sqrt(1+x) - sqrt(x)

Percentage Accurate: 99.9% → 99.9%
Time: 3.3s
Alternatives: 6
Speedup: 1.0×

Specification

?
\[-1.79 \cdot 10^{+308} \leq x \land x \leq 1\]
\[\begin{array}{l} \\ \sqrt{1 + x} - \sqrt{x} \end{array} \]
(FPCore (x) :precision binary64 (- (sqrt (+ 1.0 x)) (sqrt x)))
double code(double x) {
	return sqrt((1.0 + x)) - sqrt(x);
}

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

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

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

function code(x)
	return Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
end

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

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

\begin{array}{l}

\\
\sqrt{1 + x} - \sqrt{x}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 6 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 99.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sqrt{1 + x} - \sqrt{x} \end{array} \]
(FPCore (x) :precision binary64 (- (sqrt (+ 1.0 x)) (sqrt x)))
double code(double x) {
	return sqrt((1.0 + x)) - sqrt(x);
}

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

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

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

function code(x)
	return Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
end

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

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

\begin{array}{l}

\\
\sqrt{1 + x} - \sqrt{x}
\end{array}

Alternative 1: 99.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sqrt{1 + x} - \sqrt{x} \end{array} \]
(FPCore (x) :precision binary64 (- (sqrt (+ 1.0 x)) (sqrt x)))
double code(double x) {
	return sqrt((1.0 + x)) - sqrt(x);
}

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

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

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

function code(x)
	return Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
end

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

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

\begin{array}{l}

\\
\sqrt{1 + x} - \sqrt{x}
\end{array}
Derivation

  1. Initial program 100.0%

    \[\sqrt{1 + x} - \sqrt{x} \]
  2. Add Preprocessing
  3. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right) \end{array} \]
(FPCore (x) :precision binary64 (fma (fma -0.125 x 0.5) x (- 1.0 (sqrt x))))
double code(double x) {
	return fma(fma(-0.125, x, 0.5), x, (1.0 - sqrt(x)));
}

function code(x)
	return fma(fma(-0.125, x, 0.5), x, Float64(1.0 - sqrt(x)))
end

code[x_] := N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)
\end{array}
Derivation

  1. Initial program 100.0%

    \[\sqrt{1 + x} - \sqrt{x} \]
  2. Add Preprocessing

  3. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)\right) - \sqrt{x}} \]
  4. Step-by-step derivation

    1. +-commutativeN/A

      \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + 1\right)} - \sqrt{x} \]

    2. associate--l+N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + \left(1 - \sqrt{x}\right)} \]

    3. *-commutativeN/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) \cdot x} + \left(1 - \sqrt{x}\right) \]

    4. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{-1}{8} \cdot x, x, 1 - \sqrt{x}\right)} \]

    5. +-commutativeN/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot x + \frac{1}{2}}, x, 1 - \sqrt{x}\right) \]

    6. lower-fma.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right)}, x, 1 - \sqrt{x}\right) \]

    7. lower--.f64N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, \color{blue}{1 - \sqrt{x}}\right) \]

    8. lower-sqrt.f6498.7

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \color{blue}{\sqrt{x}}\right) \]

  5. Applied rewrites98.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)} \]
  6. Add Preprocessing

Alternative 3: 98.9% accurate, 1.2× speedup?

\[\begin{array}{l} \\ 1 - \left(\sqrt{x} - 0.5 \cdot x\right) \end{array} \]
(FPCore (x) :precision binary64 (- 1.0 (- (sqrt x) (* 0.5 x))))
double code(double x) {
	return 1.0 - (sqrt(x) - (0.5 * x));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (sqrt(x) - (0.5d0 * x))
end function

public static double code(double x) {
	return 1.0 - (Math.sqrt(x) - (0.5 * x));
}

def code(x):
	return 1.0 - (math.sqrt(x) - (0.5 * x))

function code(x)
	return Float64(1.0 - Float64(sqrt(x) - Float64(0.5 * x)))
end

function tmp = code(x)
	tmp = 1.0 - (sqrt(x) - (0.5 * x));
end

code[x_] := N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
1 - \left(\sqrt{x} - 0.5 \cdot x\right)
\end{array}
Derivation

  1. Initial program 100.0%

    \[\sqrt{1 + x} - \sqrt{x} \]
  2. Add Preprocessing

  3. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}} \]
  4. Step-by-step derivation

    1. +-commutativeN/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x} \]

    2. associate--l+N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)} \]

    3. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right)} \]

    4. lower--.f64N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \color{blue}{1 - \sqrt{x}}\right) \]

    5. lower-sqrt.f6498.4

      \[\leadsto \mathsf{fma}\left(0.5, x, 1 - \color{blue}{\sqrt{x}}\right) \]

