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

Percentage Accurate: 99.8% → 99.8%

Time: 5.6s

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. add-sqr-sqrt 52.5%

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

   2. sqrt-unprod 66.1%

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

   3. pow1/2 66.1%

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

   4. *-commutative 66.1%

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

   5. *-commutative 66.1%

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

   6. swap-sqr 62.7%

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

   7. add-sqr-sqrt 62.8%

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

5. Applied egg-rr 62.8%

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

   1. unpow1/2 62.8%

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

   2. *-commutative 62.8%

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

   3. associate-*l* 66.1%

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

7. Simplified 66.1%

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

   1. associate-*r* 62.8%

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

   2. sqrt-prod 67.9%

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

   3. sqrt-prod 48.6%

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

   4. add-sqr-sqrt 99.8%

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

   5. add-cbrt-cube 80.0%

      \[\leadsto 0.5 \cdot \left(x + y \cdot \color{blue}{\sqrt[3]{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \sqrt{z}}}\right) \]

   6. add-sqr-sqrt 80.0%

      \[\leadsto 0.5 \cdot \left(x + y \cdot \sqrt[3]{\color{blue}{z} \cdot \sqrt{z}}\right) \]

   7. cbrt-prod 99.5%

      \[\leadsto 0.5 \cdot \left(x + y \cdot \color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{\sqrt{z}}\right)}\right) \]

   8. associate-*r* 99.5%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left(y \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{\sqrt{z}}}\right) \]

9. Applied egg-rr 99.5%

   \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left(y \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{\sqrt{z}}}\right) \]
10. Step-by-step derivation

    1. associate-*l* 99.5%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{\sqrt{z}}\right)}\right) \]

    2. cbrt-prod 80.0%

       \[\leadsto 0.5 \cdot \left(x + y \cdot \color{blue}{\sqrt[3]{z \cdot \sqrt{z}}}\right) \]

    3. add-sqr-sqrt 80.0%

       \[\leadsto 0.5 \cdot \left(x + y \cdot \sqrt[3]{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)} \cdot \sqrt{z}}\right) \]

    4. add-cbrt-cube 99.8%

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

    5. /-rgt-identity 99.8%

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

    6. associate-/r/ 99.8%

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

    7. pow1/2 99.8%

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

    8. pow-flip 99.9%

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

    9. metadata-eval 99.9%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+70} \lor \neg \left(t_0 \leq 2 \cdot 10^{-43}\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -2e+70) (not (<= t_0 2e-43))) (* 0.5 t_0) (* 0.5 x))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if ((t_0 <= -2e+70) || !(t_0 <= 2e-43)) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * x;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(z)
    if ((t_0 <= (-2d+70)) .or. (.not. (t_0 <= 2d-43))) then
        tmp = 0.5d0 * t_0
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.sqrt(z);
	double tmp;
	if ((t_0 <= -2e+70) || !(t_0 <= 2e-43)) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	tmp = 0
	if (t_0 <= -2e+70) or not (t_0 <= 2e-43):
		tmp = 0.5 * t_0
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if ((t_0 <= -2e+70) || !(t_0 <= 2e-43))
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	tmp = 0.0;
	if ((t_0 <= -2e+70) || ~((t_0 <= 2e-43)))
		tmp = 0.5 * t_0;
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+70], N[Not[LessEqual[t$95$0, 2e-43]], $MachinePrecision]], N[(0.5 * t$95$0), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (sqrt.f64 z)) < -2.00000000000000015e70 or 2.00000000000000015e-43 < (*.f64 y (sqrt.f64 z))

   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 83.4%

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

   if -2.00000000000000015e70 < (*.f64 y (sqrt.f64 z)) < 2.00000000000000015e-43

   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 83.2%

      \[\leadsto 0.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -2 \cdot 10^{+70} \lor \neg \left(y \cdot \sqrt{z} \leq 2 \cdot 10^{-43}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

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: 50.5% 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 54.2%

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

5. Final simplification 54.2%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023195 
(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)))))