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

Percentage Accurate: 99.8% → 99.8%

Time: 6.6s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.5e-11)
   (* 0.5 x)
   (if (<= x 2.1e+62) (* 0.5 (* y (sqrt z))) (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e-11) {
		tmp = 0.5 * x;
	} else if (x <= 2.1e+62) {
		tmp = 0.5 * (y * sqrt(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 (x <= (-2.5d-11)) then
        tmp = 0.5d0 * x
    else if (x <= 2.1d+62) then
        tmp = 0.5d0 * (y * sqrt(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 (x <= -2.5e-11) {
		tmp = 0.5 * x;
	} else if (x <= 2.1e+62) {
		tmp = 0.5 * (y * Math.sqrt(z));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.5e-11:
		tmp = 0.5 * x
	elif x <= 2.1e+62:
		tmp = 0.5 * (y * math.sqrt(z))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.5e-11)
		tmp = Float64(0.5 * x);
	elseif (x <= 2.1e+62)
		tmp = Float64(0.5 * Float64(y * sqrt(z)));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.5e-11)
		tmp = 0.5 * x;
	elseif (x <= 2.1e+62)
		tmp = 0.5 * (y * sqrt(z));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.5e-11], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 2.1e+62], N[(0.5 * N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.50000000000000009e-11 or 2.1e62 < x

   1. Initial program 99.9%

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

      1. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 83.0%

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

   if -2.50000000000000009e-11 < x < 2.1e62

   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 82.2%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3: 50.8% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+192}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \left(y \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.5e+192) (* 0.5 (* y (* y (/ z x)))) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.5e+192) {
		tmp = 0.5 * (y * (y * (z / x)));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-6.5d+192)) then
        tmp = 0.5d0 * (y * (y * (z / x)))
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.5e+192) {
		tmp = 0.5 * (y * (y * (z / x)));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6.5e+192:
		tmp = 0.5 * (y * (y * (z / x)))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.5e+192)
		tmp = Float64(0.5 * Float64(y * Float64(y * Float64(z / x))));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -6.5e+192)
		tmp = 0.5 * (y * (y * (z / x)));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.5e+192], N[(0.5 * N[(y * N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.50000000000000033e192

   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-+ 16.4%

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

      2. div-sub 16.4%

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

      3. *-commutative 16.4%

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

      4. *-commutative 16.4%

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

      5. swap-sqr 3.2%

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

      6. add-sqr-sqrt 3.2%

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

   5. Applied egg-rr 3.2%

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

      1. +-rgt-identity 3.2%

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

      2. div-sub 3.2%

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

      3. +-rgt-identity 3.2%

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

      4. *-commutative 3.2%

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

      5. associate-*l* 16.4%

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

   7. Simplified 16.4%

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

   8. Taylor expanded in x around 0 4.1%

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

      1. mul-1-neg 4.1%

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

      2. unpow2 4.1%

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

      3. associate-*r* 16.8%

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

      4. *-commutative 16.8%

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

      5. distribute-rgt-neg-out 16.8%

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

      6. associate-*l* 16.8%

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

   10. Simplified 16.8%

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

   11. Taylor expanded in y around 0 15.1%

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

       1. associate-*r/ 15.1%

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

       2. mul-1-neg 15.1%

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

       3. unpow2 15.1%

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

       4. *-commutative 15.1%

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

       5. distribute-rgt-neg-in 15.1%

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

       6. associate-/l* 15.1%

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

   13. Simplified 15.1%

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

       1. associate-/r/ 15.1%

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

       2. distribute-rgt-neg-in 15.1%

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

       3. associate-*r* 15.3%

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

       4. add-sqr-sqrt 15.3%

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

       5. sqrt-unprod 15.1%

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

       6. sqr-neg 15.1%

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

       7. sqrt-prod 0.0%

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

       8. add-sqr-sqrt 22.6%

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

   15. Applied egg-rr 22.6%

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

   if -6.50000000000000033e192 < 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 52.5%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+192}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \left(y \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 4: 50.9% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \frac{y}{\frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.85e+193) (* 0.5 (* z (/ y (/ x y)))) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.85e+193) {
		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 <= (-1.85d+193)) 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 <= -1.85e+193) {
		tmp = 0.5 * (z * (y / (x / y)));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.85e+193:
		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 <= -1.85e+193)
		tmp = Float64(0.5 * Float64(z * 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 <= -1.85e+193)
		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, -1.85e+193], 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 < -1.8500000000000001e193

