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

Percentage Accurate: 99.8% → 99.8%

Time: 10.3s

Alternatives: 8

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 74.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-85} \lor \neg \left(t_0 \leq 7 \cdot 10^{+75}\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x - \frac{y \cdot z}{\frac{x}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -1e-85) (not (<= t_0 7e+75)))
     (* 0.5 t_0)
     (* 0.5 (- x (/ (* y z) (/ x y)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if ((t_0 <= -1e-85) || !(t_0 <= 7e+75)) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * (x - ((y * z) / (x / 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 <= (-1d-85)) .or. (.not. (t_0 <= 7d+75))) then
        tmp = 0.5d0 * t_0
    else
        tmp = 0.5d0 * (x - ((y * z) / (x / 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 <= -1e-85) || !(t_0 <= 7e+75)) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * (x - ((y * z) / (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	tmp = 0
	if (t_0 <= -1e-85) or not (t_0 <= 7e+75):
		tmp = 0.5 * t_0
	else:
		tmp = 0.5 * (x - ((y * z) / (x / y)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if ((t_0 <= -1e-85) || !(t_0 <= 7e+75))
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(0.5 * Float64(x - Float64(Float64(y * z) / Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	tmp = 0.0;
	if ((t_0 <= -1e-85) || ~((t_0 <= 7e+75)))
		tmp = 0.5 * t_0;
	else
		tmp = 0.5 * (x - ((y * z) / (x / y)));
	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, -1e-85], N[Not[LessEqual[t$95$0, 7e+75]], $MachinePrecision]], N[(0.5 * t$95$0), $MachinePrecision], N[(0.5 * N[(x - N[(N[(y * z), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (sqrt.f64 z)) < -9.9999999999999998e-86 or 6.9999999999999997e75 < (*.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 81.8%

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

   if -9.9999999999999998e-86 < (*.f64 y (sqrt.f64 z)) < 6.9999999999999997e75

   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-+ 60.3%

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

      2. div-inv 60.1%

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

      3. *-commutative 60.1%

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

      4. *-commutative 60.1%

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

      5. swap-sqr 56.3%

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

      6. add-sqr-sqrt 56.3%

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

   5. Applied egg-rr 56.3%

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

   6. Taylor expanded in x around inf 42.8%

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

   7. Taylor expanded in z around 0 42.8%

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

      1. unpow2 42.8%

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

      2. associate-*r* 43.6%

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

   9. Simplified 43.6%

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

   10. Taylor expanded in x around 0 74.3%

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

       1. mul-1-neg 74.3%

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

       2. unpow2 74.3%

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

       3. *-commutative 74.3%

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

       4. rem-3cbrt-lft 74.3%

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

       5. unpow2 74.3%

          \[\leadsto 0.5 \cdot \left(x + \left(-\frac{\color{blue}{{\left(\sqrt[3]{z \cdot \left(y \cdot y\right)}\right)}^{2}} \cdot \sqrt[3]{z \cdot \left(y \cdot y\right)}}{x}\right)\right) \]

       6. associate-*r/ 74.3%

          \[\leadsto 0.5 \cdot \left(x + \left(-\color{blue}{{\left(\sqrt[3]{z \cdot \left(y \cdot y\right)}\right)}^{2} \cdot \frac{\sqrt[3]{z \cdot \left(y \cdot y\right)}}{x}}\right)\right) \]

       7. distribute-lft-neg-in 74.3%

          \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left(-{\left(\sqrt[3]{z \cdot \left(y \cdot y\right)}\right)}^{2}\right) \cdot \frac{\sqrt[3]{z \cdot \left(y \cdot y\right)}}{x}}\right) \]

       8. cancel-sign-sub-inv 74.3%

          \[\leadsto 0.5 \cdot \color{blue}{\left(x - {\left(\sqrt[3]{z \cdot \left(y \cdot y\right)}\right)}^{2} \cdot \frac{\sqrt[3]{z \cdot \left(y \cdot y\right)}}{x}\right)} \]

       9. associate-*r/ 74.3%

          \[\leadsto 0.5 \cdot \left(x - \color{blue}{\frac{{\left(\sqrt[3]{z \cdot \left(y \cdot y\right)}\right)}^{2} \cdot \sqrt[3]{z \cdot \left(y \cdot y\right)}}{x}}\right) \]

