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

Percentage Accurate: 99.8% → 99.8%

Time: 6.5s

Alternatives: 6

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 99.9%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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. Final simplification 99.9%

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

Alternative 2: 74.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+144} \lor \neg \left(t_0 \leq -1 \cdot 10^{+93}\right) \land \left(t_0 \leq -4 \cdot 10^{-23} \lor \neg \left(t_0 \leq 10^{+34}\right)\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -2e+144)
           (and (not (<= t_0 -1e+93))
                (or (<= t_0 -4e-23) (not (<= t_0 1e+34)))))
     (* 0.5 t_0)
     (* 0.5 x))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if ((t_0 <= -2e+144) || (!(t_0 <= -1e+93) && ((t_0 <= -4e-23) || !(t_0 <= 1e+34)))) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(z)
    if ((t_0 <= (-2d+144)) .or. (.not. (t_0 <= (-1d+93))) .and. (t_0 <= (-4d-23)) .or. (.not. (t_0 <= 1d+34))) then
        tmp = 0.5d0 * t_0
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.sqrt(z);
	double tmp;
	if ((t_0 <= -2e+144) || (!(t_0 <= -1e+93) && ((t_0 <= -4e-23) || !(t_0 <= 1e+34)))) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	tmp = 0
	if (t_0 <= -2e+144) or (not (t_0 <= -1e+93) and ((t_0 <= -4e-23) or not (t_0 <= 1e+34))):
		tmp = 0.5 * t_0
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if ((t_0 <= -2e+144) || (!(t_0 <= -1e+93) && ((t_0 <= -4e-23) || !(t_0 <= 1e+34))))
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	tmp = 0.0;
	if ((t_0 <= -2e+144) || (~((t_0 <= -1e+93)) && ((t_0 <= -4e-23) || ~((t_0 <= 1e+34)))))
		tmp = 0.5 * t_0;
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+144], And[N[Not[LessEqual[t$95$0, -1e+93]], $MachinePrecision], Or[LessEqual[t$95$0, -4e-23], N[Not[LessEqual[t$95$0, 1e+34]], $MachinePrecision]]]], N[(0.5 * t$95$0), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (sqrt.f64 z)) < -2.00000000000000005e144 or -1.00000000000000004e93 < (*.f64 y (sqrt.f64 z)) < -3.99999999999999984e-23 or 9.99999999999999946e33 < (*.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 82.3%

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

   if -2.00000000000000005e144 < (*.f64 y (sqrt.f64 z)) < -1.00000000000000004e93 or -3.99999999999999984e-23 < (*.f64 y (sqrt.f64 z)) < 9.99999999999999946e33

   1. Initial program 99.9%

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

      1. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 79.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -2 \cdot 10^{+144} \lor \neg \left(y \cdot \sqrt{z} \leq -1 \cdot 10^{+93}\right) \land \left(y \cdot \sqrt{z} \leq -4 \cdot 10^{-23} \lor \neg \left(y \cdot \sqrt{z} \leq 10^{+34}\right)\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3: 51.2% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+164}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{y}{\frac{x}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3e+164) (* 0.5 x) (* -0.5 (* z (/ y (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+164) {
		tmp = 0.5 * x;
	} else {
		tmp = -0.5 * (z * (y / (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3d+164) then
        tmp = 0.5d0 * x
    else
        tmp = (-0.5d0) * (z * (y / (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+164) {
		tmp = 0.5 * x;
	} else {
		tmp = -0.5 * (z * (y / (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3e+164:
		tmp = 0.5 * x
	else:
		tmp = -0.5 * (z * (y / (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3e+164)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(-0.5 * Float64(z * Float64(y / Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3e+164)
		tmp = 0.5 * x;
	else
		tmp = -0.5 * (z * (y / (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3e+164], N[(0.5 * x), $MachinePrecision], N[(-0.5 * N[(z * N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.00000000000000001e164

   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 59.2%

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

   if 3.00000000000000001e164 < 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-+ 20.9%

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

      2. div-sub 20.9%

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

      3. *-commutative 20.9%

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

      4. *-commutative 20.9%

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

      5. swap-sqr 3.5%

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

      6. add-sqr-sqrt 3.5%

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

   5. Applied egg-rr 3.5%

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

      1. +-rgt-identity 3.5%

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

      2. div-sub 3.5%

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

      3. +-rgt-identity 3.5%

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

      4. *-commutative 3.5%

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

      5. associate-*l* 20.9%

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

   7. Simplified 20.9%

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

   8. Taylor expanded in x around 0 4.2%

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

      1. mul-1-neg 4.2%

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

      2. unpow2 4.2%

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

      3. associate-*r* 21.2%

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

      4. *-commutative 21.2%

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

      5. distribute-rgt-neg-out 21.2%

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

      6. associate-*l* 21.2%

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

   10. Simplified 21.2%

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

   11. Taylor expanded in y around 0 21.4%

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

       1. *-commutative 21.4%

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

       2. unpow2 21.4%

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

       3. *-commutative 21.4%

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

       4. associate-*r* 21.4%

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

       5. *-commutative 21.4%

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

   13. Simplified 21.4%

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

   14. Taylor expanded in y around 0 21.4%

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

       1. unpow2 21.4%

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

       2. *-commutative 21.4%

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

       3. associate-*r* 21.4%

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

       4. associate-/l* 21.4%

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

       5. *-commutative 21.4%

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

       6. associate-*l/ 21.4%

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

       7. *-commutative 21.4%

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

   16. Simplified 21.4%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+164}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{y}{\frac{x}{y}}\right)\\ \end{array} \]

