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

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 6

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-cube-cbrt 99.1%

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

   2. pow3 99.1%

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

5. Applied egg-rr 99.1%

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

   1. rem-cube-cbrt 99.8%

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

   2. *-commutative 99.8%

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

   3. add-sqr-sqrt 99.5%

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

   4. associate-*l* 99.5%

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

   5. pow1/2 99.5%

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

   6. sqrt-pow1 99.6%

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

   7. metadata-eval 99.6%

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

   8. pow1/2 99.6%

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

   9. sqrt-pow1 99.5%

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

   10. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

   1. associate-*r* 99.5%

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

   2. *-commutative 99.5%

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

9. Simplified 99.5%

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

    1. *-un-lft-identity 99.5%

       \[\leadsto 0.5 \cdot \left(x + y \cdot \color{blue}{\left(1 \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right)}\right) \]

    2. associate-*r* 99.5%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left(y \cdot 1\right) \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)}\right) \]

    3. metadata-eval 99.5%

       \[\leadsto 0.5 \cdot \left(x + \left(y \cdot \color{blue}{\frac{1}{1}}\right) \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

    4. div-inv 99.5%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{\frac{y}{1}} \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

    5. associate-/r/ 99.6%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{\frac{y}{\frac{1}{{z}^{0.25} \cdot {z}^{0.25}}}}\right) \]

    6. metadata-eval 99.6%

       \[\leadsto 0.5 \cdot \left(x + \frac{y}{\frac{\color{blue}{\sqrt{1}}}{{z}^{0.25} \cdot {z}^{0.25}}}\right) \]

    7. pow-prod-up 99.7%

       \[\leadsto 0.5 \cdot \left(x + \frac{y}{\frac{\sqrt{1}}{\color{blue}{{z}^{\left(0.25 + 0.25\right)}}}}\right) \]

    8. metadata-eval 99.7%

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

    9. pow1/2 99.7%

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

    10. sqrt-div 99.8%

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

    11. inv-pow 99.8%

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

    12. sqrt-pow1 99.8%

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

    13. 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: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -50:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{{z}^{-0.5}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (<= t_0 -50.0)
     (* 0.5 t_0)
     (if (<= t_0 2e-46) (* 0.5 x) (/ 0.5 (/ (pow z -0.5) y))))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if (t_0 <= -50.0) {
		tmp = 0.5 * t_0;
	} else if (t_0 <= 2e-46) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 / (pow(z, -0.5) / y);
	}
	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 <= (-50.0d0)) then
        tmp = 0.5d0 * t_0
    else if (t_0 <= 2d-46) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 / ((z ** (-0.5d0)) / y)
    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 <= -50.0) {
		tmp = 0.5 * t_0;
	} else if (t_0 <= 2e-46) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 / (Math.pow(z, -0.5) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	tmp = 0
	if t_0 <= -50.0:
		tmp = 0.5 * t_0
	elif t_0 <= 2e-46:
		tmp = 0.5 * x
	else:
		tmp = 0.5 / (math.pow(z, -0.5) / y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if (t_0 <= -50.0)
		tmp = Float64(0.5 * t_0);
	elseif (t_0 <= 2e-46)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 / Float64((z ^ -0.5) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	tmp = 0.0;
	if (t_0 <= -50.0)
		tmp = 0.5 * t_0;
	elseif (t_0 <= 2e-46)
		tmp = 0.5 * x;
	else
		tmp = 0.5 / ((z ^ -0.5) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50.0], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$0, 2e-46], N[(0.5 * x), $MachinePrecision], N[(0.5 / N[(N[Power[z, -0.5], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (sqrt.f64 z)) < -50

   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 78.5%

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

   if -50 < (*.f64 y (sqrt.f64 z)) < 2.00000000000000005e-46

   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 80.6%

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

   if 2.00000000000000005e-46 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.6%

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

      1. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. pow1/2 79.2%

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

      3. metadata-eval 79.2%

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

      4. pow-flip 79.2%

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

      5. associate-/r/ 79.2%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{{z}^{-0.5}}{y}}} \]

      6. *-un-lft-identity 79.2%

         \[\leadsto 0.5 \cdot \frac{1}{\frac{\color{blue}{1 \cdot {z}^{-0.5}}}{y}} \]

      7. associate-*l/ 79.3%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot {z}^{-0.5}}} \]

      8. un-div-inv 79.3%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{y} \cdot {z}^{-0.5}}} \]

      9. associate-*l/ 79.2%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot {z}^{-0.5}}{y}}} \]

      10. *-un-lft-identity 79.2%

          \[\leadsto \frac{0.5}{\frac{\color{blue}{{z}^{-0.5}}}{y}} \]

