sqrt sqr

Average Error: 32.1 → 0.0

Time: 14.7s

Precision: binary64

Cost: 516

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{x - \left(-x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - x}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310) (/ (- x (- x)) x) (/ (- x x) x)))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = (x - -x) / x;
	} else {
		tmp = (x - x) / x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-4d-310)) then
        tmp = (x - -x) / x
    else
        tmp = (x - x) / x
    end if
    code = tmp
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) {
	double tmp;
	if (x <= -4e-310) {
		tmp = (x - -x) / x;
	} else {
		tmp = (x - x) / x;
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -4e-310:
		tmp = (x - -x) / x
	else:
		tmp = (x - x) / x
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(Float64(x - Float64(-x)) / x);
	else
		tmp = Float64(Float64(x - x) / x);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e-310)
		tmp = (x - -x) / x;
	else
		tmp = (x - x) / x;
	end
	tmp_2 = tmp;
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_] := If[LessEqual[x, -4e-310], N[(N[(x - (-x)), $MachinePrecision] / x), $MachinePrecision], N[(N[(x - x), $MachinePrecision] / x), $MachinePrecision]]

Target

Original	32.1
Target	0
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 28.6

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

   2. Simplified 28.6

      \[\leadsto \color{blue}{\frac{x - \sqrt{x \cdot x}}{x}} \]
      Proof

      [Start] 28.6	\[ \frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
      rational_best-simplify-1 [=>] 28.6	\[ \frac{x}{x} - \color{blue}{\sqrt{x \cdot x} \cdot \frac{1}{x}} \]
      rational_best-simplify-55 [=>] 28.6	\[ \frac{x}{x} - \color{blue}{1 \cdot \frac{\sqrt{x \cdot x}}{x}} \]
      rational_best-simplify-1 [=>] 28.6	\[ \frac{x}{x} - \color{blue}{\frac{\sqrt{x \cdot x}}{x} \cdot 1} \]
      rational_best-simplify-7 [=>] 28.6	\[ \frac{x}{x} - \color{blue}{\frac{\sqrt{x \cdot x}}{x}} \]
      rational_best-simplify-66 [=>] 28.6	\[ \color{blue}{\frac{x - \sqrt{x \cdot x}}{x}} \]

   3. Taylor expanded in x around -inf 0.1

      \[\leadsto \frac{x - \color{blue}{-1 \cdot x}}{x} \]

   4. Simplified 0.1

      \[\leadsto \frac{x - \color{blue}{\left(-x\right)}}{x} \]
      Proof

      [Start] 0.1	\[ \frac{x - -1 \cdot x}{x} \]
      rational_best-simplify-1 [=>] 0.1	\[ \frac{x - \color{blue}{x \cdot -1}}{x} \]
      rational_best-simplify-11 [<=] 0.1	\[ \frac{x - \color{blue}{\left(-x\right)}}{x} \]

   if -3.999999999999988e-310 < x

   1. Initial program 35.4

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

   2. Simplified 30.5

      \[\leadsto \color{blue}{\frac{x - \sqrt{x \cdot x}}{x}} \]
      Proof

      [Start] 35.4	\[ \frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
      rational_best-simplify-1 [=>] 35.4	\[ \frac{x}{x} - \color{blue}{\sqrt{x \cdot x} \cdot \frac{1}{x}} \]
      rational_best-simplify-55 [=>] 30.5	\[ \frac{x}{x} - \color{blue}{1 \cdot \frac{\sqrt{x \cdot x}}{x}} \]
      rational_best-simplify-1 [=>] 30.5	\[ \frac{x}{x} - \color{blue}{\frac{\sqrt{x \cdot x}}{x} \cdot 1} \]
      rational_best-simplify-7 [=>] 30.5	\[ \frac{x}{x} - \color{blue}{\frac{\sqrt{x \cdot x}}{x}} \]
      rational_best-simplify-66 [=>] 30.5	\[ \color{blue}{\frac{x - \sqrt{x \cdot x}}{x}} \]

   3. Taylor expanded in x around 0 0

      \[\leadsto \frac{x - \color{blue}{x}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{x - \left(-x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - x}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	30.3
Cost	320

\[\frac{x - x}{x} \]

Reproduce

herbie shell --seed 2023099 
(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)))))