Alternative 4: 51.2% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+164}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(y \cdot z\right) \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3e+164) (* 0.5 x) (* -0.5 (* (* y z) (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+164) {
		tmp = 0.5 * x;
	} else {
		tmp = -0.5 * ((y * z) * (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 (y <= 3d+164) then
        tmp = 0.5d0 * x
    else
        tmp = (-0.5d0) * ((y * z) * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+164) {
		tmp = 0.5 * x;
	} else {
		tmp = -0.5 * ((y * z) * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3e+164:
		tmp = 0.5 * x
	else:
		tmp = -0.5 * ((y * z) * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3e+164)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(-0.5 * Float64(Float64(y * z) * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3e+164)
		tmp = 0.5 * x;
	else
		tmp = -0.5 * ((y * z) * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3e+164], N[(0.5 * x), $MachinePrecision], N[(-0.5 * N[(N[(y * z), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.00000000000000001e164

   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 59.2%

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

   if 3.00000000000000001e164 < 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-+ 20.9%

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

      2. div-sub 20.9%

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

      3. *-commutative 20.9%

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

      4. *-commutative 20.9%

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

      5. swap-sqr 3.5%

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

      6. add-sqr-sqrt 3.5%

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

   5. Applied egg-rr 3.5%

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

      1. +-rgt-identity 3.5%

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

      2. div-sub 3.5%

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

      3. +-rgt-identity 3.5%

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

      4. *-commutative 3.5%

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

      5. associate-*l* 20.9%

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

   7. Simplified 20.9%

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

   8. Taylor expanded in x around 0 4.2%

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

      1. mul-1-neg 4.2%

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

      2. unpow2 4.2%

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

      3. associate-*r* 21.2%

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

      4. *-commutative 21.2%

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

      5. distribute-rgt-neg-out 21.2%

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

      6. associate-*l* 21.2%

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

   10. Simplified 21.2%

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

   11. Taylor expanded in y around 0 21.4%

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

       1. *-commutative 21.4%

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

       2. unpow2 21.4%

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

       3. *-commutative 21.4%

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

       4. associate-*r* 21.4%

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

       5. *-commutative 21.4%

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

   13. Simplified 21.4%

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

   14. Taylor expanded in y around 0 21.4%

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

       1. unpow2 21.4%

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

       2. *-commutative 21.4%

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

       3. associate-*r* 21.4%

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

       4. associate-*r/ 21.4%

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

       5. *-commutative 21.4%

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

   16. Simplified 21.4%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+164}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(y \cdot z\right) \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 5: 51.2% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+164}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{y \cdot z}{x}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3e+164) (* 0.5 x) (* (* y (/ (* y z) x)) -0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+164) {
		tmp = 0.5 * x;
	} else {
		tmp = (y * ((y * z) / x)) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3d+164) then
        tmp = 0.5d0 * x
    else
        tmp = (y * ((y * z) / x)) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+164) {
		tmp = 0.5 * x;
	} else {
		tmp = (y * ((y * z) / x)) * -0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3e+164:
		tmp = 0.5 * x
	else:
		tmp = (y * ((y * z) / x)) * -0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3e+164)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(Float64(y * Float64(Float64(y * z) / x)) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3e+164)
		tmp = 0.5 * x;
	else
		tmp = (y * ((y * z) / x)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3e+164], N[(0.5 * x), $MachinePrecision], N[(N[(y * N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.00000000000000001e164

   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 59.2%

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

   if 3.00000000000000001e164 < 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-+ 20.9%

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

      2. div-sub 20.9%

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

      3. *-commutative 20.9%

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

      4. *-commutative 20.9%

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

      5. swap-sqr 3.5%

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

      6. add-sqr-sqrt 3.5%

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

   5. Applied egg-rr 3.5%

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

      1. +-rgt-identity 3.5%

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

      2. div-sub 3.5%

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

      3. +-rgt-identity 3.5%

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

      4. *-commutative 3.5%

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

      5. associate-*l* 20.9%

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

   7. Simplified 20.9%

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

   8. Taylor expanded in x around 0 4.2%

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

      1. mul-1-neg 4.2%

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

      2. unpow2 4.2%

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

      3. associate-*r* 21.2%

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

      4. *-commutative 21.2%

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

      5. distribute-rgt-neg-out 21.2%

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

      6. associate-*l* 21.2%

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

   10. Simplified 21.2%

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

   11. Taylor expanded in y around 0 21.4%

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

       1. *-commutative 21.4%

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

       2. unpow2 21.4%

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

       3. *-commutative 21.4%

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

       4. associate-*r* 21.4%

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

       5. *-commutative 21.4%

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

   13. Simplified 21.4%

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

       1. associate-/l* 21.4%

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

       2. associate-/r/ 21.4%

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

       3. *-commutative 21.4%

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

   15. Applied egg-rr 21.4%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+164}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{y \cdot z}{x}\right) \cdot -0.5\\ \end{array} \]

Alternative 6: 51.0% 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 99.9%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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 53.3%

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

5. Final simplification 53.3%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023271 
(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)))))