?

Average Error: 58.7 → 0.1
Time: 10.5s
Precision: binary64
Cost: 20224

?

\[-1 \leq x \land x \leq 1\]
\[\sqrt{1 + x} - \sqrt{1 - x} \]
↓
\[x + \left(0.125 \cdot {x}^{3} + \left(0.0546875 \cdot {x}^{5} + 0.0322265625 \cdot {x}^{7}\right)\right) \]
(FPCore (x) :precision binary64 (- (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))
↓
(FPCore (x)
 :precision binary64
 (+
  x
  (+
   (* 0.125 (pow x 3.0))
   (+ (* 0.0546875 (pow x 5.0)) (* 0.0322265625 (pow x 7.0))))))
double code(double x) {
	return sqrt((1.0 + x)) - sqrt((1.0 - x));
}
↓
double code(double x) {
	return x + ((0.125 * pow(x, 3.0)) + ((0.0546875 * pow(x, 5.0)) + (0.0322265625 * pow(x, 7.0))));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 + x)) - sqrt((1.0d0 - x))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((0.125d0 * (x ** 3.0d0)) + ((0.0546875d0 * (x ** 5.0d0)) + (0.0322265625d0 * (x ** 7.0d0))))
end function

public static double code(double x) {
	return Math.sqrt((1.0 + x)) - Math.sqrt((1.0 - x));
}
↓
public static double code(double x) {
	return x + ((0.125 * Math.pow(x, 3.0)) + ((0.0546875 * Math.pow(x, 5.0)) + (0.0322265625 * Math.pow(x, 7.0))));
}

def code(x):
	return math.sqrt((1.0 + x)) - math.sqrt((1.0 - x))
↓
def code(x):
	return x + ((0.125 * math.pow(x, 3.0)) + ((0.0546875 * math.pow(x, 5.0)) + (0.0322265625 * math.pow(x, 7.0))))

function code(x)
	return Float64(sqrt(Float64(1.0 + x)) - sqrt(Float64(1.0 - x)))
end
↓
function code(x)
	return Float64(x + Float64(Float64(0.125 * (x ^ 3.0)) + Float64(Float64(0.0546875 * (x ^ 5.0)) + Float64(0.0322265625 * (x ^ 7.0)))))
end

function tmp = code(x)
	tmp = sqrt((1.0 + x)) - sqrt((1.0 - x));
end
↓
function tmp = code(x)
	tmp = x + ((0.125 * (x ^ 3.0)) + ((0.0546875 * (x ^ 5.0)) + (0.0322265625 * (x ^ 7.0))));
end

code[x_] := N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(x + N[(N[(0.125 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0546875 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.0322265625 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\sqrt{1 + x} - \sqrt{1 - x}
↓
x + \left(0.125 \cdot {x}^{3} + \left(0.0546875 \cdot {x}^{5} + 0.0322265625 \cdot {x}^{7}\right)\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target0.0
Herbie0.1
\[\frac{2 \cdot x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

Derivation?

  1. Initial program 58.7

    \[\sqrt{1 + x} - \sqrt{1 - x} \]

  2. Simplified58.7

    \[\leadsto \color{blue}{\sqrt{x - -1} - \sqrt{1 - x}} \]
    Proof

    [Start]58.7

    \[ \sqrt{1 + x} - \sqrt{1 - x} \]

    rational.json-simplify-17 [=>]58.7

    \[ \sqrt{\color{blue}{x - -1}} - \sqrt{1 - x} \]

  3. Taylor expanded in x around 0 0.1

    \[\leadsto \color{blue}{0.0546875 \cdot {x}^{5} + \left(0.0322265625 \cdot {x}^{7} + \left(0.125 \cdot {x}^{3} + x\right)\right)} \]

  4. Simplified0.1

    \[\leadsto \color{blue}{x + \left(0.125 \cdot {x}^{3} + \left(0.0546875 \cdot {x}^{5} + 0.0322265625 \cdot {x}^{7}\right)\right)} \]
    Proof

    [Start]0.1

    \[ 0.0546875 \cdot {x}^{5} + \left(0.0322265625 \cdot {x}^{7} + \left(0.125 \cdot {x}^{3} + x\right)\right) \]

    rational.json-simplify-41 [<=]0.1

    \[ \color{blue}{\left(0.125 \cdot {x}^{3} + x\right) + \left(0.0546875 \cdot {x}^{5} + 0.0322265625 \cdot {x}^{7}\right)} \]

    rational.json-simplify-1 [=>]0.1

    \[ \color{blue}{\left(0.0546875 \cdot {x}^{5} + 0.0322265625 \cdot {x}^{7}\right) + \left(0.125 \cdot {x}^{3} + x\right)} \]

    rational.json-simplify-1 [=>]0.1

    \[ \left(0.0546875 \cdot {x}^{5} + 0.0322265625 \cdot {x}^{7}\right) + \color{blue}{\left(x + 0.125 \cdot {x}^{3}\right)} \]

    rational.json-simplify-41 [=>]0.1

    \[ \color{blue}{x + \left(0.125 \cdot {x}^{3} + \left(0.0546875 \cdot {x}^{5} + 0.0322265625 \cdot {x}^{7}\right)\right)} \]

  5. Final simplification0.1

    \[\leadsto x + \left(0.125 \cdot {x}^{3} + \left(0.0546875 \cdot {x}^{5} + 0.0322265625 \cdot {x}^{7}\right)\right) \]

Alternatives

Alternative 1
Error0.2
Cost13504
\[0.125 \cdot {x}^{3} + \left(x + 0.0546875 \cdot {x}^{5}\right) \]
Alternative 2
Error0.2
Cost6784
\[0.125 \cdot {x}^{3} + x \]
Alternative 3
Error0.5
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023077 
(FPCore (x)
  :name "bug333 (missed optimization)"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :herbie-target
  (/ (* 2.0 x) (+ (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

  (- (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))