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

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 4

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: 72.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+72} \lor \neg \left(t_0 \leq 10^{-147}\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -1e+72) (not (<= t_0 1e-147))) (* 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 <= -1e+72) || !(t_0 <= 1e-147)) {
		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 <= (-1d+72)) .or. (.not. (t_0 <= 1d-147))) 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 <= -1e+72) || !(t_0 <= 1e-147)) {
		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 <= -1e+72) or not (t_0 <= 1e-147):
		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 <= -1e+72) || !(t_0 <= 1e-147))
		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 <= -1e+72) || ~((t_0 <= 1e-147)))
		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, -1e+72], N[Not[LessEqual[t$95$0, 1e-147]], $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)) < -9.99999999999999944e71 or 9.9999999999999997e-148 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.8%

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

      1. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 77.3%

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

   if -9.99999999999999944e71 < (*.f64 y (sqrt.f64 z)) < 9.9999999999999997e-148

   1. Initial program 99.9%

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

      1. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 80.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -1 \cdot 10^{+72} \lor \neg \left(y \cdot \sqrt{z} \leq 10^{-147}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3: 52.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+220} \lor \neg \left(y \leq 5.2 \cdot 10^{+199}\right):\\ \;\;\;\;\left(z \cdot \frac{y}{x}\right) \cdot \left(0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.25e+220) (not (<= y 5.2e+199)))
   (* (* z (/ y x)) (* 0.5 y))
   (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e+220) || !(y <= 5.2e+199)) {
		tmp = (z * (y / x)) * (0.5 * 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 <= (-1.25d+220)) .or. (.not. (y <= 5.2d+199))) then
        tmp = (z * (y / x)) * (0.5d0 * 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 <= -1.25e+220) || !(y <= 5.2e+199)) {
		tmp = (z * (y / x)) * (0.5 * y);
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.25e+220) or not (y <= 5.2e+199):
		tmp = (z * (y / x)) * (0.5 * y)
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.25e+220) || !(y <= 5.2e+199))
		tmp = Float64(Float64(z * Float64(y / x)) * Float64(0.5 * y));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.25e+220) || ~((y <= 5.2e+199)))
		tmp = (z * (y / x)) * (0.5 * y);
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.25e+220], N[Not[LessEqual[y, 5.2e+199]], $MachinePrecision]], N[(N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(0.5 * y), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2500000000000001e220 or 5.2000000000000003e199 < 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-+ 10.6%

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

      2. div-sub 10.6%

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

      3. *-commutative 10.6%

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

      4. *-commutative 10.6%

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

      5. swap-sqr 3.1%

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

      6. add-sqr-sqrt 3.1%

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

   5. Applied egg-rr 3.1%

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

      1. +-rgt-identity 3.1%

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

      2. div-sub 3.1%

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

      3. +-rgt-identity 3.1%

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

      4. *-commutative 3.1%

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

      5. associate-*l* 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in x around 0 3.6%

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

      1. mul-1-neg 3.6%

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

      2. unpow2 3.6%

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

      3. distribute-rgt-neg-out 3.6%

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

      4. associate-*l* 11.1%

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

   10. Simplified 11.1%

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

   11. Taylor expanded in y around 0 14.5%

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

       1. *-commutative 14.5%

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

       2. unpow2 14.5%

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

       3. associate-/l* 14.5%

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

   13. Simplified 14.5%

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

       1. add-sqr-sqrt 8.6%

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

       2. sqrt-unprod 25.7%

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

       3. *-commutative 25.7%

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

       4. *-commutative 25.7%

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

       5. swap-sqr 25.7%

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

       6. metadata-eval 25.7%

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

       7. metadata-eval 25.7%

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

       8. pow2 25.7%

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

       9. add-sqr-sqrt 25.7%

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

       10. sqrt-unprod 25.3%

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

       11. sqr-neg 25.3%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 25.7%

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

       14. associate-/l* 25.9%

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

       15. *-commutative 25.9%

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

       16. pow2 25.9%

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

   15. Applied egg-rr 33.9%

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

       1. associate-/r/ 33.9%

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

       2. associate-*l/ 33.9%

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

       3. metadata-eval 33.9%

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

       4. distribute-lft-neg-in 33.9%

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

       5. *-commutative 33.9%

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

       6. associate-*r* 33.9%

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

       7. associate-*r* 33.9%

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

       8. distribute-rgt-neg-in 33.9%

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

       9. metadata-eval 33.9%

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

       10. associate-*l/ 33.9%

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

       11. associate-*l/ 34.0%

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

       12. associate-/l* 34.1%

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

       13. associate-*r/ 34.0%

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

       14. associate-*l* 34.0%

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

       15. associate-*r/ 34.1%

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

       16. associate-*l/ 34.1%

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

       17. *-commutative 34.1%

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

   17. Simplified 34.1%

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

   if -1.2500000000000001e220 < y < 5.2000000000000003e199

   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 61.1%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+220} \lor \neg \left(y \leq 5.2 \cdot 10^{+199}\right):\\ \;\;\;\;\left(z \cdot \frac{y}{x}\right) \cdot \left(0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 4: 50.9% 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 52.8%

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

5. Final simplification 52.8%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023293 
(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)))))