   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-+ 16.4%

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

      2. div-sub 16.4%

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

      3. *-commutative 16.4%

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

      4. *-commutative 16.4%

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

      5. swap-sqr 3.2%

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

      6. add-sqr-sqrt 3.2%

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

   5. Applied egg-rr 3.2%

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

      1. +-rgt-identity 3.2%

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

      2. div-sub 3.2%

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

      3. +-rgt-identity 3.2%

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

      4. *-commutative 3.2%

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

      5. associate-*l* 16.4%

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

   7. Simplified 16.4%

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

   8. Taylor expanded in x around 0 4.1%

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

      1. mul-1-neg 4.1%

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

      2. unpow2 4.1%

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

      3. associate-*r* 16.8%

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

      4. *-commutative 16.8%

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

      5. distribute-rgt-neg-out 16.8%

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

      6. associate-*l* 16.8%

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

   10. Simplified 16.8%

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

   11. Taylor expanded in y around 0 15.1%

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

       1. associate-*r/ 15.1%

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

       2. mul-1-neg 15.1%

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

       3. unpow2 15.1%

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

       4. *-commutative 15.1%

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

       5. distribute-rgt-neg-in 15.1%

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

       6. associate-/l* 15.1%

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

   13. Simplified 15.1%

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

       1. clear-num 15.1%

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

       2. associate-/r/ 15.1%

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

       3. clear-num 15.1%

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

       4. distribute-lft-neg-in 15.1%

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

       5. associate-/l* 15.3%

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

       6. add-sqr-sqrt 15.3%

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

       7. sqrt-unprod 15.1%

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

       8. sqr-neg 15.1%

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

       9. sqrt-prod 0.0%

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

       10. add-sqr-sqrt 22.8%

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

   15. Applied egg-rr 22.8%

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

   if -1.8500000000000001e193 < 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 52.5%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \frac{y}{\frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 5: 50.9% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{y \cdot z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.85e+193) (* 0.5 (* y (/ (* y z) x))) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.85e+193) {
		tmp = 0.5 * (y * ((y * z) / x));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.85d+193)) then
        tmp = 0.5d0 * (y * ((y * z) / x))
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.85e+193) {
		tmp = 0.5 * (y * ((y * z) / x));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.85e+193:
		tmp = 0.5 * (y * ((y * z) / x))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.85e+193)
		tmp = Float64(0.5 * Float64(y * Float64(Float64(y * z) / x)));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.85e+193)
		tmp = 0.5 * (y * ((y * z) / x));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.85e+193], N[(0.5 * N[(y * N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8500000000000001e193

