{\frac{x}{x} + \left(-\frac{1}{x}\right) \cdot \sqrt{x \cdot x}} \]
   *-commutative [=>] 50.48	\[ \frac{x}{x} + \color{blue}{\sqrt{x \cdot x} \cdot \left(-\frac{1}{x}\right)} \]
   cancel-sign-sub [<=] 50.48	\[ \color{blue}{\frac{x}{x} - \left(-\sqrt{x \cdot x}\right) \cdot \left(-\frac{1}{x}\right)} \]
   *-inverses [=>] 50.48	\[ \color{blue}{1} - \left(-\sqrt{x \cdot x}\right) \cdot \left(-\frac{1}{x}\right) \]
   distribute-rgt-neg-in [<=] 50.48	\[ 1 - \color{blue}{\left(-\left(-\sqrt{x \cdot x}\right) \cdot \frac{1}{x}\right)} \]
   distribute-lft-neg-in [<=] 50.48	\[ 1 - \left(-\color{blue}{\left(-\sqrt{x \cdot x} \cdot \frac{1}{x}\right)}\right) \]
   *-commutative [<=] 50.48	\[ 1 - \left(-\left(-\color{blue}{\frac{1}{x} \cdot \sqrt{x \cdot x}}\right)\right) \]
   remove-double-neg [=>] 50.48	\[ 1 - \color{blue}{\frac{1}{x} \cdot \sqrt{x \cdot x}} \]
   associate-*l/ [=>] 46.77	\[ 1 - \color{blue}{\frac{1 \cdot \sqrt{x \cdot x}}{x}} \]
   *-lft-identity [=>] 46.77	\[ 1 - \frac{\color{blue}{\sqrt{x \cdot x}}}{x} \]
   rem-sqrt-square [=>] 0	\[ 1 - \frac{\color{blue}{\left|x\right|}}{x} \]

3. Final simplification 0

   \[\leadsto 1 - \frac{\left|x\right|}{x} \]

Reproduce

herbie shell --seed 2023089 
(FPCore (x)
  :name "sqrt sqr"
  :precision binary64

  :herbie-target
  (if (< x 0.0) 2.0 0.0)

  (- (/ x x) (* (/ 1.0 x) (sqrt (* x x)))))