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

Percentage Accurate: 99.8% → 99.8%

Time: 6.3s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+39} \lor \neg \left(t_0 \leq 2 \cdot 10^{-55}\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -1e+39) (not (<= t_0 2e-55))) (* 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+39) || !(t_0 <= 2e-55)) {
		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+39)) .or. (.not. (t_0 <= 2d-55))) 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+39) || !(t_0 <= 2e-55)) {
		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+39) or not (t_0 <= 2e-55):
		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+39) || !(t_0 <= 2e-55))
		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+39) || ~((t_0 <= 2e-55)))
		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+39], N[Not[LessEqual[t$95$0, 2e-55]], $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)) < -9.9999999999999994e38 or 1.99999999999999999e-55 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.6%

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

      1. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 77.6%

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

   if -9.9999999999999994e38 < (*.f64 y (sqrt.f64 z)) < 1.99999999999999999e-55

   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 76.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -1 \cdot 10^{+39} \lor \neg \left(y \cdot \sqrt{z} \leq 2 \cdot 10^{-55}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3: 51.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+246}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{x}{z \cdot \left(-y\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.1e+246) (* 0.5 (/ y (/ x (* z (- y))))) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e+246) {
		tmp = 0.5 * (y / (x / (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 (y <= (-1.1d+246)) then
        tmp = 0.5d0 * (y / (x / (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 (y <= -1.1e+246) {
		tmp = 0.5 * (y / (x / (z * -y)));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.1e+246:
		tmp = 0.5 * (y / (x / (z * -y)))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.1e+246)
		tmp = Float64(0.5 * Float64(y / Float64(x / Float64(z * Float64(-y)))));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.1e+246)
		tmp = 0.5 * (y / (x / (z * -y)));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.1e+246], N[(0.5 * N[(y / N[(x / N[(z * (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.09999999999999994e246

   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. +-commutative 99.9%

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

      2. flip-+ 9.7%

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

      3. *-commutative 9.7%

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

      4. *-commutative 9.7%

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

      5. swap-sqr 2.4%

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

      6. add-sqr-sqrt 2.4%

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

   5. Applied egg-rr 2.4%

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

      1. *-commutative 2.4%

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

      2. associate-*r* 9.6%

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

      3. fma-neg 9.6%

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

      4. *-commutative 9.6%

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

      5. distribute-rgt-neg-in 9.6%

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

   7. Applied egg-rr 9.6%

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

      1. fma-udef 9.6%

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

      2. distribute-rgt-neg-out 9.6%

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

      3. unsub-neg 9.6%

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

   9. Simplified 9.6%

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

   10. Taylor expanded in y around inf 2.9%

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

       1. *-commutative 2.9%

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

       2. unpow2 2.9%

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

   12. Simplified 2.9%

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

   13. Taylor expanded in z around 0 33.0%

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

       1. associate-*r/ 33.0%

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

       2. mul-1-neg 33.0%

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

       3. unpow2 33.0%

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

       4. distribute-rgt-neg-out 33.0%

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

       5. associate-*l* 33.1%

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

       6. associate-/l* 33.1%

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

   15. Simplified 33.1%

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

   if -1.09999999999999994e246 < 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 52.9%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+246}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{x}{z \cdot \left(-y\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 4: 50.9% accurate, 36.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around inf 50.3%

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

5. Final simplification 50.3%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023268 
(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)))))