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

Percentage Accurate: 99.8% → 99.8%

Time: 8.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 52.5%

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

   2. sqrt-unprod 60.8%

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

   3. pow1/2 60.8%

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

   4. *-commutative 60.8%

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

   5. *-commutative 60.8%

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

   6. swap-sqr 58.8%

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

   7. add-sqr-sqrt 58.8%

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

5. Applied egg-rr 58.8%

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

   1. unpow1/2 58.8%

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

   2. *-commutative 58.8%

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

   3. associate-*l* 60.8%

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

7. Simplified 60.8%

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

   1. sqrt-prod 45.5%

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

   2. sqrt-prod 49.0%

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

   3. associate-*r* 49.0%

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

   4. add-sqr-sqrt 99.8%

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

   5. *-commutative 99.8%

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

   6. pow1/2 99.8%

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

   7. metadata-eval 99.8%

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

   8. pow-prod-up 99.6%

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

   9. associate-*r* 99.6%

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

9. Applied egg-rr 99.6%

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

    1. associate-*r* 99.6%

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

    2. remove-double-div 99.6%

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

    3. metadata-eval 99.6%

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

    4. pow-prod-up 99.7%

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

    5. metadata-eval 99.7%

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

    6. pow1/2 99.7%

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

    7. sqrt-div 99.7%

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

    8. associate-/r/ 99.8%

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

    9. clear-num 99.8%

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

    10. inv-pow 99.8%

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

    11. sqrt-pow1 99.8%

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

    12. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-12} \lor \neg \left(t_0 \leq 2 \cdot 10^{-34}\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -5e-12) (not (<= t_0 2e-34))) (* 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 <= -5e-12) || !(t_0 <= 2e-34)) {
		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 <= (-5d-12)) .or. (.not. (t_0 <= 2d-34))) 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 <= -5e-12) || !(t_0 <= 2e-34)) {
		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 <= -5e-12) or not (t_0 <= 2e-34):
		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 <= -5e-12) || !(t_0 <= 2e-34))
		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 <= -5e-12) || ~((t_0 <= 2e-34)))
		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, -5e-12], N[Not[LessEqual[t$95$0, 2e-34]], $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)) < -4.9999999999999997e-12 or 1.99999999999999986e-34 < (*.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 80.1%

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

   if -4.9999999999999997e-12 < (*.f64 y (sqrt.f64 z)) < 1.99999999999999986e-34

   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 79.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -5 \cdot 10^{-12} \lor \neg \left(y \cdot \sqrt{z} \leq 2 \cdot 10^{-34}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 4: 52.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{-y}{\frac{x}{z}}\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{x}{y \cdot \left(y \cdot z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.8e+193)
   (* 0.5 (* y (/ (- y) (/ x z))))
   (if (<= y 3.4e+190) (* 0.5 x) (* 0.5 (/ 1.0 (/ x (* y (* y z))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.8e+193) {
		tmp = 0.5 * (y * (-y / (x / z)));
	} else if (y <= 3.4e+190) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (1.0 / (x / (y * (y * z))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5.8d+193)) then
        tmp = 0.5d0 * (y * (-y / (x / z)))
    else if (y <= 3.4d+190) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * (1.0d0 / (x / (y * (y * z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.8e+193) {
		tmp = 0.5 * (y * (-y / (x / z)));
	} else if (y <= 3.4e+190) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (1.0 / (x / (y * (y * z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.8e+193:
		tmp = 0.5 * (y * (-y / (x / z)))
	elif y <= 3.4e+190:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * (1.0 / (x / (y * (y * z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.8e+193)
		tmp = Float64(0.5 * Float64(y * Float64(Float64(-y) / Float64(x / z))));
	elseif (y <= 3.4e+190)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(1.0 / Float64(x / Float64(y * Float64(y * z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.8e+193)
		tmp = 0.5 * (y * (-y / (x / z)));
	elseif (y <= 3.4e+190)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * (1.0 / (x / (y * (y * z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.8e+193], N[(0.5 * N[(y * N[((-y) / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e+190], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(1.0 / N[(x / N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.80000000000000026e193

