sqrt times

Percentage Accurate: 99.2% → 99.4%

Time: 3.8s

Alternatives: 3

Speedup: 29.3×

Specification

Math

\[\begin{array}{l} \\ \sqrt{x - 1} \cdot \sqrt{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x - 1} \cdot \sqrt{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \left(x + \frac{-0.125}{x}\right) + -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (+ x (/ -0.125 x)) -0.5))

C

double code(double x) {
	return (x + (-0.125 / x)) + -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + ((-0.125d0) / x)) + (-0.5d0)
end function

Java

public static double code(double x) {
	return (x + (-0.125 / x)) + -0.5;
}

Python

def code(x):
	return (x + (-0.125 / x)) + -0.5

Julia

function code(x)
	return Float64(Float64(x + Float64(-0.125 / x)) + -0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x + (-0.125 / x)) + -0.5;
end

Wolfram

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

Derivation

1. Initial program 99.2%

   \[\sqrt{x - 1} \cdot \sqrt{x} \]
2. Step-by-step derivation

   1. expm1-log1p-u 90.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x - 1} \cdot \sqrt{x}\right)\right)} \]

   2. expm1-udef 90.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{x - 1} \cdot \sqrt{x}\right)} - 1} \]

   3. log1p-udef 90.9%

      \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt{x - 1} \cdot \sqrt{x}\right)}} - 1 \]

   4. rem-exp-log 99.2%

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

   5. *-commutative 99.2%

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

   6. sqrt-unprod 50.9%

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

   7. sub-neg 50.9%

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

   8. metadata-eval 50.9%

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

3. Applied egg-rr 50.9%

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

4. Taylor expanded in x around inf 99.4%

   \[\leadsto \left(1 + \color{blue}{\left(x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)\right)}\right) - 1 \]
5. Step-by-step derivation

   1. associate-*r/ 99.4%

      \[\leadsto \left(1 + \left(x - \left(0.5 + \color{blue}{\frac{0.125 \cdot 1}{x}}\right)\right)\right) - 1 \]

   2. metadata-eval 99.4%

      \[\leadsto \left(1 + \left(x - \left(0.5 + \frac{\color{blue}{0.125}}{x}\right)\right)\right) - 1 \]

6. Simplified 99.4%

   \[\leadsto \left(1 + \color{blue}{\left(x - \left(0.5 + \frac{0.125}{x}\right)\right)}\right) - 1 \]

7. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{\left(\left(0.5 + x\right) - 0.125 \cdot \frac{1}{x}\right)} - 1 \]
8. Step-by-step derivation

   1. associate-*r/ 99.4%

      \[\leadsto \left(\left(0.5 + x\right) - \color{blue}{\frac{0.125 \cdot 1}{x}}\right) - 1 \]

   2. metadata-eval 99.4%

      \[\leadsto \left(\left(0.5 + x\right) - \frac{\color{blue}{0.125}}{x}\right) - 1 \]

   3. metadata-eval 99.4%

      \[\leadsto \left(\left(\color{blue}{\left(1 + -0.5\right)} + x\right) - \frac{0.125}{x}\right) - 1 \]

   4. associate-+r+ 99.4%

      \[\leadsto \left(\color{blue}{\left(1 + \left(-0.5 + x\right)\right)} - \frac{0.125}{x}\right) - 1 \]

   5. +-commutative 99.4%

      \[\leadsto \left(\left(1 + \color{blue}{\left(x + -0.5\right)}\right) - \frac{0.125}{x}\right) - 1 \]

   6. sub-neg 99.4%

      \[\leadsto \color{blue}{\left(\left(1 + \left(x + -0.5\right)\right) + \left(-\frac{0.125}{x}\right)\right)} - 1 \]

   7. +-commutative 99.4%

      \[\leadsto \color{blue}{\left(\left(-\frac{0.125}{x}\right) + \left(1 + \left(x + -0.5\right)\right)\right)} - 1 \]

   8. distribute-neg-frac 99.4%

      \[\leadsto \left(\color{blue}{\frac{-0.125}{x}} + \left(1 + \left(x + -0.5\right)\right)\right) - 1 \]

   9. metadata-eval 99.4%

      \[\leadsto \left(\frac{\color{blue}{-0.125}}{x} + \left(1 + \left(x + -0.5\right)\right)\right) - 1 \]

   10. +-commutative 99.4%

       \[\leadsto \left(\frac{-0.125}{x} + \color{blue}{\left(\left(x + -0.5\right) + 1\right)}\right) - 1 \]

   11. associate-+l+ 99.4%

       \[\leadsto \left(\frac{-0.125}{x} + \color{blue}{\left(x + \left(-0.5 + 1\right)\right)}\right) - 1 \]

   12. metadata-eval 99.4%

       \[\leadsto \left(\frac{-0.125}{x} + \left(x + \color{blue}{0.5}\right)\right) - 1 \]

9. Simplified 99.4%

   \[\leadsto \color{blue}{\left(\frac{-0.125}{x} + \left(x + 0.5\right)\right)} - 1 \]
10. Step-by-step derivation

    1. associate-+r+ 99.4%

       \[\leadsto \color{blue}{\left(\left(\frac{-0.125}{x} + x\right) + 0.5\right)} - 1 \]

    2. +-commutative 99.4%

       \[\leadsto \left(\color{blue}{\left(x + \frac{-0.125}{x}\right)} + 0.5\right) - 1 \]

    3. associate--l+ 99.4%

       \[\leadsto \color{blue}{\left(x + \frac{-0.125}{x}\right) + \left(0.5 - 1\right)} \]

    4. metadata-eval 99.4%

       \[\leadsto \left(x + \frac{-0.125}{x}\right) + \color{blue}{-0.5} \]

11. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\left(x + \frac{-0.125}{x}\right) + -0.5} \]

12. Final simplification 99.4%

    \[\leadsto \left(x + \frac{-0.125}{x}\right) + -0.5 \]

Alternative 2: 99.1% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ x - 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- x 0.5))

C

double code(double x) {
	return x - 0.5;
}

Fortran

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

Java

public static double code(double x) {
	return x - 0.5;
}

Python

def code(x):
	return x - 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf 99.1%

   \[\leadsto \color{blue}{x - 0.5} \]

3. Final simplification 99.1%

   \[\leadsto x - 0.5 \]

Alternative 3: 98.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{x} \]

3. Final simplification 98.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023334 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))