   6. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{{z}^{-0.5}}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -50:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{elif}\;y \cdot \sqrt{z} \leq 2 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{{z}^{-0.5}}{y}}\\ \end{array} \]

Alternative 3: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -50:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{{z}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (<= t_0 -50.0)
     (* 0.5 t_0)
     (if (<= t_0 2e-46) (* 0.5 x) (/ (* 0.5 y) (pow z -0.5))))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if (t_0 <= -50.0) {
		tmp = 0.5 * t_0;
	} else if (t_0 <= 2e-46) {
		tmp = 0.5 * x;
	} else {
		tmp = (0.5 * y) / pow(z, -0.5);
	}
	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 <= (-50.0d0)) then
        tmp = 0.5d0 * t_0
    else if (t_0 <= 2d-46) then
        tmp = 0.5d0 * x
    else
        tmp = (0.5d0 * y) / (z ** (-0.5d0))
    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 <= -50.0) {
		tmp = 0.5 * t_0;
	} else if (t_0 <= 2e-46) {
		tmp = 0.5 * x;
	} else {
		tmp = (0.5 * y) / Math.pow(z, -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	tmp = 0
	if t_0 <= -50.0:
		tmp = 0.5 * t_0
	elif t_0 <= 2e-46:
		tmp = 0.5 * x
	else:
		tmp = (0.5 * y) / math.pow(z, -0.5)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if (t_0 <= -50.0)
		tmp = Float64(0.5 * t_0);
	elseif (t_0 <= 2e-46)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(Float64(0.5 * y) / (z ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	tmp = 0.0;
	if (t_0 <= -50.0)
		tmp = 0.5 * t_0;
	elseif (t_0 <= 2e-46)
		tmp = 0.5 * x;
	else
		tmp = (0.5 * y) / (z ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50.0], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$0, 2e-46], N[(0.5 * x), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] / N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (sqrt.f64 z)) < -50

   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 78.5%

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

   if -50 < (*.f64 y (sqrt.f64 z)) < 2.00000000000000005e-46

   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 80.6%

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

   if 2.00000000000000005e-46 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.6%

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

      1. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. *-commutative 79.2%

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

      3. pow1/2 79.2%

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

      4. metadata-eval 79.2%

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

      5. pow-flip 79.2%

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

      6. associate-/r/ 79.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{{z}^{-0.5}}{y}}} \cdot 0.5 \]

      7. *-un-lft-identity 79.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot {z}^{-0.5}}}{y}} \cdot 0.5 \]

      8. associate-*l/ 79.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{y} \cdot {z}^{-0.5}}} \cdot 0.5 \]

      9. associate-*l/ 79.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot {z}^{-0.5}}{y}}} \cdot 0.5 \]

      10. *-un-lft-identity 79.2%

          \[\leadsto \frac{1}{\frac{\color{blue}{{z}^{-0.5}}}{y}} \cdot 0.5 \]

      11. clear-num 79.3%

          \[\leadsto \color{blue}{\frac{y}{{z}^{-0.5}}} \cdot 0.5 \]

      12. associate-*l/ 79.3%

          \[\leadsto \color{blue}{\frac{y \cdot 0.5}{{z}^{-0.5}}} \]

   6. Applied egg-rr 79.3%

      \[\leadsto \color{blue}{\frac{y \cdot 0.5}{{z}^{-0.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -50:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{elif}\;y \cdot \sqrt{z} \leq 2 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{{z}^{-0.5}}\\ \end{array} \]

Alternative 4: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -0.5 \lor \neg \left(t_0 \leq 7.1 \cdot 10^{-29}\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -0.5) (not (<= t_0 7.1e-29))) (* 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 <= -0.5) || !(t_0 <= 7.1e-29)) {
		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 <= (-0.5d0)) .or. (.not. (t_0 <= 7.1d-29))) 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 <= -0.5) || !(t_0 <= 7.1e-29)) {
		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 <= -0.5) or not (t_0 <= 7.1e-29):
		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 <= -0.5) || !(t_0 <= 7.1e-29))
		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 <= -0.5) || ~((t_0 <= 7.1e-29)))
		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, -0.5], N[Not[LessEqual[t$95$0, 7.1e-29]], $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)) < -0.5 or 7.10000000000000003e-29 < (*.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 78.8%

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

   if -0.5 < (*.f64 y (sqrt.f64 z)) < 7.10000000000000003e-29

   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 80.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -0.5 \lor \neg \left(y \cdot \sqrt{z} \leq 7.1 \cdot 10^{-29}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 5: 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 6: 50.2% 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 49.0%

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

5. Final simplification 49.0%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023258 
(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)))))