sqrt E (should all be same)

Percentage Accurate: 54.8% → 100.0%

Time: 4.1s

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: 54.8% 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 50.3%

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

   1. unpow2 50.3%

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

   2. unpow2 50.3%

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

   3. hypot-define 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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 11.2% 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 50.3%

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

   1. unpow2 50.3%

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

   2. unpow2 50.3%

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

   3. hypot-define 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. Add Preprocessing

5. Taylor expanded in x around 0 55.0%

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

7. Simplified 10.6%

   \[\leadsto \color{blue}{x \cdot -2} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 9.3%

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

   2. sqrt-unprod 13.6%

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

   3. swap-sqr 13.6%

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

   4. metadata-eval 13.6%

      \[\leadsto \sqrt{\left(x \cdot x\right) \cdot \color{blue}{4}} \]

   5. metadata-eval 13.6%

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

   6. swap-sqr 13.6%

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

   7. sqrt-unprod 11.4%

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

   8. add-sqr-sqrt 12.4%

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

   9. add-log-exp 4.9%

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

   10. exp-lft-sqr 4.9%

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

   11. log-prod 4.9%

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

   12. add-log-exp 8.7%

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

   13. add-log-exp 12.4%

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

9. Applied egg-rr 12.4%

   \[\leadsto \color{blue}{x + x} \]
10. Add Preprocessing

Alternative 3: 5.4% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 8 \end{array} \]

FPCore

(FPCore (x) :precision binary64 8.0)

C

double code(double x) {
	return 8.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 8.0d0
end function

Java

public static double code(double x) {
	return 8.0;
}

Python

def code(x):
	return 8.0

Julia

function code(x)
	return 8.0
end

MATLAB

function tmp = code(x)
	tmp = 8.0;
end

Wolfram

code[x_] := 8.0

Derivation

1. Initial program 50.3%

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

   1. unpow2 50.3%

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

   2. unpow2 50.3%

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

   3. hypot-define 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. Add Preprocessing

5. Taylor expanded in x around 0 55.0%

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

7. Simplified 10.6%

   \[\leadsto \color{blue}{x \cdot -2} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 9.3%

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

   2. sqrt-unprod 13.6%

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

   3. swap-sqr 13.6%

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

   4. metadata-eval 13.6%

      \[\leadsto \sqrt{\left(x \cdot x\right) \cdot \color{blue}{4}} \]

   5. metadata-eval 13.6%

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

   6. swap-sqr 13.6%

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

   7. sqrt-unprod 11.4%

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

   8. add-sqr-sqrt 12.4%

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

   9. expm1-log1p-u 12.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 2\right)\right)} \]

   10. expm1-undefine 8.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot 2\right)} - 1} \]

   11. log1p-undefine 8.5%

       \[\leadsto e^{\color{blue}{\log \left(1 + x \cdot 2\right)}} - 1 \]

   12. rem-exp-log 8.6%

       \[\leadsto \color{blue}{\left(1 + x \cdot 2\right)} - 1 \]

   13. *-commutative 8.6%

       \[\leadsto \left(1 + \color{blue}{2 \cdot x}\right) - 1 \]

9. Applied egg-rr 8.6%

   \[\leadsto \color{blue}{\left(1 + 2 \cdot x\right) - 1} \]

10. Taylor expanded in x around inf 8.5%

    \[\leadsto \color{blue}{x \cdot \left(2 + \frac{1}{x}\right)} - 1 \]

12. Simplified 6.8%

    \[\leadsto \color{blue}{x \cdot \left(2 + \frac{-1}{x}\right)} - 1 \]

14. Applied egg-rr 5.3%

    \[\leadsto \color{blue}{8} \]
15. Add Preprocessing

Alternative 4: 5.4% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 50.3%

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

   1. unpow2 50.3%

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

   2. unpow2 50.3%

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

   3. hypot-define 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. Add Preprocessing

5. Taylor expanded in x around 0 55.0%

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

7. Simplified 10.6%

   \[\leadsto \color{blue}{x \cdot -2} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 9.3%

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

   2. sqrt-unprod 13.6%

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

   3. swap-sqr 13.6%

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

   4. metadata-eval 13.6%

      \[\leadsto \sqrt{\left(x \cdot x\right) \cdot \color{blue}{4}} \]

   5. metadata-eval 13.6%

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

   6. swap-sqr 13.6%

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

   7. sqrt-unprod 11.4%

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

   8. add-sqr-sqrt 12.4%

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

   9. expm1-log1p-u 12.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 2\right)\right)} \]

   10. expm1-undefine 8.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot 2\right)} - 1} \]

   11. log1p-undefine 8.5%

       \[\leadsto e^{\color{blue}{\log \left(1 + x \cdot 2\right)}} - 1 \]

   12. rem-exp-log 8.6%

       \[\leadsto \color{blue}{\left(1 + x \cdot 2\right)} - 1 \]

   13. *-commutative 8.6%

       \[\leadsto \left(1 + \color{blue}{2 \cdot x}\right) - 1 \]

9. Applied egg-rr 8.6%

   \[\leadsto \color{blue}{\left(1 + 2 \cdot x\right) - 1} \]

10. Taylor expanded in x around inf 8.5%

    \[\leadsto \color{blue}{x \cdot \left(2 + \frac{1}{x}\right)} - 1 \]

12. Simplified 6.8%

    \[\leadsto \color{blue}{x \cdot \left(2 + \frac{-1}{x}\right)} - 1 \]

14. Applied egg-rr 5.3%

    \[\leadsto \color{blue}{0.5} \]
15. Add Preprocessing

Alternative 5: 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 50.3%

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

   1. unpow2 50.3%

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

   2. unpow2 50.3%

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

   3. hypot-define 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. Add Preprocessing

5. Taylor expanded in x around 0 55.0%

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

7. Simplified 10.6%

   \[\leadsto \color{blue}{x \cdot -2} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 9.3%

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

   2. sqrt-unprod 13.6%

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

   3. pow1/2 13.6%

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

9. Applied egg-rr 3.9%

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

    1. pow-base-1 3.9%

       \[\leadsto {\log \color{blue}{1}}^{0.5} \]

    2. metadata-eval 3.9%

       \[\leadsto {\color{blue}{0}}^{0.5} \]

    3. metadata-eval 3.9%

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

11. Simplified 3.9%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))