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

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

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

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

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 2: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|x\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.9e-11)
   (* 0.5 x)
   (if (<= x 4.9e-18) (* 0.5 (* y (sqrt z))) (* 0.5 (fabs x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e-11) {
		tmp = 0.5 * x;
	} else if (x <= 4.9e-18) {
		tmp = 0.5 * (y * sqrt(z));
	} else {
		tmp = 0.5 * fabs(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) :: tmp
    if (x <= (-1.9d-11)) then
        tmp = 0.5d0 * x
    else if (x <= 4.9d-18) then
        tmp = 0.5d0 * (y * sqrt(z))
    else
        tmp = 0.5d0 * abs(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e-11) {
		tmp = 0.5 * x;
	} else if (x <= 4.9e-18) {
		tmp = 0.5 * (y * Math.sqrt(z));
	} else {
		tmp = 0.5 * Math.abs(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.9e-11:
		tmp = 0.5 * x
	elif x <= 4.9e-18:
		tmp = 0.5 * (y * math.sqrt(z))
	else:
		tmp = 0.5 * math.fabs(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.9e-11)
		tmp = Float64(0.5 * x);
	elseif (x <= 4.9e-18)
		tmp = Float64(0.5 * Float64(y * sqrt(z)));
	else
		tmp = Float64(0.5 * abs(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.9e-11)
		tmp = 0.5 * x;
	elseif (x <= 4.9e-18)
		tmp = 0.5 * (y * sqrt(z));
	else
		tmp = 0.5 * abs(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.9e-11], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 4.9e-18], N[(0.5 * N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Abs[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8999999999999999e-11

   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 76.0%

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

   if -1.8999999999999999e-11 < x < 4.9000000000000001e-18

   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 4.9000000000000001e-18 < 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. Step-by-step derivation

      1. flip-+ 48.0%

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

      2. div-inv 47.9%

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

      3. *-commutative 47.9%

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

      4. *-commutative 47.9%

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

      5. swap-sqr 45.9%

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

      6. add-sqr-sqrt 45.9%

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

   5. Applied egg-rr 45.9%

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

   6. Taylor expanded in x around inf 44.5%

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

   7. Taylor expanded in x around inf 42.6%

      \[\leadsto 0.5 \cdot \left(\color{blue}{{x}^{2}} \cdot \frac{1}{x}\right) \]
   8. Step-by-step derivation

      1. unpow2 42.6%

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

   9. Simplified 42.6%

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

       1. pow2 42.6%

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

       2. inv-pow 42.6%

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

       3. pow-prod-up 73.6%

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

       4. metadata-eval 73.6%

          \[\leadsto 0.5 \cdot {x}^{\color{blue}{1}} \]

       5. pow1 73.6%

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

       6. add-sqr-sqrt 73.0%

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

       7. sqrt-prod 42.7%

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

       8. rem-sqrt-square 73.6%

          \[\leadsto 0.5 \cdot \color{blue}{\left|x\right|} \]

   11. Applied egg-rr 73.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|x\right|\\ \end{array} \]

Alternative 3: 53.1% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x - z \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.9e-55) (* 0.5 x) (* 0.5 (- x (* z (/ (* y y) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.9e-55) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (x - (z * ((y * y) / 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) :: tmp
    if (z <= 2.9d-55) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * (x - (z * ((y * y) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.9e-55) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (x - (z * ((y * y) / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.9e-55:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * (x - (z * ((y * y) / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.9e-55)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(x - Float64(z * Float64(Float64(y * y) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.9e-55)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * (x - (z * ((y * y) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.9e-55], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(x - N[(z * N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.9e-55

   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 57.3%

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

   if 2.9e-55 < z

   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. flip-+ 45.3%

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

      2. div-inv 45.2%

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

      3. *-commutative 45.2%

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

      4. *-commutative 45.2%

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

      5. swap-sqr 41.1%

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

      6. add-sqr-sqrt 41.1%

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

   5. Applied egg-rr 41.1%

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

   6. Taylor expanded in x around inf 28.5%

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

   7. Taylor expanded in x around 0 41.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-1 \cdot \frac{{y}^{2} \cdot z}{x} + x\right)} \]
   8. Step-by-step derivation

      1. +-commutative 41.5%

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

      2. mul-1-neg 41.5%

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

      3. unsub-neg 41.5%

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

      4. unpow2 41.5%

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

      5. associate-/l* 46.6%

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

      6. associate-/r/ 47.0%

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

   9. Simplified 47.0%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x - z \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \]