   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-+ 16.4%

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

      2. div-sub 16.4%

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

      3. *-commutative 16.4%

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

      4. *-commutative 16.4%

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

      5. swap-sqr 3.2%

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

      6. add-sqr-sqrt 3.2%

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

   5. Applied egg-rr 3.2%

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

      1. +-rgt-identity 3.2%

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

      2. div-sub 3.2%

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

      3. +-rgt-identity 3.2%

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

      4. *-commutative 3.2%

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

      5. associate-*l* 16.4%

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

   7. Simplified 16.4%

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

   8. Taylor expanded in x around 0 4.1%

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

      1. mul-1-neg 4.1%

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

      2. unpow2 4.1%

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

      3. associate-*r* 16.8%

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

      4. *-commutative 16.8%

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

      5. distribute-rgt-neg-out 16.8%

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

      6. associate-*l* 16.8%

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

   10. Simplified 16.8%

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

       1. add-sqr-sqrt 16.7%

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

       2. times-frac 50.8%

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

       3. add-sqr-sqrt 50.6%

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

       4. sqrt-unprod 4.1%

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

       5. sqr-neg 4.1%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 0.4%

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

       8. *-commutative 0.4%

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

       9. times-frac 0.3%

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

       10. add-sqr-sqrt 0.5%

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

       11. associate-*l/ 0.6%

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

       12. *-commutative 0.6%

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

       13. associate-*r* 1.1%

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

   12. Applied egg-rr 59.7%

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

   13. Taylor expanded in y around 0 22.9%

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

   if -1.8500000000000001e193 < 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 52.5%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{y \cdot z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 6: 50.9% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \frac{z}{\frac{x}{y \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.3e+193) (* 0.5 (/ z (/ x (* y y)))) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e+193) {
		tmp = 0.5 * (z / (x / (y * 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 <= (-1.3d+193)) then
        tmp = 0.5d0 * (z / (x / (y * 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 <= -1.3e+193) {
		tmp = 0.5 * (z / (x / (y * y)));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.3e+193:
		tmp = 0.5 * (z / (x / (y * y)))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.3e+193)
		tmp = Float64(0.5 * Float64(z / Float64(x / Float64(y * y))));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.3e+193)
		tmp = 0.5 * (z / (x / (y * y)));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.3e+193], N[(0.5 * N[(z / N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.30000000000000007e193

   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-+ 16.4%

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

      2. div-sub 16.4%

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

      3. *-commutative 16.4%

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

      4. *-commutative 16.4%

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

      5. swap-sqr 3.2%

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

      6. add-sqr-sqrt 3.2%

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

   5. Applied egg-rr 3.2%

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

      1. +-rgt-identity 3.2%

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

      2. div-sub 3.2%

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

      3. +-rgt-identity 3.2%

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

      4. *-commutative 3.2%

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

      5. associate-*l* 16.4%

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

   7. Simplified 16.4%

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

   8. Taylor expanded in x around 0 4.1%

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

      1. mul-1-neg 4.1%

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

      2. unpow2 4.1%

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

      3. associate-*r* 16.8%

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

      4. *-commutative 16.8%

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

      5. distribute-rgt-neg-out 16.8%

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

      6. associate-*l* 16.8%

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

   10. Simplified 16.8%

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

   11. Taylor expanded in y around 0 15.1%

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

       1. associate-*r/ 15.1%

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

       2. mul-1-neg 15.1%

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

       3. unpow2 15.1%

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

       4. *-commutative 15.1%

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

       5. distribute-rgt-neg-in 15.1%

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

       6. associate-/l* 15.1%

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

   13. Simplified 15.1%

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

       1. expm1-log1p-u 15.1%

          \[\leadsto 0.5 \cdot \frac{z}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{-y \cdot y}\right)\right)}} \]

       2. expm1-udef 0.5%

          \[\leadsto 0.5 \cdot \frac{z}{\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{-y \cdot y}\right)} - 1}} \]

       3. distribute-rgt-neg-in 0.5%

          \[\leadsto 0.5 \cdot \frac{z}{e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{y \cdot \left(-y\right)}}\right)} - 1} \]

       4. distribute-rgt-neg-in 0.5%

          \[\leadsto 0.5 \cdot \frac{z}{e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{-y \cdot y}}\right)} - 1} \]

       5. add-sqr-sqrt 0.0%

          \[\leadsto 0.5 \cdot \frac{z}{e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\sqrt{-y \cdot y} \cdot \sqrt{-y \cdot y}}}\right)} - 1} \]

       6. sqrt-unprod 0.5%

          \[\leadsto 0.5 \cdot \frac{z}{e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\sqrt{\left(-y \cdot y\right) \cdot \left(-y \cdot y\right)}}}\right)} - 1} \]

       7. sqr-neg 0.5%

          \[\leadsto 0.5 \cdot \frac{z}{e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}}}\right)} - 1} \]

       8. sqrt-unprod 0.5%

          \[\leadsto 0.5 \cdot \frac{z}{e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\sqrt{y \cdot y} \cdot \sqrt{y \cdot y}}}\right)} - 1} \]

       9. add-sqr-sqrt 0.5%

          \[\leadsto 0.5 \cdot \frac{z}{e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{y \cdot y}}\right)} - 1} \]

   15. Applied egg-rr 0.5%

       \[\leadsto 0.5 \cdot \frac{z}{\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y \cdot y}\right)} - 1}} \]
   16. Step-by-step derivation

       1. expm1-def 23.1%

          \[\leadsto 0.5 \cdot \frac{z}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y \cdot y}\right)\right)}} \]

       2. expm1-log1p 23.1%

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

   17. Simplified 23.1%

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

   if -1.30000000000000007e193 < 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 52.5%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \frac{z}{\frac{x}{y \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 7: 50.5% accurate, 36.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around inf 47.5%

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

5. Final simplification 47.5%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023297 
(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)))))