sqrt E (should all be same)

Percentage Accurate: 53.7% → 100.0%

Time: 4.1s

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: 53.7% 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 50.9%

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

   1. unpow2 50.9%

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

   2. unpow2 50.9%

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

   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: 6.9% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x 2.0)))

C

double code(double x) {
	return x * (x * 2.0);
}

Fortran

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

Java

public static double code(double x) {
	return x * (x * 2.0);
}

Python

def code(x):
	return x * (x * 2.0)

Julia

function code(x)
	return Float64(x * Float64(x * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * 2.0);
end

Wolfram

code[x_] := N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

   1. unpow2 50.9%

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

   2. unpow2 50.9%

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

   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 50.9%

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

   2. flip-+ 0.0%

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

   3. difference-of-squares 0.0%

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

   4. +-inverses 0.0%

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

   5. metadata-eval 0.0%

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

   6. +-inverses 0.0%

      \[\leadsto \sqrt{\frac{\left(x \cdot x + x \cdot x\right) \cdot \log 1}{\color{blue}{0}}} \]

   7. metadata-eval 0.0%

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

   8. associate-*r/ 0.0%

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

   9. metadata-eval 0.0%

      \[\leadsto \sqrt{\left(x \cdot x + x \cdot x\right) \cdot \frac{\color{blue}{0}}{\log 1}} \]

   10. +-inverses 0.0%

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

   11. metadata-eval 0.0%

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

   12. +-inverses 0.0%

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

   13. flip-+ 6.6%

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

   14. sqrt-unprod 6.9%

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

   15. add-sqr-sqrt 6.9%

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

   16. distribute-lft-out 6.9%

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

   17. *-commutative 6.9%

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

   18. *-un-lft-identity 6.9%

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

   19. distribute-rgt-out 6.9%

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

   20. distribute-lft-out 6.9%

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

   21. metadata-eval 6.9%

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

5. Applied egg-rr 6.9%

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

6. Final simplification 6.9%

   \[\leadsto x \cdot \left(x \cdot 2\right) \]

Alternative 3: 11.5% accurate, 101.7× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x -2.0))

C

double code(double x) {
	return x * -2.0;
}

Fortran

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

Java

public static double code(double x) {
	return x * -2.0;
}

Python

def code(x):
	return x * -2.0

Julia

function code(x)
	return Float64(x * -2.0)
end

MATLAB

function tmp = code(x)
	tmp = x * -2.0;
end

Wolfram

code[x_] := N[(x * -2.0), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

   1. unpow2 50.9%

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

   2. unpow2 50.9%

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

   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. Taylor expanded in x around 0 52.1%

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

6. Simplified 11.0%

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

7. Final simplification 11.0%

   \[\leadsto x \cdot -2 \]

Alternative 4: 1.7% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -2.0d0
end function

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

Derivation

1. Initial program 50.9%

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

   1. unpow2 50.9%

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

   2. unpow2 50.9%

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

   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 50.9%

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

   2. flip-+ 0.0%

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

   3. difference-of-squares 0.0%

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

   4. +-inverses 0.0%

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

   5. metadata-eval 0.0%

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

   6. +-inverses 0.0%

      \[\leadsto \sqrt{\frac{\left(x \cdot x + x \cdot x\right) \cdot \log 1}{\color{blue}{0}}} \]

   7. metadata-eval 0.0%

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

   8. associate-*r/ 0.0%

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

   9. metadata-eval 0.0%

      \[\leadsto \sqrt{\left(x \cdot x + x \cdot x\right) \cdot \frac{\color{blue}{0}}{\log 1}} \]

   10. +-inverses 0.0%

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

   11. metadata-eval 0.0%

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

   12. +-inverses 0.0%

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

   13. flip-+ 6.6%

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

   14. sqrt-unprod 6.9%

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

   15. add-sqr-sqrt 6.9%

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

   16. pow2 6.9%

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

   17. pow2 6.9%

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

5. Applied egg-rr 6.9%

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

7. Simplified 1.7%

   \[\leadsto \color{blue}{-2} \]

8. Final simplification 1.7%

   \[\leadsto -2 \]

Reproduce

herbie shell --seed 2023332 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))