?

Average Error: 0.1 → 0.1
Time: 9.3s
Precision: binary64
Cost: 13056

?

\[-1000 \leq x \land x \leq 1000\]
\[\left|x\right| - \sqrt{{x}^{2} + 1} \]
↓
\[\left|x\right| - \mathsf{hypot}\left(1, x\right) \]
(FPCore (x) :precision binary64 (- (fabs x) (sqrt (+ (pow x 2.0) 1.0))))
↓
(FPCore (x) :precision binary64 (- (fabs x) (hypot 1.0 x)))
double code(double x) {
	return fabs(x) - sqrt((pow(x, 2.0) + 1.0));
}
↓
double code(double x) {
	return fabs(x) - hypot(1.0, x);
}

public static double code(double x) {
	return Math.abs(x) - Math.sqrt((Math.pow(x, 2.0) + 1.0));
}
↓
public static double code(double x) {
	return Math.abs(x) - Math.hypot(1.0, x);
}

def code(x):
	return math.fabs(x) - math.sqrt((math.pow(x, 2.0) + 1.0))
↓
def code(x):
	return math.fabs(x) - math.hypot(1.0, x)

function code(x)
	return Float64(abs(x) - sqrt(Float64((x ^ 2.0) + 1.0)))
end
↓
function code(x)
	return Float64(abs(x) - hypot(1.0, x))
end

function tmp = code(x)
	tmp = abs(x) - sqrt(((x ^ 2.0) + 1.0));
end
↓
function tmp = code(x)
	tmp = abs(x) - hypot(1.0, x);
end

code[x_] := N[(N[Abs[x], $MachinePrecision] - N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(N[Abs[x], $MachinePrecision] - N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]

\left|x\right| - \sqrt{{x}^{2} + 1}
↓
\left|x\right| - \mathsf{hypot}\left(1, x\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.1

    \[\left|x\right| - \sqrt{{x}^{2} + 1} \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} \]
    Proof

    [Start]0.1

    \[ \left|x\right| - \sqrt{{x}^{2} + 1} \]

    +-commutative [=>]0.1

    \[ \left|x\right| - \sqrt{\color{blue}{1 + {x}^{2}}} \]

    unpow2 [=>]0.1

    \[ \left|x\right| - \sqrt{1 + \color{blue}{x \cdot x}} \]

    hypot-1-def [=>]0.1

    \[ \left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)} \]

  3. Final simplification0.1

    \[\leadsto \left|x\right| - \mathsf{hypot}\left(1, x\right) \]

Alternatives

Alternative 1
Error0.8
Cost6976
\[\left|x\right| - \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \]
Alternative 2
Error0.9
Cost6784
\[\frac{-1}{x + \mathsf{hypot}\left(1, x\right)} \]
Alternative 3
Error0.9
Cost6656
\[x - \mathsf{hypot}\left(1, x\right) \]
Alternative 4
Error1.2
Cost6592
\[\left|x\right| + -1 \]
Alternative 5
Error1.3
Cost576
\[x - \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \]
Alternative 6
Error1.5
Cost192
\[x + -1 \]
Alternative 7
Error1.9
Cost64
\[-1 \]

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x)
  :name "abs(x) - sqrt(x^2 + 1)"
  :precision binary64
  :pre (and (<= -1000.0 x) (<= x 1000.0))
  (- (fabs x) (sqrt (+ (pow x 2.0) 1.0))))