   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-+ 23.3%

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

      2. clear-num 23.2%

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

      3. *-commutative 23.2%

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

      4. *-commutative 23.2%

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

      5. swap-sqr 3.1%

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

      6. add-sqr-sqrt 3.1%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in x around 0 89.2%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} + -1 \cdot \frac{x}{{y}^{2} \cdot z}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 89.2%

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

      2. unsub-neg 89.2%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} - \frac{x}{{y}^{2} \cdot z}}} \]

      3. associate-*l/ 89.1%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{\frac{1}{z}}}{y}} - \frac{x}{{y}^{2} \cdot z}} \]

      4. *-lft-identity 89.1%

         \[\leadsto 0.5 \cdot \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{z}}}}{y} - \frac{x}{{y}^{2} \cdot z}} \]

      5. unpow2 89.1%

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

      6. associate-/r* 89.1%

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

      7. associate-/r* 89.1%

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

   8. Simplified 89.1%

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

   9. Taylor expanded in z around 0 38.4%

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

       1. mul-1-neg 38.4%

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

       2. unpow2 38.4%

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

       3. associate-/l* 38.3%

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

   11. Simplified 38.3%

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

       1. *-un-lft-identity 38.3%

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

       2. times-frac 38.5%

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

   13. Applied egg-rr 38.5%

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

   if -5.80000000000000026e193 < y < 3.3999999999999999e190

   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 56.3%

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

   if 3.3999999999999999e190 < 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. Step-by-step derivation

      1. flip-+ 10.7%

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

      2. clear-num 10.7%

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

      3. *-commutative 10.7%

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

      4. *-commutative 10.7%

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

      5. swap-sqr 3.7%

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

      6. add-sqr-sqrt 3.7%

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

   5. Applied egg-rr 3.7%

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

   6. Taylor expanded in x around 0 99.5%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} + -1 \cdot \frac{x}{{y}^{2} \cdot z}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} - \frac{x}{{y}^{2} \cdot z}}} \]

      3. associate-*l/ 99.6%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{\frac{1}{z}}}{y}} - \frac{x}{{y}^{2} \cdot z}} \]

      4. *-lft-identity 99.6%

         \[\leadsto 0.5 \cdot \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{z}}}}{y} - \frac{x}{{y}^{2} \cdot z}} \]

      5. unpow2 99.6%

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

      6. associate-/r* 99.6%

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

      7. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in z around 0 16.7%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{-1 \cdot \frac{x}{{y}^{2} \cdot z}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 16.7%

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

       2. unpow2 16.7%

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

       3. *-commutative 16.7%

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

       4. times-frac 16.7%

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

       5. metadata-eval 16.7%

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

       6. distribute-neg-frac 16.7%

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

       7. distribute-lft-neg-out 16.7%

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

       8. *-inverses 16.7%

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

       9. associate-/r* 16.7%

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

       10. times-frac 16.8%

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

       11. associate-/r/ 16.7%

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

       12. associate-/r/ 16.8%

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

       13. *-commutative 16.8%

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

       14. *-inverses 16.8%

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

       15. metadata-eval 16.8%

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

       16. times-frac 16.8%

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

       17. *-lft-identity 16.8%

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

       18. associate-/l/ 16.8%

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

       19. /-rgt-identity 16.8%

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

   11. Simplified 16.8%

       \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{-\frac{\frac{x}{y}}{y \cdot z}}} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.1%

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

       2. sqrt-unprod 0.1%

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

       3. sqr-neg 0.1%

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

       4. sqrt-unprod 0.1%

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

       5. add-sqr-sqrt 21.1%

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

       6. neg-sub0 0.7%

          \[\leadsto 0.5 \cdot \frac{1}{-\color{blue}{\left(0 - \frac{\frac{x}{y}}{y \cdot z}\right)}} \]

   13. Applied egg-rr 0.7%

       \[\leadsto 0.5 \cdot \frac{1}{-\color{blue}{\left(0 - \frac{\frac{x}{y}}{y \cdot z}\right)}} \]
   14. Step-by-step derivation

