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

Percentage Accurate: 99.8% → 99.8%

Time: 9.1s

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, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (sqrt(z) * (y * 0.5)) + (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 = (sqrt(z) * (y * 0.5d0)) + (0.5d0 * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. distribute-rgt-in 99.5%

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

   2. +-commutative 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.9%

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

   5. *-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{+33} \lor \neg \left(x \leq -2 \cdot 10^{-16}\right) \land x \leq 5 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{z} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.8e+59)
   (* 0.5 x)
   (if (or (<= x -1.25e+33) (and (not (<= x -2e-16)) (<= x 5e-31)))
     (* 0.5 (* (sqrt z) y))
     (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e+59) {
		tmp = 0.5 * x;
	} else if ((x <= -1.25e+33) || (!(x <= -2e-16) && (x <= 5e-31))) {
		tmp = 0.5 * (sqrt(z) * 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 (x <= (-1.8d+59)) then
        tmp = 0.5d0 * x
    else if ((x <= (-1.25d+33)) .or. (.not. (x <= (-2d-16))) .and. (x <= 5d-31)) then
        tmp = 0.5d0 * (sqrt(z) * 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 (x <= -1.8e+59) {
		tmp = 0.5 * x;
	} else if ((x <= -1.25e+33) || (!(x <= -2e-16) && (x <= 5e-31))) {
		tmp = 0.5 * (Math.sqrt(z) * y);
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.8e+59:
		tmp = 0.5 * x
	elif (x <= -1.25e+33) or (not (x <= -2e-16) and (x <= 5e-31)):
		tmp = 0.5 * (math.sqrt(z) * y)
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.8e+59)
		tmp = Float64(0.5 * x);
	elseif ((x <= -1.25e+33) || (!(x <= -2e-16) && (x <= 5e-31)))
		tmp = Float64(0.5 * Float64(sqrt(z) * y));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.8e+59)
		tmp = 0.5 * x;
	elseif ((x <= -1.25e+33) || (~((x <= -2e-16)) && (x <= 5e-31)))
		tmp = 0.5 * (sqrt(z) * y);
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.8e+59], N[(0.5 * x), $MachinePrecision], If[Or[LessEqual[x, -1.25e+33], And[N[Not[LessEqual[x, -2e-16]], $MachinePrecision], LessEqual[x, 5e-31]]], N[(0.5 * N[(N[Sqrt[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7999999999999999e59 or -1.24999999999999993e33 < x < -2e-16 or 5e-31 < 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 82.3%

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

   if -1.7999999999999999e59 < x < -1.24999999999999993e33 or -2e-16 < x < 5e-31

   1. Initial program 99.1%

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

      1. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 82.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{+33} \lor \neg \left(x \leq -2 \cdot 10^{-16}\right) \land x \leq 5 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{z} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+34}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{z} \cdot y\right)\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{z} \cdot \left(y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.3e+59)
   (* 0.5 x)
   (if (<= x -9e+34)
     (* 0.5 (* (sqrt z) y))
     (if (<= x -4.1e-16)
       (* 0.5 x)
       (if (<= x 5.5e-35) (* (sqrt z) (* y 0.5)) (* 0.5 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e+59) {
		tmp = 0.5 * x;
	} else if (x <= -9e+34) {
		tmp = 0.5 * (sqrt(z) * y);
	} else if (x <= -4.1e-16) {
		tmp = 0.5 * x;
	} else if (x <= 5.5e-35) {
		tmp = sqrt(z) * (y * 0.5);
	} 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 <= (-1.3d+59)) then
        tmp = 0.5d0 * x
    else if (x <= (-9d+34)) then
        tmp = 0.5d0 * (sqrt(z) * y)
    else if (x <= (-4.1d-16)) then
        tmp = 0.5d0 * x
    else if (x <= 5.5d-35) then
        tmp = sqrt(z) * (y * 0.5d0)
    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 <= -1.3e+59) {
		tmp = 0.5 * x;
	} else if (x <= -9e+34) {
		tmp = 0.5 * (Math.sqrt(z) * y);
	} else if (x <= -4.1e-16) {
		tmp = 0.5 * x;
	} else if (x <= 5.5e-35) {
		tmp = Math.sqrt(z) * (y * 0.5);
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.3e+59:
		tmp = 0.5 * x
	elif x <= -9e+34:
		tmp = 0.5 * (math.sqrt(z) * y)
	elif x <= -4.1e-16:
		tmp = 0.5 * x
	elif x <= 5.5e-35:
		tmp = math.sqrt(z) * (y * 0.5)
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.3e+59)
		tmp = Float64(0.5 * x);
	elseif (x <= -9e+34)
		tmp = Float64(0.5 * Float64(sqrt(z) * y));
	elseif (x <= -4.1e-16)
		tmp = Float64(0.5 * x);
	elseif (x <= 5.5e-35)
		tmp = Float64(sqrt(z) * Float64(y * 0.5));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.3e+59)
		tmp = 0.5 * x;
	elseif (x <= -9e+34)
		tmp = 0.5 * (sqrt(z) * y);
	elseif (x <= -4.1e-16)
		tmp = 0.5 * x;
	elseif (x <= 5.5e-35)
		tmp = sqrt(z) * (y * 0.5);
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.3e+59], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, -9e+34], N[(0.5 * N[(N[Sqrt[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.1e-16], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 5.5e-35], N[(N[Sqrt[z], $MachinePrecision] * N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3e59 or -9.0000000000000001e34 < x < -4.10000000000000006e-16 or 5.4999999999999997e-35 < 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 82.3%