       10. unpow2 74.3%

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

       11. rem-3cbrt-lft 74.3%

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

       12. associate-*r/ 74.3%

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

       13. associate-/l* 75.0%

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

       14. associate-*r/ 75.9%

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

   12. Simplified 75.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -1 \cdot 10^{-85} \lor \neg \left(y \cdot \sqrt{z} \leq 7 \cdot 10^{+75}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x - \frac{y \cdot z}{\frac{x}{y}}\right)\\ \end{array} \]

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: 52.1% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.08 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x - z \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.08e+17) (* 0.5 x) (* 0.5 (- x (* z (/ (* y y) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.08e+17) {
		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 <= 1.08d+17) 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 <= 1.08e+17) {
		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 <= 1.08e+17:
		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 <= 1.08e+17)
		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 <= 1.08e+17)
		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, 1.08e+17], 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 < 1.08e17

   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 57.0%

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

   if 1.08e17 < 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-+ 44.6%

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

      2. div-inv 44.4%

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

      3. *-commutative 44.4%

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

      4. *-commutative 44.4%

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

      5. swap-sqr 41.3%

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

      6. add-sqr-sqrt 41.4%

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

   5. Applied egg-rr 41.4%

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

   6. Taylor expanded in x around inf 24.4%

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

   7. Taylor expanded in x around 0 42.1%

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

      1. mul-1-neg 42.1%

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

      2. unsub-neg 42.1%

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

      3. unpow2 42.1%

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

      4. associate-/l* 42.3%

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

      5. associate-/r/ 43.5%

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

   9. Simplified 43.5%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.08 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x - z \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \]

Alternative 5: 50.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+218}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \frac{-y}{\frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3e+218) (* 0.5 (* z (/ (- y) (/ x y)))) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3e+218) {
		tmp = 0.5 * (z * (-y / (x / y)));
	} 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 <= (-3d+218)) then
        tmp = 0.5d0 * (z * (-y / (x / y)))
    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 <= -3e+218) {
		tmp = 0.5 * (z * (-y / (x / y)));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3e+218:
		tmp = 0.5 * (z * (-y / (x / y)))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3e+218)
		tmp = Float64(0.5 * Float64(z * Float64(Float64(-y) / Float64(x / y))));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3e+218)
		tmp = 0.5 * (z * (-y / (x / y)));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3e+218], N[(0.5 * N[(z * N[((-y) / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.0000000000000001e218

   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. Step-by-step derivation

      1. flip-+ 10.8%

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

      2. div-inv 10.8%

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

      3. *-commutative 10.8%

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

      4. *-commutative 10.8%

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

      5. swap-sqr 2.5%

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

      6. add-sqr-sqrt 2.5%

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

   5. Applied egg-rr 2.5%

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

   6. Taylor expanded in x around inf 9.5%

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

   7. Taylor expanded in x around 0 27.1%

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

      1. associate-*r/ 27.1%

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

      2. mul-1-neg 27.1%

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

      3. unpow2 27.1%

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

      4. *-commutative 27.1%

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

      5. distribute-lft-neg-in 27.1%

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

   9. Simplified 27.1%

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

   10. Taylor expanded in z around 0 27.1%

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

       1. mul-1-neg 27.1%

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

       2. unpow2 27.1%

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

       3. *-commutative 27.1%

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

       4. associate-*r/ 27.1%

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

       5. distribute-rgt-neg-in 27.1%

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

       6. associate-/l* 27.1%

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

   12. Simplified 27.1%

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

   if -3.0000000000000001e218 < 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 50.7%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+218}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \frac{-y}{\frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 6: 50.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+218}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(z \cdot \left(-y\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e+218) (* 0.5 (/ (* y (* z (- y))) x)) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e+218) {
		tmp = 0.5 * ((y * (z * -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 <= (-3.2d+218)) then
        tmp = 0.5d0 * ((y * (z * -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 <= -3.2e+218) {
		tmp = 0.5 * ((y * (z * -y)) / x);
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.2e+218:
		tmp = 0.5 * ((y * (z * -y)) / x)
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.2e+218)
		tmp = Float64(0.5 * Float64(Float64(y * Float64(z * Float64(-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 <= -3.2e+218)
		tmp = 0.5 * ((y * (z * -y)) / x);
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.2e+218], N[(0.5 * N[(N[(y * N[(z * (-y)), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.19999999999999987e218

