sqrt sqr

Average Accuracy: 49.8% → 100.0%

Time: 2.4s

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	49.8%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 49.8%

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

2. Simplified 100.0%

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

   [Start] 49.8	\[ \frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
   sub-neg [=>] 49.8	\[ \color{blue}{\frac{x}{x} + \left(-\frac{1}{x} \cdot \sqrt{x \cdot x}\right)} \]
   distribute-rgt-neg-in [=>] 49.8	\[ \frac{x}{x} + \color{blue}{\frac{1}{x} \cdot \left(-\sqrt{x \cdot x}\right)} \]
   cancel-sign-sub [<=] 49.8	\[ \color{blue}{\frac{x}{x} - \left(-\frac{1}{x}\right) \cdot \left(-\sqrt{x \cdot x}\right)} \]
   *-inverses [=>] 49.8	\[ \color{blue}{1} - \left(-\frac{1}{x}\right) \cdot \left(-\sqrt{x \cdot x}\right) \]
   *-inverses [<=] 49.8	\[ 1 - \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \left(-\sqrt{x \cdot x}\right) \]
   distribute-neg-frac [=>] 49.8	\[ 1 - \color{blue}{\frac{-\frac{x}{x}}{x}} \cdot \left(-\sqrt{x \cdot x}\right) \]
   *-inverses [=>] 49.8	\[ 1 - \frac{-\color{blue}{1}}{x} \cdot \left(-\sqrt{x \cdot x}\right) \]
   metadata-eval [=>] 49.8	\[ 1 - \frac{\color{blue}{-1}}{x} \cdot \left(-\sqrt{x \cdot x}\right) \]
   associate-*l/ [=>] 53.6	\[ 1 - \color{blue}{\frac{-1 \cdot \left(-\sqrt{x \cdot x}\right)}{x}} \]
   neg-mul-1 [<=] 53.6	\[ 1 - \frac{\color{blue}{-\left(-\sqrt{x \cdot x}\right)}}{x} \]
   remove-double-neg [=>] 53.6	\[ 1 - \frac{\color{blue}{\sqrt{x \cdot x}}}{x} \]
   rem-sqrt-square [=>] 100.0	\[ 1 - \frac{\color{blue}{\left|x\right|}}{x} \]

3. Final simplification 100.0%

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

Reproduce

herbie shell --seed 2023146 
(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)))))