       1. neg-sub0 21.1%

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

       2. associate-/l/ 21.0%

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

       3. *-commutative 21.0%

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

       4. associate-*r* 21.0%

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

       5. distribute-neg-frac 21.0%

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

       6. *-commutative 21.0%

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

       7. associate-*r* 21.0%

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

   15. Simplified 21.0%

       \[\leadsto 0.5 \cdot \frac{1}{-\color{blue}{\frac{-x}{y \cdot \left(y \cdot z\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{-y}{\frac{x}{z}}\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{x}{y \cdot \left(y \cdot z\right)}}\\ \end{array} \]

Alternative 5: 52.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+191}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{-y}{\frac{x}{z}}\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+188}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\frac{x}{y}}{y \cdot z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.15e+191)
   (* 0.5 (* y (/ (- y) (/ x z))))
   (if (<= y 2e+188) (* 0.5 x) (* 0.5 (/ 1.0 (/ (/ x y) (* y z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.15e+191) {
		tmp = 0.5 * (y * (-y / (x / z)));
	} else if (y <= 2e+188) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (1.0 / ((x / y) / (y * z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.15d+191)) then
        tmp = 0.5d0 * (y * (-y / (x / z)))
    else if (y <= 2d+188) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * (1.0d0 / ((x / y) / (y * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.15e+191) {
		tmp = 0.5 * (y * (-y / (x / z)));
	} else if (y <= 2e+188) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (1.0 / ((x / y) / (y * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.15e+191:
		tmp = 0.5 * (y * (-y / (x / z)))
	elif y <= 2e+188:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * (1.0 / ((x / y) / (y * z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.15e+191)
		tmp = Float64(0.5 * Float64(y * Float64(Float64(-y) / Float64(x / z))));
	elseif (y <= 2e+188)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(1.0 / Float64(Float64(x / y) / Float64(y * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.15e+191)
		tmp = 0.5 * (y * (-y / (x / z)));
	elseif (y <= 2e+188)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * (1.0 / ((x / y) / (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.15e+191], N[(0.5 * N[(y * N[((-y) / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+188], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(1.0 / N[(N[(x / y), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15e191

   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-+ 23.3%

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

      2. clear-num 23.2%

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

      3. *-commutative 23.2%

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

      4. *-commutative 23.2%

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

      5. swap-sqr 3.1%

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

      6. add-sqr-sqrt 3.1%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in x around 0 89.2%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} + -1 \cdot \frac{x}{{y}^{2} \cdot z}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 89.2%

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

      2. unsub-neg 89.2%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} - \frac{x}{{y}^{2} \cdot z}}} \]

      3. associate-*l/ 89.1%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{\frac{1}{z}}}{y}} - \frac{x}{{y}^{2} \cdot z}} \]

      4. *-lft-identity 89.1%

         \[\leadsto 0.5 \cdot \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{z}}}}{y} - \frac{x}{{y}^{2} \cdot z}} \]

      5. unpow2 89.1%

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

      6. associate-/r* 89.1%

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

      7. associate-/r* 89.1%

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

   8. Simplified 89.1%

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

   9. Taylor expanded in z around 0 38.4%

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

       1. mul-1-neg 38.4%

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

       2. unpow2 38.4%

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

       3. associate-/l* 38.3%

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

   11. Simplified 38.3%

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

       1. *-un-lft-identity 38.3%

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

       2. times-frac 38.5%

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

   13. Applied egg-rr 38.5%

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

   if -1.15e191 < y < 2e188

   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 56.3%

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

   if 2e188 < 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. Step-by-step derivation

      1. flip-+ 10.7%

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

      2. clear-num 10.7%

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

      3. *-commutative 10.7%

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

      4. *-commutative 10.7%

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

      5. swap-sqr 3.7%

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

      6. add-sqr-sqrt 3.7%

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

   5. Applied egg-rr 3.7%

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

   6. Taylor expanded in x around 0 99.5%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} + -1 \cdot \frac{x}{{y}^{2} \cdot z}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} - \frac{x}{{y}^{2} \cdot z}}} \]