Alternative 4: 51.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+218}:\\ \;\;\;\;0.5 \cdot \left(\left(-y\right) \cdot \frac{y \cdot z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4e+218) (* 0.5 (* (- y) (/ (* y z) x))) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+218) {
		tmp = 0.5 * (-y * ((y * z) / x));
	} 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) :: tmp
    if (y <= (-2.4d+218)) then
        tmp = 0.5d0 * (-y * ((y * z) / x))
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+218) {
		tmp = 0.5 * (-y * ((y * z) / x));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.4e+218:
		tmp = 0.5 * (-y * ((y * z) / x))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.4e+218)
		tmp = Float64(0.5 * Float64(Float64(-y) * Float64(Float64(y * z) / x)));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.4e+218)
		tmp = 0.5 * (-y * ((y * z) / x));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.4e+218], N[(0.5 * N[((-y) * N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999981e218

   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. Step-by-step derivation

      1. flip-+ 16.9%

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

      2. div-inv 16.9%

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

      3. *-commutative 16.9%

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

      4. *-commutative 16.9%

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

      5. swap-sqr 3.1%

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

      6. add-sqr-sqrt 3.1%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in x around inf 25.7%

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

   7. Taylor expanded in x around 0 26.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-1 \cdot \frac{{y}^{2} \cdot z}{x}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 26.2%

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

      2. mul-1-neg 26.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{-{y}^{2} \cdot z}}{x} \]

      3. unpow2 26.2%

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

      4. *-commutative 26.2%

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

      5. distribute-lft-neg-in 26.2%

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

   9. Simplified 26.2%

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

   10. Taylor expanded in z around 0 26.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-1 \cdot \frac{{y}^{2} \cdot z}{x}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 26.2%

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

       2. unpow2 26.2%

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

       3. associate-*r* 26.2%

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

       4. *-rgt-identity 26.2%

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

       5. associate-*r/ 26.2%

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

       6. associate-*l* 26.2%

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

       7. distribute-lft-neg-in 26.2%

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

       8. associate-*r/ 26.2%

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

       9. *-rgt-identity 26.2%

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

   12. Simplified 26.2%

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

   if -2.39999999999999981e218 < y

   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 51.6%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+218}:\\ \;\;\;\;0.5 \cdot \left(\left(-y\right) \cdot \frac{y \cdot z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 5: 51.3% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+218}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot \left(y \cdot y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.95e+218) (* -0.5 (/ (* z (* y y)) x)) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.95e+218) {
		tmp = -0.5 * ((z * (y * y)) / x);
	} 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) :: tmp
    if (y <= (-1.95d+218)) then
        tmp = (-0.5d0) * ((z * (y * y)) / x)
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.95e+218) {
		tmp = -0.5 * ((z * (y * y)) / x);
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.95e+218:
		tmp = -0.5 * ((z * (y * y)) / x)
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.95e+218)
		tmp = Float64(-0.5 * Float64(Float64(z * Float64(y * y)) / x));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.95e+218)
		tmp = -0.5 * ((z * (y * y)) / x);
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.95e+218], N[(-0.5 * N[(N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9500000000000001e218

   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. Step-by-step derivation

      1. flip-+ 16.9%

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

      2. div-inv 16.9%

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

      3. *-commutative 16.9%

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

      4. *-commutative 16.9%

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

      5. swap-sqr 3.1%

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

      6. add-sqr-sqrt 3.1%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in x around inf 25.7%

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

   7. Taylor expanded in x around 0 26.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-1 \cdot \frac{{y}^{2} \cdot z}{x}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 26.2%

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

      2. mul-1-neg 26.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{-{y}^{2} \cdot z}}{x} \]

      3. unpow2 26.2%

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

      4. *-commutative 26.2%

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

      5. distribute-lft-neg-in 26.2%

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

   9. Simplified 26.2%

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

   10. Taylor expanded in z around 0 26.2%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{{y}^{2} \cdot z}{x}} \]
   11. Step-by-step derivation

       1. unpow2 26.2%

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

       2. *-commutative 26.2%

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

   12. Simplified 26.2%

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

   if -1.9500000000000001e218 < y

   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 51.6%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+218}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot \left(y \cdot y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 6: 50.7% 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 47.9%

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

5. Final simplification 47.9%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023224 
(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)))))