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

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (* 0.5 (+ (* (/ z z) x) (/ y (pow z -0.5)))))

C

double code(double x, double y, double z) {
	return 0.5 * (((z / z) * x) + (y / pow(z, -0.5)));
}

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 * (((z / z) * x) + (y / (z ** (-0.5d0))))
end function

Java

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

Python

def code(x, y, z):
	return 0.5 * (((z / z) * x) + (y / math.pow(z, -0.5)))

Julia

function code(x, y, z)
	return Float64(0.5 * Float64(Float64(Float64(z / z) * x) + Float64(y / (z ^ -0.5))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * (((z / z) * x) + (y / (z ^ -0.5)));
end

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(N[(N[(z / z), $MachinePrecision] * x), $MachinePrecision] + N[(y / N[Power[z, -0.5], $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. Add Preprocessing

5. Taylor expanded in z around inf 84.2%

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

   1. sqrt-div 84.1%

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

   2. metadata-eval 84.1%

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

   3. un-div-inv 84.2%

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

7. Applied egg-rr 84.2%

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

   1. +-commutative 84.2%

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

   2. distribute-rgt-in 84.2%

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

   3. *-commutative 84.2%

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

   4. clear-num 83.8%

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

   5. un-div-inv 84.7%

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

   6. associate-*l/ 83.6%

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

9. Applied egg-rr 83.6%

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

    1. associate-/r/ 92.8%

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

    2. associate-/l* 99.7%

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

11. Simplified 99.7%

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

    1. clear-num 99.7%

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

    2. un-div-inv 99.8%

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

    3. pow1/2 99.8%

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

    4. pow1 99.8%

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

    5. pow-div 99.8%

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

    6. metadata-eval 99.8%

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

13. Applied egg-rr 99.8%

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

Alternative 2: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+40} \lor \neg \left(y \leq 5 \cdot 10^{+87}\right):\\ \;\;\;\;\sqrt{z} \cdot \left(0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8.5e+40) (not (<= y 5e+87))) (* (sqrt z) (* 0.5 y)) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8.5e+40) || !(y <= 5e+87)) {
		tmp = sqrt(z) * (0.5 * y);
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-8.5d+40)) .or. (.not. (y <= 5d+87))) then
        tmp = sqrt(z) * (0.5d0 * y)
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8.5e+40) || !(y <= 5e+87)) {
		tmp = Math.sqrt(z) * (0.5 * y);
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8.5e+40) or not (y <= 5e+87):
		tmp = math.sqrt(z) * (0.5 * y)
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8.5e+40) || !(y <= 5e+87))
		tmp = Float64(sqrt(z) * Float64(0.5 * y));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -8.5e+40) || ~((y <= 5e+87)))
		tmp = sqrt(z) * (0.5 * y);
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8.5e+40], N[Not[LessEqual[y, 5e+87]], $MachinePrecision]], N[(N[Sqrt[z], $MachinePrecision] * N[(0.5 * y), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.49999999999999996e40 or 4.9999999999999998e87 < 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. Add Preprocessing

   5. Taylor expanded in x around 0 88.4%

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

      1. associate-*r* 88.4%

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

      2. *-commutative 88.4%

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

      3. *-commutative 88.4%

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

   7. Simplified 88.4%

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

   if -8.49999999999999996e40 < y < 4.9999999999999998e87

   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. Add Preprocessing

   5. Taylor expanded in x around inf 70.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+40} \lor \neg \left(y \leq 5 \cdot 10^{+87}\right):\\ \;\;\;\;\sqrt{z} \cdot \left(0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing
5. Add Preprocessing

Alternative 4: 51.4% 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 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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. Add Preprocessing

5. Taylor expanded in x around inf 50.0%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(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)))))