2sqrt (example 3.1)

Percentage Accurate: 6.5% → 99.6%
Time: 15.1s
Alternatives: 5
Speedup: 2.0×

Specification

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

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

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

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

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

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

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

\begin{array}{l}

\\
\sqrt{x + 1} - \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 5 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: 6.5% accurate, 1.0× speedup?

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

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

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

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

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

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

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

\begin{array}{l}

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

Alternative 1: 99.6% accurate, 1.0× speedup?

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

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

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

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

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

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

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

\begin{array}{l}

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

  1. Initial program 5.2%

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

  4. Applied egg-rr0

    \[\leadsto expr\]

  5. Taylor expanded in x around 0 0

    \[\leadsto expr\]

  7. Simplified0

    \[\leadsto expr\]
  8. Add Preprocessing

Alternative 2: 98.3% accurate, 2.0× speedup?

\[\begin{array}{l} \\ {x}^{-0.5} \cdot 0.5 \end{array} \]
(FPCore (x) :precision binary64 (* (pow x -0.5) 0.5))
double code(double x) {
	return pow(x, -0.5) * 0.5;
}

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

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

def code(x):
	return math.pow(x, -0.5) * 0.5

function code(x)
	return Float64((x ^ -0.5) * 0.5)
end

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

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

\begin{array}{l}

\\
{x}^{-0.5} \cdot 0.5
\end{array}
Derivation

  1. Initial program 5.2%

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

  3. Taylor expanded in x around inf 0

    \[\leadsto expr\]

  5. Simplified0

    \[\leadsto expr\]

  7. Applied egg-rr0

    \[\leadsto expr\]
  8. Add Preprocessing

Alternative 3: 97.9% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \frac{0.5}{\sqrt{x}} \end{array} \]
(FPCore (x) :precision binary64 (/ 0.5 (sqrt x)))
double code(double x) {
	return 0.5 / sqrt(x);
}

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

public static double code(double x) {
	return 0.5 / Math.sqrt(x);
}

def code(x):
	return 0.5 / math.sqrt(x)

function code(x)
	return Float64(0.5 / sqrt(x))
end

function tmp = code(x)
	tmp = 0.5 / sqrt(x);
end

code[x_] := N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{0.5}{\sqrt{x}}
\end{array}
Derivation

  1. Initial program 5.2%

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

  3. Taylor expanded in x around inf 0

    \[\leadsto expr\]

  5. Simplified0

    \[\leadsto expr\]

  7. Applied egg-rr0

    \[\leadsto expr\]
  8. Add Preprocessing

Alternative 4: 1.6% accurate, 2.0× 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 5.2%

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

  3. Taylor expanded in x around 0 0

    \[\leadsto expr\]

  5. Simplified0

    \[\leadsto expr\]
  6. Add Preprocessing

Alternative 5: 1.6% accurate, 2.0× speedup?

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

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

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

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

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

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

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

\begin{array}{l}

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

  1. Initial program 5.2%

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

  3. Taylor expanded in x around 0 0

    \[\leadsto expr\]

  5. Simplified0

    \[\leadsto expr\]

  6. Taylor expanded in x around inf 0

    \[\leadsto expr\]

  8. Simplified0

    \[\leadsto expr\]
  9. Add Preprocessing

Developer target: 99.6% accurate, 1.0× speedup?

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

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

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

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

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

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

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

\begin{array}{l}

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

Reproduce

?
herbie shell --seed 2024110 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

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