      3. associate-*l/ 99.6%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{\frac{1}{z}}}{y}} - \frac{x}{{y}^{2} \cdot z}} \]

      4. *-lft-identity 99.6%

         \[\leadsto 0.5 \cdot \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{z}}}}{y} - \frac{x}{{y}^{2} \cdot z}} \]

      5. unpow2 99.6%

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

      6. associate-/r* 99.6%

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

      7. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in z around 0 16.7%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{-1 \cdot \frac{x}{{y}^{2} \cdot z}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 16.7%

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

       2. unpow2 16.7%

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

       3. *-commutative 16.7%

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

       4. times-frac 16.7%

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

       5. metadata-eval 16.7%

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

       6. distribute-neg-frac 16.7%

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

       7. distribute-lft-neg-out 16.7%

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

       8. *-inverses 16.7%

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

       9. associate-/r* 16.7%

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

       10. times-frac 16.8%

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

       11. associate-/r/ 16.7%

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

       12. associate-/r/ 16.8%

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

       13. *-commutative 16.8%

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

       14. *-inverses 16.8%

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

       15. metadata-eval 16.8%

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

       16. times-frac 16.8%

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

       17. *-lft-identity 16.8%

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

       18. associate-/l/ 16.8%

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

       19. /-rgt-identity 16.8%

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

   11. Simplified 16.8%

       \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{-\frac{\frac{x}{y}}{y \cdot z}}} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.1%

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

       2. sqrt-unprod 0.1%

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

       3. sqr-neg 0.1%

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

       4. sqrt-unprod 0.1%

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

       5. add-sqr-sqrt 21.1%

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

       6. neg-sub0 0.7%

          \[\leadsto 0.5 \cdot \frac{1}{-\color{blue}{\left(0 - \frac{\frac{x}{y}}{y \cdot z}\right)}} \]

   13. Applied egg-rr 0.7%

       \[\leadsto 0.5 \cdot \frac{1}{-\color{blue}{\left(0 - \frac{\frac{x}{y}}{y \cdot z}\right)}} \]
   14. Step-by-step derivation

       1. neg-sub0 21.1%

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

       2. distribute-neg-frac 21.1%

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

   15. Simplified 21.1%

       \[\leadsto 0.5 \cdot \frac{1}{-\color{blue}{\frac{-\frac{x}{y}}{y \cdot z}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+191}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{-y}{\frac{x}{z}}\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+188}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\frac{x}{y}}{y \cdot z}}\\ \end{array} \]

Alternative 6: 51.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+194}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \frac{y \cdot \left(-y\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.5e+194) (* 0.5 (* z (/ (* y (- y)) x))) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e+194) {
		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 <= (-2.5d+194)) 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 <= -2.5e+194) {
		tmp = 0.5 * (z * ((y * -y) / x));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.5e+194:
		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 <= -2.5e+194)
		tmp = Float64(0.5 * Float64(z * Float64(Float64(y * 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 <= -2.5e+194)
		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, -2.5e+194], N[(0.5 * N[(z * N[(N[(y * (-y)), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.49999999999999994e194

   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-+ 23.3%

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

      2. clear-num 23.2%

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

      3. *-commutative 23.2%

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

      4. *-commutative 23.2%

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

      5. swap-sqr 3.1%

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

      6. add-sqr-sqrt 3.1%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in x around 0 89.2%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} + -1 \cdot \frac{x}{{y}^{2} \cdot z}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 89.2%

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

      2. unsub-neg 89.2%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} - \frac{x}{{y}^{2} \cdot z}}} \]

      3. associate-*l/ 89.1%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{\frac{1}{z}}}{y}} - \frac{x}{{y}^{2} \cdot z}} \]

      4. *-lft-identity 89.1%

         \[\leadsto 0.5 \cdot \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{z}}}}{y} - \frac{x}{{y}^{2} \cdot z}} \]

