sqrt E (should all be same)

Percentage Accurate: 54.0% → 100.0%

Time: 3.2s

Alternatives: 4

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: 54.0% 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 47.9%

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

   1. unpow2 47.9%

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

   2. unpow2 47.9%

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

   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: 8.3% accurate, 60.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.0) 0.0 (+ x -2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 0.0;
	} else {
		tmp = x + -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 0.0d0
    else
        tmp = x + (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 0.0;
	} else {
		tmp = x + -2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 0.0
	else:
		tmp = x + -2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = 0.0;
	else
		tmp = Float64(x + -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = 0.0;
	else
		tmp = x + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], 0.0, N[(x + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 50.2%

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

      1. unpow2 50.2%

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

      2. unpow2 50.2%

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

      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. Step-by-step derivation

      1. hypot-udef 50.2%

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

      2. pow1/2 50.2%

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

      3. distribute-lft-out 50.2%

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

      4. unpow-prod-down 37.9%

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

      5. pow1/2 37.9%

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

   5. Applied egg-rr 37.9%

      \[\leadsto \color{blue}{\sqrt{x} \cdot {\left(x + x\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. unpow1/2 37.8%

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

   7. Simplified 37.8%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x + x}} \]
   8. Step-by-step derivation

      1. sqrt-unprod 50.2%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

         \[\leadsto \sqrt{x \cdot \frac{\color{blue}{0}}{x - x}} \]

      4. +-inverses 0.0%

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

      5. associate-*r/ 0.0%

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

      6. distribute-lft-out-- 0.0%

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

      7. difference-of-squares 0.0%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

         \[\leadsto \sqrt{\left(x + x\right) \cdot \frac{\color{blue}{0}}{x - x}} \]

      10. +-inverses 0.0%

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

      11. flip-+ 13.4%

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

      12. sqrt-unprod 7.7%

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

      13. add-sqr-sqrt 9.2%

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

   9. Applied egg-rr 9.2%

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

   11. Simplified 4.4%

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

   if 2 < x

   1. Initial program 40.0%

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

      1. unpow2 40.0%

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

      2. unpow2 40.0%

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

      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. Step-by-step derivation

      1. hypot-udef 40.0%

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

      2. pow1/2 40.0%

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

      3. distribute-lft-out 40.0%

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

      4. unpow-prod-down 99.4%

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

      5. pow1/2 99.4%

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

   5. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot {\left(x + x\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. unpow1/2 99.4%

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

   7. Simplified 99.4%

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

   9. Applied egg-rr 20.3%

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

       1. add-exp-log 20.3%

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

       2. log1p-udef 20.3%

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

       3. expm1-udef 20.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + x\right)\right)} \]

       4. expm1-log1p-u 20.3%

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

       5. *-un-lft-identity 20.3%

          \[\leadsto \color{blue}{1 \cdot x} + x \]

       6. fma-def 20.3%

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

   11. Applied egg-rr 20.3%

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

   13. Simplified 20.3%

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

4. Final simplification 7.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x + -2\\ \end{array} \]

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 47.9%

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

   1. unpow2 47.9%

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

   2. unpow2 47.9%

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

   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. Step-by-step derivation

   1. hypot-udef 47.9%

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

   2. pow1/2 47.9%

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

   3. distribute-lft-out 47.9%

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

   4. unpow-prod-down 51.3%

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

   5. pow1/2 51.3%

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

5. Applied egg-rr 51.3%

   \[\leadsto \color{blue}{\sqrt{x} \cdot {\left(x + x\right)}^{0.5}} \]
6. Step-by-step derivation

   1. unpow1/2 51.3%

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

7. Simplified 51.3%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x + x}} \]
8. Step-by-step derivation

   1. sqrt-unprod 47.9%

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

   2. flip-+ 0.0%

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

   3. +-inverses 0.0%

      \[\leadsto \sqrt{x \cdot \frac{\color{blue}{0}}{x - x}} \]

   4. +-inverses 0.0%

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

   5. associate-*r/ 0.0%

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

   6. distribute-lft-out-- 0.0%

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

   7. difference-of-squares 0.0%

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

   8. associate-*r/ 0.0%

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

   9. +-inverses 0.0%

      \[\leadsto \sqrt{\left(x + x\right) \cdot \frac{\color{blue}{0}}{x - x}} \]

   10. +-inverses 0.0%

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

   11. flip-+ 13.0%

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

   12. sqrt-unprod 10.5%

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

   13. add-sqr-sqrt 11.6%

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

9. Applied egg-rr 11.6%

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

10. Final simplification 11.6%

    \[\leadsto x + x \]

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 47.9%

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

   1. unpow2 47.9%

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

   2. unpow2 47.9%

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

   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. Step-by-step derivation

   1. hypot-udef 47.9%

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

   2. pow1/2 47.9%

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

   3. distribute-lft-out 47.9%

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

   4. unpow-prod-down 51.3%

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

   5. pow1/2 51.3%

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

5. Applied egg-rr 51.3%

   \[\leadsto \color{blue}{\sqrt{x} \cdot {\left(x + x\right)}^{0.5}} \]
6. Step-by-step derivation

   1. unpow1/2 51.3%

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

7. Simplified 51.3%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x + x}} \]
8. Step-by-step derivation

   1. sqrt-unprod 47.9%

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

   2. flip-+ 0.0%

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

   3. +-inverses 0.0%

      \[\leadsto \sqrt{x \cdot \frac{\color{blue}{0}}{x - x}} \]

   4. +-inverses 0.0%

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

   5. associate-*r/ 0.0%

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

   6. distribute-lft-out-- 0.0%

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

   7. difference-of-squares 0.0%

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

   8. associate-*r/ 0.0%

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

   9. +-inverses 0.0%

      \[\leadsto \sqrt{\left(x + x\right) \cdot \frac{\color{blue}{0}}{x - x}} \]

   10. +-inverses 0.0%

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

   11. flip-+ 13.0%

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

   12. sqrt-unprod 10.5%

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

   13. add-sqr-sqrt 11.6%

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

9. Applied egg-rr 11.6%

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

11. Simplified 3.9%

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

12. Final simplification 3.9%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023215 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))