   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. Step-by-step derivation

      1. flip-+ 10.8%

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

      2. div-inv 10.8%

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

      3. *-commutative 10.8%

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

      4. *-commutative 10.8%

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

      5. swap-sqr 2.5%

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

      6. add-sqr-sqrt 2.5%

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

   5. Applied egg-rr 2.5%

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

   6. Taylor expanded in x around inf 9.5%

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

   7. Taylor expanded in x around 0 27.1%

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

      1. associate-*r/ 27.1%

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

      2. mul-1-neg 27.1%

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

      3. unpow2 27.1%

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

      4. *-commutative 27.1%

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

      5. distribute-lft-neg-in 27.1%

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

   9. Simplified 27.1%

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

   10. Taylor expanded in z around 0 27.1%

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

       1. mul-1-neg 27.1%

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

       2. unpow2 27.1%

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

       3. associate-*r* 27.1%

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

       4. *-commutative 27.1%

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

       5. distribute-rgt-neg-in 27.1%

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

       6. *-commutative 27.1%

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

   12. Simplified 27.1%

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

   if -3.19999999999999987e218 < 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 50.7%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+218}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(z \cdot \left(-y\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 7: 52.0% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

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) / (x / y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(x - N[(N[(y * z), $MachinePrecision] / N[(x / y), $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. flip-+ 51.1%

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

   2. div-inv 51.0%

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

   3. *-commutative 51.0%

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

   4. *-commutative 51.0%

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

   5. swap-sqr 45.2%

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

   6. add-sqr-sqrt 45.3%

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

5. Applied egg-rr 45.3%

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

6. Taylor expanded in x around inf 26.5%

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

7. Taylor expanded in z around 0 26.5%

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

   1. unpow2 26.5%

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

   2. associate-*r* 27.4%

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

9. Simplified 27.4%

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

10. Taylor expanded in x around 0 47.1%

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

    1. mul-1-neg 47.1%

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

    2. unpow2 47.1%

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

    3. *-commutative 47.1%

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

    4. rem-3cbrt-lft 47.1%

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

    5. unpow2 47.1%

       \[\leadsto 0.5 \cdot \left(x + \left(-\frac{\color{blue}{{\left(\sqrt[3]{z \cdot \left(y \cdot y\right)}\right)}^{2}} \cdot \sqrt[3]{z \cdot \left(y \cdot y\right)}}{x}\right)\right) \]

    6. associate-*r/ 47.1%

       \[\leadsto 0.5 \cdot \left(x + \left(-\color{blue}{{\left(\sqrt[3]{z \cdot \left(y \cdot y\right)}\right)}^{2} \cdot \frac{\sqrt[3]{z \cdot \left(y \cdot y\right)}}{x}}\right)\right) \]

    7. distribute-lft-neg-in 47.1%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left(-{\left(\sqrt[3]{z \cdot \left(y \cdot y\right)}\right)}^{2}\right) \cdot \frac{\sqrt[3]{z \cdot \left(y \cdot y\right)}}{x}}\right) \]

    8. cancel-sign-sub-inv 47.1%

       \[\leadsto 0.5 \cdot \color{blue}{\left(x - {\left(\sqrt[3]{z \cdot \left(y \cdot y\right)}\right)}^{2} \cdot \frac{\sqrt[3]{z \cdot \left(y \cdot y\right)}}{x}\right)} \]

    9. associate-*r/ 47.1%

       \[\leadsto 0.5 \cdot \left(x - \color{blue}{\frac{{\left(\sqrt[3]{z \cdot \left(y \cdot y\right)}\right)}^{2} \cdot \sqrt[3]{z \cdot \left(y \cdot y\right)}}{x}}\right) \]

    10. unpow2 47.1%

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

    11. rem-3cbrt-lft 47.1%

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

    12. associate-*r/ 47.8%

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

    13. associate-/l* 49.0%

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

    14. associate-*r/ 49.9%

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

12. Simplified 49.9%

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

13. Final simplification 49.9%

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

Alternative 8: 49.6% 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 46.9%

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

5. Final simplification 46.9%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023279 
(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)))))