      5. unpow2 89.1%

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

      6. associate-/r* 89.1%

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

      7. associate-/r* 89.1%

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

   8. Simplified 89.1%

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

   9. Taylor expanded in z around 0 38.4%

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

       1. mul-1-neg 38.4%

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

       2. unpow2 38.4%

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

       3. associate-/l* 38.3%

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

   11. Simplified 38.3%

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

       1. associate-/r/ 38.4%

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

   13. Applied egg-rr 38.4%

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

   if -2.49999999999999994e194 < 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 50.2%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+194}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \frac{y \cdot \left(-y\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 7: 51.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+195}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \left(z \cdot \left(-\frac{y}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.8e+195) (* 0.5 (* y (* z (- (/ y x))))) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.8e+195) {
		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 <= (-5.8d+195)) 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 <= -5.8e+195) {
		tmp = 0.5 * (y * (z * -(y / x)));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.8e+195:
		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 <= -5.8e+195)
		tmp = Float64(0.5 * Float64(y * Float64(z * Float64(-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 <= -5.8e+195)
		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, -5.8e+195], N[(0.5 * N[(y * N[(z * (-N[(y / x), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.79999999999999984e195

   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-+ 23.3%

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

      2. clear-num 23.2%

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

      3. *-commutative 23.2%

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

      4. *-commutative 23.2%

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

      5. swap-sqr 3.1%

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

      6. add-sqr-sqrt 3.1%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in x around 0 89.2%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} + -1 \cdot \frac{x}{{y}^{2} \cdot z}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 89.2%

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

      2. unsub-neg 89.2%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} - \frac{x}{{y}^{2} \cdot z}}} \]

      3. associate-*l/ 89.1%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{\frac{1}{z}}}{y}} - \frac{x}{{y}^{2} \cdot z}} \]

      4. *-lft-identity 89.1%

         \[\leadsto 0.5 \cdot \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{z}}}}{y} - \frac{x}{{y}^{2} \cdot z}} \]

      5. unpow2 89.1%

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

      6. associate-/r* 89.1%

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

      7. associate-/r* 89.1%

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

   8. Simplified 89.1%

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

   9. Taylor expanded in z around 0 38.4%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{-1 \cdot \frac{x}{{y}^{2} \cdot z}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 38.4%

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

       2. unpow2 38.4%

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

       3. *-commutative 38.4%

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

       4. times-frac 38.4%

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

       5. metadata-eval 38.4%

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

       6. distribute-neg-frac 38.4%

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

       7. distribute-lft-neg-out 38.4%

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

       8. *-inverses 38.4%

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

       9. associate-/r* 38.4%

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

       10. times-frac 38.4%

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

       11. associate-/r/ 38.4%

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

       12. associate-/r/ 38.4%

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

       13. *-commutative 38.4%

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

       14. *-inverses 38.4%

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

       15. metadata-eval 38.4%

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

       16. times-frac 38.4%

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

       17. *-lft-identity 38.4%

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

       18. associate-/l/ 38.4%

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

       19. /-rgt-identity 38.4%

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

   11. Simplified 38.4%

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

   12. Taylor expanded in x around 0 38.4%

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

       1. mul-1-neg 38.4%

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

       2. unpow2 38.4%

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

       3. associate-/l* 38.3%

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

       4. associate-*r/ 38.5%

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

       5. distribute-rgt-neg-in 38.5%

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

       6. associate-/r/ 38.4%

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

   14. Simplified 38.4%

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

   if -5.79999999999999984e195 < 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 50.2%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+195}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \left(z \cdot \left(-\frac{y}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 8: 51.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+192}:\\ \;\;\;\;0.5 \cdot \frac{\left(y \cdot z\right) \cdot \left(-y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.8e+192) (* 0.5 (/ (* (* y z) (- y)) x)) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.8e+192) {
		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 <= (-8.8d+192)) 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 <= -8.8e+192) {
		tmp = 0.5 * (((y * z) * -y) / x);
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.8e+192:
		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 <= -8.8e+192)
		tmp = Float64(0.5 * Float64(Float64(Float64(y * 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 <= -8.8e+192)
		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, -8.8e+192], N[(0.5 * N[(N[(N[(y * z), $MachinePrecision] * (-y)), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.8000000000000003e192

