sqrt E (should all be same)

Percentage Accurate: 54.1% → 100.0%

Time: 2.7s

Alternatives: 5

Speedup: 3.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{{x}^{2} + {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))

C

double code(double x) {
	return sqrt((pow(x, 2.0) + pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((x ** 2.0d0) + (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt((Math.pow(x, 2.0) + Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.sqrt((math.pow(x, 2.0) + math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64((x ^ 2.0) + (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((x ^ 2.0) + (x ^ 2.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 54.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{{x}^{2} + {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))

C

double code(double x) {
	return sqrt((pow(x, 2.0) + pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((x ** 2.0d0) + (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt((Math.pow(x, 2.0) + Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.sqrt((math.pow(x, 2.0) + math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64((x ^ 2.0) + (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((x ^ 2.0) + (x ^ 2.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (hypot x x))

C

double code(double x) {
	return hypot(x, x);
}

Java

public static double code(double x) {
	return Math.hypot(x, x);
}

Python

def code(x):
	return math.hypot(x, x)

Julia

function code(x)
	return hypot(x, x)
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[x ^ 2 + x ^ 2], $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
2. Step-by-step derivation

   1. unpow2 53.7%

      \[\leadsto \sqrt{\color{blue}{x \cdot x} + {x}^{2}} \]

   2. unpow2 53.7%

      \[\leadsto \sqrt{x \cdot x + \color{blue}{x \cdot x}} \]

   3. hypot-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 20.3% accurate, 50.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-x\right) - x\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -5e-310) (- (- x) x) (+ x x)))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -x - x;
	} else {
		tmp = x + x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = -x - x
    else
        tmp = x + x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -x - x;
	} else {
		tmp = x + x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = -x - x
	else:
		tmp = x + x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(-x) - x);
	else
		tmp = Float64(x + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -x - x;
	else
		tmp = x + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[((-x) - x), $MachinePrecision], N[(x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 52.6%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   2. Step-by-step derivation

      1. unpow2 52.6%

         \[\leadsto \sqrt{\color{blue}{x \cdot x} + {x}^{2}} \]

      2. unpow2 52.6%

         \[\leadsto \sqrt{x \cdot x + \color{blue}{x \cdot x}} \]

      3. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 99.3%

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

      1. mul-1-neg 99.3%

         \[\leadsto \color{blue}{-\sqrt{2} \cdot x} \]

      2. unpow1/2 99.3%

         \[\leadsto -\color{blue}{{2}^{0.5}} \cdot x \]

      3. *-commutative 99.3%

         \[\leadsto -\color{blue}{x \cdot {2}^{0.5}} \]

      4. distribute-rgt-neg-in 99.3%

         \[\leadsto \color{blue}{x \cdot \left(-{2}^{0.5}\right)} \]

      5. unpow1/2 99.3%

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

   6. Simplified 99.3%

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

   8. Applied egg-rr 20.3%

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

   if -4.999999999999985e-310 < x

   1. Initial program 55.2%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   2. Step-by-step derivation

      1. unpow2 55.2%

         \[\leadsto \sqrt{\color{blue}{x \cdot x} + {x}^{2}} \]

      2. unpow2 55.2%

         \[\leadsto \sqrt{x \cdot x + \color{blue}{x \cdot x}} \]

      3. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around -inf 2.3%

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

      1. mul-1-neg 2.3%

         \[\leadsto \color{blue}{-\sqrt{2} \cdot x} \]

      2. unpow1/2 2.3%

         \[\leadsto -\color{blue}{{2}^{0.5}} \cdot x \]

      3. *-commutative 2.3%

         \[\leadsto -\color{blue}{x \cdot {2}^{0.5}} \]

      4. distribute-rgt-neg-in 2.3%

         \[\leadsto \color{blue}{x \cdot \left(-{2}^{0.5}\right)} \]

      5. unpow1/2 2.3%

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

   6. Simplified 2.3%

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

   8. Applied egg-rr 21.0%

      \[\leadsto \color{blue}{x + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-x\right) - x\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \]

Alternative 3: 11.3% accurate, 101.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (+ x x))

C

double code(double x) {
	return x + x;
}

Fortran

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

Java

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

Python

def code(x):
	return x + x

Julia

function code(x)
	return Float64(x + x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.7%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
2. Step-by-step derivation

   1. unpow2 53.7%

      \[\leadsto \sqrt{\color{blue}{x \cdot x} + {x}^{2}} \]

   2. unpow2 53.7%

      \[\leadsto \sqrt{x \cdot x + \color{blue}{x \cdot x}} \]

   3. hypot-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around -inf 58.4%

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

   1. mul-1-neg 58.4%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot x} \]

   2. unpow1/2 58.4%

      \[\leadsto -\color{blue}{{2}^{0.5}} \cdot x \]

   3. *-commutative 58.4%

      \[\leadsto -\color{blue}{x \cdot {2}^{0.5}} \]

   4. distribute-rgt-neg-in 58.4%

      \[\leadsto \color{blue}{x \cdot \left(-{2}^{0.5}\right)} \]

   5. unpow1/2 58.4%

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

6. Simplified 58.4%

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

8. Applied egg-rr 10.1%

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

9. Final simplification 10.1%

   \[\leadsto x + x \]

Alternative 4: 3.8% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 53.7%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
2. Step-by-step derivation

   1. unpow2 53.7%

      \[\leadsto \sqrt{\color{blue}{x \cdot x} + {x}^{2}} \]

   2. unpow2 53.7%

      \[\leadsto \sqrt{x \cdot x + \color{blue}{x \cdot x}} \]

   3. hypot-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around -inf 58.4%

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

   1. mul-1-neg 58.4%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot x} \]

   2. unpow1/2 58.4%

      \[\leadsto -\color{blue}{{2}^{0.5}} \cdot x \]

   3. *-commutative 58.4%

      \[\leadsto -\color{blue}{x \cdot {2}^{0.5}} \]

   4. distribute-rgt-neg-in 58.4%

      \[\leadsto \color{blue}{x \cdot \left(-{2}^{0.5}\right)} \]

   5. unpow1/2 58.4%

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

6. Simplified 58.4%

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

8. Applied egg-rr 10.1%

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

10. Simplified 3.8%

    \[\leadsto \color{blue}{0} \]

11. Final simplification 3.8%

    \[\leadsto 0 \]

Alternative 5: 11.3% accurate, 305.0× 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 53.7%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
2. Step-by-step derivation

   1. unpow2 53.7%

      \[\leadsto \sqrt{\color{blue}{x \cdot x} + {x}^{2}} \]

   2. unpow2 53.7%

      \[\leadsto \sqrt{x \cdot x + \color{blue}{x \cdot x}} \]

   3. hypot-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around -inf 58.4%

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

   1. mul-1-neg 58.4%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot x} \]

   2. unpow1/2 58.4%

      \[\leadsto -\color{blue}{{2}^{0.5}} \cdot x \]

   3. *-commutative 58.4%

      \[\leadsto -\color{blue}{x \cdot {2}^{0.5}} \]

   4. distribute-rgt-neg-in 58.4%

      \[\leadsto \color{blue}{x \cdot \left(-{2}^{0.5}\right)} \]

   5. unpow1/2 58.4%

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

6. Simplified 58.4%

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

8. Applied egg-rr 10.1%

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

10. Simplified 9.8%

    \[\leadsto \color{blue}{0 + x} \]

11. Final simplification 9.8%

    \[\leadsto x \]

Reproduce

herbie shell --seed 2023193 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))