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

   if -1.3e59 < x < -9.0000000000000001e34

   1. Initial program 99.4%

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

      1. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 83.3%

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

   if -4.10000000000000006e-16 < x < 5.4999999999999997e-35

   1. Initial program 99.1%

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

      1. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. flip-+ 62.0%

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

      2. associate-*r/ 62.0%

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

      3. *-commutative 62.0%

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

      4. *-commutative 62.0%

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

      5. swap-sqr 52.4%

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

      6. add-sqr-sqrt 52.5%

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

   5. Applied egg-rr 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-/l* 52.6%

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

   7. Simplified 52.6%

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

   8. Taylor expanded in x around 0 40.4%

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

      1. mul-1-neg 40.4%

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

      2. unpow2 40.4%

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

      3. distribute-rgt-neg-in 40.4%

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

   10. Simplified 40.4%

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

   11. Taylor expanded in y around inf 81.9%

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

       1. associate-*r* 82.6%

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

   13. Simplified 82.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+34}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{z} \cdot y\right)\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{z} \cdot \left(y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + (sqrt(z) * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 5: 50.8% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+203}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \left(y \cdot \frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.5e+203) (* -0.5 (* z (* y (/ y x)))) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5e+203) {
		tmp = -0.5 * (z * (y * (y / x)));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-7.5d+203)) then
        tmp = (-0.5d0) * (z * (y * (y / x)))
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5e+203) {
		tmp = -0.5 * (z * (y * (y / x)));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.5e+203:
		tmp = -0.5 * (z * (y * (y / x)))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.5e+203)
		tmp = Float64(-0.5 * Float64(z * Float64(y * Float64(y / x))));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.5e+203)
		tmp = -0.5 * (z * (y * (y / x)));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.5e+203], N[(-0.5 * N[(z * N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.49999999999999957e203

   1. Initial program 97.0%

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

      1. metadata-eval 97.0%

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

   3. Simplified 97.0%

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

      1. flip-+ 9.3%

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

      2. associate-*r/ 9.3%

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

      3. *-commutative 9.3%

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

      4. *-commutative 9.3%

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

      5. swap-sqr 2.3%

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

      6. add-sqr-sqrt 2.3%

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

   5. Applied egg-rr 2.3%

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

      1. *-commutative 2.3%

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

      2. associate-/l* 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in x around 0 3.0%

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

      1. mul-1-neg 3.0%

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

      2. unpow2 3.0%

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

      3. distribute-rgt-neg-in 3.0%

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

   10. Simplified 3.0%

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

   11. Taylor expanded in y around 0 34.2%

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

       1. unpow2 34.2%

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

       2. associate-*r* 34.2%

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

       3. associate-/l* 34.5%

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

       4. associate-/r* 34.5%

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

   13. Simplified 34.5%

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

       1. associate-/r/ 34.4%

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

       2. div-inv 34.4%

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

       3. clear-num 34.4%

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

   15. Applied egg-rr 34.4%

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

   if -7.49999999999999957e203 < y

   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 58.5%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+203}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \left(y \cdot \frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 6: 50.8% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+203}:\\ \;\;\;\;-0.5 \cdot \frac{y}{\frac{\frac{x}{y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7e+203) (* -0.5 (/ y (/ (/ x y) z))) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e+203) {
		tmp = -0.5 * (y / ((x / y) / z));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-7d+203)) then
        tmp = (-0.5d0) * (y / ((x / y) / z))
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e+203) {
		tmp = -0.5 * (y / ((x / y) / z));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7e+203:
		tmp = -0.5 * (y / ((x / y) / z))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7e+203)
		tmp = Float64(-0.5 * Float64(y / Float64(Float64(x / y) / z)));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7e+203)
		tmp = -0.5 * (y / ((x / y) / z));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7e+203], N[(-0.5 * N[(y / N[(N[(x / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.00000000000000062e203

   1. Initial program 97.0%

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

      1. metadata-eval 97.0%

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

   3. Simplified 97.0%

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

      1. flip-+ 9.3%

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

      2. associate-*r/ 9.3%

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

      3. *-commutative 9.3%

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

      4. *-commutative 9.3%

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

      5. swap-sqr 2.3%

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

      6. add-sqr-sqrt 2.3%

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

   5. Applied egg-rr 2.3%

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

      1. *-commutative 2.3%

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

      2. associate-/l* 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in x around 0 3.0%

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

      1. mul-1-neg 3.0%

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

      2. unpow2 3.0%

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

      3. distribute-rgt-neg-in 3.0%

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

   10. Simplified 3.0%

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

   11. Taylor expanded in y around 0 34.2%

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

       1. unpow2 34.2%

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

       2. associate-*r* 34.2%

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

       3. associate-/l* 34.5%

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

       4. associate-/r* 34.5%

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

   13. Simplified 34.5%

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

   if -7.00000000000000062e203 < y

   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 58.5%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+203}:\\ \;\;\;\;-0.5 \cdot \frac{y}{\frac{\frac{x}{y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 7: 50.4% 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 99.5%

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

   1. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around inf 53.1%

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

5. Final simplification 53.1%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023274 
(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)))))