   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-+ 23.3%

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

      2. clear-num 23.2%

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

      3. *-commutative 23.2%

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

      4. *-commutative 23.2%

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

      5. swap-sqr 3.1%

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

      6. add-sqr-sqrt 3.1%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in x around 0 89.2%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} + -1 \cdot \frac{x}{{y}^{2} \cdot z}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 89.2%

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

      2. unsub-neg 89.2%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} - \frac{x}{{y}^{2} \cdot z}}} \]

      3. associate-*l/ 89.1%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{\frac{1}{z}}}{y}} - \frac{x}{{y}^{2} \cdot z}} \]

      4. *-lft-identity 89.1%

         \[\leadsto 0.5 \cdot \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{z}}}}{y} - \frac{x}{{y}^{2} \cdot z}} \]

      5. unpow2 89.1%

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

      6. associate-/r* 89.1%

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

      7. associate-/r* 89.1%

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

   8. Simplified 89.1%

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

   9. Taylor expanded in z around 0 38.4%

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

       1. associate-*r/ 38.4%

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

       2. unpow2 38.4%

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

       3. associate-*l* 38.4%

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

       4. *-commutative 38.4%

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

       5. neg-mul-1 38.4%

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

       6. distribute-rgt-neg-in 38.4%

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

       7. distribute-lft-neg-in 38.4%

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

       8. *-commutative 38.4%

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

   11. Simplified 38.4%

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

   if -8.8000000000000003e192 < 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 50.2%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+192}:\\ \;\;\;\;0.5 \cdot \frac{\left(y \cdot z\right) \cdot \left(-y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 9: 51.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{+191}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{-y}{\frac{x}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -9.6e+191) (* 0.5 (* y (/ (- y) (/ x z)))) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -9.6e+191) {
		tmp = 0.5 * (y * (-y / (x / z)));
	} 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 <= (-9.6d+191)) then
        tmp = 0.5d0 * (y * (-y / (x / z)))
    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 <= -9.6e+191) {
		tmp = 0.5 * (y * (-y / (x / z)));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -9.6e+191:
		tmp = 0.5 * (y * (-y / (x / z)))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -9.6e+191)
		tmp = Float64(0.5 * Float64(y * Float64(Float64(-y) / Float64(x / z))));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -9.6e+191)
		tmp = 0.5 * (y * (-y / (x / z)));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -9.6e+191], N[(0.5 * N[(y * N[((-y) / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.59999999999999943e191

   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-+ 23.3%

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

      2. clear-num 23.2%

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

      3. *-commutative 23.2%

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

      4. *-commutative 23.2%

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

      5. swap-sqr 3.1%

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

      6. add-sqr-sqrt 3.1%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in x around 0 89.2%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} + -1 \cdot \frac{x}{{y}^{2} \cdot z}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 89.2%

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

      2. unsub-neg 89.2%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{y} \cdot \sqrt{\frac{1}{z}} - \frac{x}{{y}^{2} \cdot z}}} \]

      3. associate-*l/ 89.1%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{\frac{1}{z}}}{y}} - \frac{x}{{y}^{2} \cdot z}} \]

      4. *-lft-identity 89.1%

         \[\leadsto 0.5 \cdot \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{z}}}}{y} - \frac{x}{{y}^{2} \cdot z}} \]

      5. unpow2 89.1%

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

      6. associate-/r* 89.1%

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

      7. associate-/r* 89.1%

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

   8. Simplified 89.1%

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

   9. Taylor expanded in z around 0 38.4%

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

       1. mul-1-neg 38.4%

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

       2. unpow2 38.4%

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

       3. associate-/l* 38.3%

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

   11. Simplified 38.3%

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

       1. *-un-lft-identity 38.3%

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

       2. times-frac 38.5%

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

   13. Applied egg-rr 38.5%

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

   if -9.59999999999999943e191 < 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 50.2%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{+191}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{-y}{\frac{x}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 10: 51.3% 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 46.6%

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

5. Final simplification 46.6%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023261 
(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)))))