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

Percentage Accurate: 99.8% → 99.8%

Time: 11.1s

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. metadata-eval 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 74.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if ((t_0 <= -1e-85) || !(t_0 <= 7e+75)) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(z)
    if ((t_0 <= (-1d-85)) .or. (.not. (t_0 <= 7d+75))) then
        tmp = 0.5d0 * t_0
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.sqrt(z);
	double tmp;
	if ((t_0 <= -1e-85) || !(t_0 <= 7e+75)) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

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

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if ((t_0 <= -1e-85) || !(t_0 <= 7e+75))
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(0.5 * 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-85) || ~((t_0 <= 7e+75)))
		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-85], N[Not[LessEqual[t$95$0, 7e+75]], $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)) < -9.9999999999999998e-86 or 6.9999999999999997e75 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.7%

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

      1. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 81.8%

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

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

   1. Initial program 99.9%

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

      1. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 75.7%

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

4. Final simplification 78.8%

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

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: 51.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + \left(y \cdot y\right) \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5.2e+101) (* 0.5 x) (* 0.5 (+ x (* (* y y) (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.2e+101) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (x + ((y * y) * (z / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 5.2d+101) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * (x + ((y * y) * (z / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.2e+101) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (x + ((y * y) * (z / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 5.2e+101:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * (x + ((y * y) * (z / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5.2e+101)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(x + Float64(Float64(y * y) * Float64(z / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 5.2e+101)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * (x + ((y * y) * (z / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5.2e+101], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(x + N[(N[(y * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.2e101

   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 52.2%

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

   if 5.2e101 < z

   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-+ 45.6%

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

      2. div-inv 45.5%

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

      3. *-commutative 45.5%

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

      4. *-commutative 45.5%

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

      5. swap-sqr 41.4%

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

      6. add-sqr-sqrt 41.5%

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

   5. Applied egg-rr 41.5%

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

   6. Taylor expanded in x around inf 25.9%

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

      1. add-sqr-sqrt 25.9%

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

      2. sqrt-unprod 26.0%

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

      3. sqr-neg 26.0%

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

      4. sqrt-unprod 14.3%

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

      5. add-sqr-sqrt 30.0%

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

      6. distribute-rgt-neg-out 30.0%

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

   8. Applied egg-rr 30.0%

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

   9. Taylor expanded in x around 0 43.8%

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

       1. +-commutative 43.8%

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

       2. unpow2 43.8%

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

       3. associate-*r/ 44.1%

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

   11. Simplified 44.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + \left(y \cdot y\right) \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternative 5: 52.1% accurate, 8.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.08e+17) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (x - (z * ((y * y) / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.08d+17) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * (x - (z * ((y * y) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.08e+17) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (x - (z * ((y * y) / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.08e+17:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * (x - (z * ((y * y) / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.08e+17)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(x - Float64(z * Float64(Float64(y * y) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.08e+17)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * (x - (z * ((y * y) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.08e+17], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(x - N[(z * N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.08e17

   1. Initial program 99.8%

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

      1. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 57.0%

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

   if 1.08e17 < z

   1. Initial program 99.8%

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

      1. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. flip-+ 44.6%

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

      2. div-inv 44.4%

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

      3. *-commutative 44.4%

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

      4. *-commutative 44.4%

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

      5. swap-sqr 41.3%

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

      6. add-sqr-sqrt 41.4%

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

   5. Applied egg-rr 41.4%

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

   6. Taylor expanded in x around inf 24.4%

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

   7. Taylor expanded in x around 0 42.1%

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

      1. +-commutative 42.1%

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

      2. mul-1-neg 42.1%

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

      3. unsub-neg 42.1%

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

      4. unpow2 42.1%

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

      5. associate-/l* 42.3%

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

      6. associate-/r/ 43.5%

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

   9. Simplified 43.5%

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

4. Final simplification 50.5%

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

Alternative 6: 50.2% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.2d+218)) then
        tmp = 0.5d0 * (y * (y * (z / -x)))
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.2e+218], N[(0.5 * N[(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 < -3.19999999999999987e218

   1. Initial program 99.7%

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

      1. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. flip-+ 10.8%

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

      2. div-inv 10.8%

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

      3. *-commutative 10.8%

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

      4. *-commutative 10.8%

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

      5. swap-sqr 2.5%

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

      6. add-sqr-sqrt 2.5%

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

   5. Applied egg-rr 2.5%

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

   6. Taylor expanded in x around inf 9.5%

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

   7. Taylor expanded in x around 0 27.1%

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

      1. mul-1-neg 27.1%

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

      2. unpow2 27.1%

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

      3. associate-/l* 26.8%

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

      4. distribute-neg-frac 26.8%

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

      5. distribute-rgt-neg-in 26.8%

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

   9. Simplified 26.8%

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

   10. Taylor expanded in y around 0 27.1%

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

       1. associate-*r/ 27.1%

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

       2. unpow2 27.1%

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

       3. *-commutative 27.1%

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

       4. associate-*l/ 27.1%

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

       5. metadata-eval 27.1%

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

       6. associate-/r* 27.1%

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

       7. neg-mul-1 27.1%

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

       8. associate-*l* 26.8%

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

       9. associate-/r/ 26.8%

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

       10. *-commutative 26.8%

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

       11. associate-*l* 27.2%

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

       12. associate-/r/ 27.1%

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

       13. associate-*l/ 27.1%

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

       14. *-lft-identity 27.1%

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

   12. Simplified 27.1%

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

   if -3.19999999999999987e218 < y

   1. Initial program 99.8%

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

      1. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 50.7%

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

4. Final simplification 48.5%

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

Alternative 7: 49.6% accurate, 36.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around inf 46.9%

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

5. Final simplification 46.9%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))