  5. Applied rewrites98.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right)} \]
  6. Step-by-step derivation

    1. Applied rewrites98.4%

      \[\leadsto 1 - \color{blue}{\left(\sqrt{x} - 0.5 \cdot x\right)} \]
    2. Add Preprocessing

    Alternative 4: 98.9% accurate, 1.4× speedup?

    \[\begin{array}{l} \\ \mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right) \end{array} \]
    (FPCore (x) :precision binary64 (fma 0.5 x (- 1.0 (sqrt x))))
    double code(double x) {
	return fma(0.5, x, (1.0 - sqrt(x)));
}

    function code(x)
	return fma(0.5, x, Float64(1.0 - sqrt(x)))
end

    code[x_] := N[(0.5 * x + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

    \begin{array}{l}

\\
\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right)
\end{array}
    Derivation

    1. Initial program 100.0%

      \[\sqrt{1 + x} - \sqrt{x} \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}} \]
    4. Step-by-step derivation

      1. +-commutativeN/A

        \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x} \]

      2. associate--l+N/A

        \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)} \]

      3. lower-fma.f64N/A

        \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right)} \]

      4. lower--.f64N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \color{blue}{1 - \sqrt{x}}\right) \]

      5. lower-sqrt.f6498.4

        \[\leadsto \mathsf{fma}\left(0.5, x, 1 - \color{blue}{\sqrt{x}}\right) \]

    5. Applied rewrites98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right)} \]
    6. Add Preprocessing

    Alternative 5: 97.8% accurate, 1.9× speedup?

    \[\begin{array}{l} \\ 1 - \sqrt{x} \end{array} \]
    (FPCore (x) :precision binary64 (- 1.0 (sqrt x)))
    double code(double x) {
	return 1.0 - sqrt(x);
}

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

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

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

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

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

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

    \begin{array}{l}

\\
1 - \sqrt{x}
\end{array}
    Derivation

    1. Initial program 100.0%

      \[\sqrt{1 + x} - \sqrt{x} \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} - \sqrt{x} \]
    4. Step-by-step derivation

      1. Applied rewrites97.4%

        \[\leadsto \color{blue}{1} - \sqrt{x} \]
      2. Add Preprocessing

      Alternative 6: 2.7% accurate, 2.5× speedup?

      \[\begin{array}{l} \\ \left(x \cdot x\right) \cdot -0.125 \end{array} \]
      (FPCore (x) :precision binary64 (* (* x x) -0.125))
      double code(double x) {
	return (x * x) * -0.125;
}

      real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (-0.125d0)
end function

      public static double code(double x) {
	return (x * x) * -0.125;
}

      def code(x):
	return (x * x) * -0.125

      function code(x)
	return Float64(Float64(x * x) * -0.125)
end

      function tmp = code(x)
	tmp = (x * x) * -0.125;
end

      code[x_] := N[(N[(x * x), $MachinePrecision] * -0.125), $MachinePrecision]

      \begin{array}{l}

\\
\left(x \cdot x\right) \cdot -0.125
\end{array}
      Derivation

      1. Initial program 100.0%

        \[\sqrt{1 + x} - \sqrt{x} \]
      2. Add Preprocessing

      3. Taylor expanded in x around 0

        \[\leadsto \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)\right) - \sqrt{x}} \]
      4. Step-by-step derivation

        1. +-commutativeN/A

          \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + 1\right)} - \sqrt{x} \]

        2. associate--l+N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + \left(1 - \sqrt{x}\right)} \]

        3. *-commutativeN/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) \cdot x} + \left(1 - \sqrt{x}\right) \]

        4. lower-fma.f64N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{-1}{8} \cdot x, x, 1 - \sqrt{x}\right)} \]

        5. +-commutativeN/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot x + \frac{1}{2}}, x, 1 - \sqrt{x}\right) \]

        6. lower-fma.f64N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right)}, x, 1 - \sqrt{x}\right) \]

        7. lower--.f64N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, \color{blue}{1 - \sqrt{x}}\right) \]

        8. lower-sqrt.f6498.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \color{blue}{\sqrt{x}}\right) \]

      5. Applied rewrites98.7%

        \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)} \]

      6. Taylor expanded in x around inf

        \[\leadsto \frac{-1}{8} \cdot \color{blue}{{x}^{2}} \]
      7. Step-by-step derivation

        1. Applied rewrites2.7%

          \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{-0.125} \]
        2. Add Preprocessing

        Reproduce

        ?
        herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(1+x) - sqrt(x)"
  :precision binary64
  :pre (and (<= -1.79e+308 x) (<= x 1.0))
  (- (sqrt (+ 1.0 x)) (sqrt x)))