Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B

Percentage Accurate: 99.8% → 99.8%

Time: 6.8s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / 2.0d0) * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return (1.0 / 2.0) * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / 2.0) * (x + (y * sqrt(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / 2.0d0) * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return (1.0 / 2.0) * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / 2.0) * (x + (y * sqrt(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x + \frac{y}{{z}^{-0.5}}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 (+ x (/ y (pow z -0.5)))))

C

double code(double x, double y, double z) {
	return 0.5 * (x + (y / pow(z, -0.5)));
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	return 0.5 * (x + (y / Math.pow(z, -0.5)));
}

Python

def code(x, y, z):
	return 0.5 * (x + (y / math.pow(z, -0.5)))

Julia

function code(x, y, z)
	return Float64(0.5 * Float64(x + Float64(y / (z ^ -0.5))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * (x + (y / (z ^ -0.5)));
end

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(x + N[(y / N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
2. Step-by-step derivation

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{{\left(y \cdot \sqrt{z}\right)}^{1}}\right) \]

   2. metadata-eval 99.8%

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

   3. pow-prod-up 48.7%

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

   4. pow-prod-down 63.9%

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

   5. *-commutative 63.9%

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

   6. *-commutative 63.9%

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

   7. swap-sqr 60.5%

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

   8. add-sqr-sqrt 60.5%

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

   9. pow2 60.5%

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

5. Applied egg-rr 60.5%

   \[\leadsto 0.5 \cdot \left(x + \color{blue}{{\left(z \cdot {y}^{2}\right)}^{0.5}}\right) \]
6. Step-by-step derivation

   1. unpow1/2 60.5%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{\sqrt{z \cdot {y}^{2}}}\right) \]

7. Simplified 60.5%

   \[\leadsto 0.5 \cdot \left(x + \color{blue}{\sqrt{z \cdot {y}^{2}}}\right) \]
8. Step-by-step derivation

   1. sqrt-prod 64.0%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{\sqrt{z} \cdot \sqrt{{y}^{2}}}\right) \]

   2. pow1/2 64.0%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{{z}^{0.5}} \cdot \sqrt{{y}^{2}}\right) \]

   3. metadata-eval 64.0%

      \[\leadsto 0.5 \cdot \left(x + {z}^{\color{blue}{\left(0.25 + 0.25\right)}} \cdot \sqrt{{y}^{2}}\right) \]

   4. pow-prod-up 63.9%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)} \cdot \sqrt{{y}^{2}}\right) \]

   5. unpow2 63.9%

      \[\leadsto 0.5 \cdot \left(x + \left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot \sqrt{\color{blue}{y \cdot y}}\right) \]

   6. sqrt-prod 45.1%

      \[\leadsto 0.5 \cdot \left(x + \left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}\right) \]

   7. add-sqr-sqrt 99.6%

      \[\leadsto 0.5 \cdot \left(x + \left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot \color{blue}{y}\right) \]

   8. associate-*r* 99.6%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{{z}^{0.25} \cdot \left({z}^{0.25} \cdot y\right)}\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto 0.5 \cdot \left(x + \color{blue}{{z}^{0.25} \cdot \left({z}^{0.25} \cdot y\right)}\right) \]
10. Step-by-step derivation

    1. associate-*r* 99.6%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot y}\right) \]

    2. pow-prod-up 99.8%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{{z}^{\left(0.25 + 0.25\right)}} \cdot y\right) \]

    3. metadata-eval 99.8%

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

    4. pow1/2 99.8%

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

    5. remove-double-div 99.8%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{\frac{1}{\frac{1}{\sqrt{z}}}} \cdot y\right) \]

    6. metadata-eval 99.8%

       \[\leadsto 0.5 \cdot \left(x + \frac{1}{\frac{\color{blue}{\sqrt{1}}}{\sqrt{z}}} \cdot y\right) \]

    7. sqrt-div 99.8%

       \[\leadsto 0.5 \cdot \left(x + \frac{1}{\color{blue}{\sqrt{\frac{1}{z}}}} \cdot y\right) \]

    8. associate-*l/ 99.8%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{\frac{1 \cdot y}{\sqrt{\frac{1}{z}}}}\right) \]

    9. *-un-lft-identity 99.8%

       \[\leadsto 0.5 \cdot \left(x + \frac{\color{blue}{y}}{\sqrt{\frac{1}{z}}}\right) \]

    10. inv-pow 99.8%

        \[\leadsto 0.5 \cdot \left(x + \frac{y}{\sqrt{\color{blue}{{z}^{-1}}}}\right) \]

    11. sqrt-pow1 99.8%

        \[\leadsto 0.5 \cdot \left(x + \frac{y}{\color{blue}{{z}^{\left(\frac{-1}{2}\right)}}}\right) \]

    12. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

    \[\leadsto 0.5 \cdot \left(x + \frac{y}{{z}^{-0.5}}\right) \]

Alternative 2: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-40} \lor \neg \left(x \leq 1.6 \cdot 10^{-22}\right):\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.9e-40) (not (<= x 1.6e-22)))
   (* 0.5 x)
   (* 0.5 (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e-40) || !(x <= 1.6e-22)) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (y * sqrt(z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2.9d-40)) .or. (.not. (x <= 1.6d-22))) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * (y * sqrt(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e-40) || !(x <= 1.6e-22)) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (y * Math.sqrt(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.9e-40) or not (x <= 1.6e-22):
		tmp = 0.5 * x
	else:
		tmp = 0.5 * (y * math.sqrt(z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.9e-40) || !(x <= 1.6e-22))
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(y * sqrt(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.9e-40) || ~((x <= 1.6e-22)))
		tmp = 0.5 * x;
	else
		tmp = 0.5 * (y * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.9e-40], N[Not[LessEqual[x, 1.6e-22]], $MachinePrecision]], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8999999999999999e-40 or 1.59999999999999994e-22 < x

   1. Initial program 99.9%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 80.7%

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

   if -2.8999999999999999e-40 < x < 1.59999999999999994e-22

   1. Initial program 99.7%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 79.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y \cdot \sqrt{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-40} \lor \neg \left(x \leq 1.6 \cdot 10^{-22}\right):\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return 0.5 * (x + (y * sqrt(z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return 0.5 * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(0.5 * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * (x + (y * sqrt(z)));
end

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
2. Step-by-step derivation

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto 0.5 \cdot \left(x + y \cdot \sqrt{z}\right) \]

Alternative 4: 51.2% accurate, 36.3× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 x))

C

double code(double x, double y, double z) {
	return 0.5 * x;
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	return 0.5 * x;
}

Python

def code(x, y, z):
	return 0.5 * x

Julia

function code(x, y, z)
	return Float64(0.5 * x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * x;
end

Wolfram

code[x_, y_, z_] := N[(0.5 * x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
2. Step-by-step derivation

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around inf 52.1%

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

5. Final simplification 52.1%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))