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

Percentage Accurate: 99.8% → 99.8%

Time: 7.1s

Alternatives: 3

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. Add Preprocessing

5. Final simplification 99.8%

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

Alternative 2: 73.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+30} \lor \neg \left(x \leq -1.4 \cdot 10^{-8}\right) \land \left(x \leq -5.2 \cdot 10^{-46} \lor \neg \left(x \leq 8.4 \cdot 10^{+64}\right)\right):\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(0.5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.02e+30)
         (and (not (<= x -1.4e-8)) (or (<= x -5.2e-46) (not (<= x 8.4e+64)))))
   (* 0.5 x)
   (* (sqrt z) (* 0.5 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.02e+30) || (!(x <= -1.4e-8) && ((x <= -5.2e-46) || !(x <= 8.4e+64)))) {
		tmp = 0.5 * x;
	} else {
		tmp = sqrt(z) * (0.5 * 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 ((x <= (-1.02d+30)) .or. (.not. (x <= (-1.4d-8))) .and. (x <= (-5.2d-46)) .or. (.not. (x <= 8.4d+64))) then
        tmp = 0.5d0 * x
    else
        tmp = sqrt(z) * (0.5d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.02e+30) || (!(x <= -1.4e-8) && ((x <= -5.2e-46) || !(x <= 8.4e+64)))) {
		tmp = 0.5 * x;
	} else {
		tmp = Math.sqrt(z) * (0.5 * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.02e+30) or (not (x <= -1.4e-8) and ((x <= -5.2e-46) or not (x <= 8.4e+64))):
		tmp = 0.5 * x
	else:
		tmp = math.sqrt(z) * (0.5 * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.02e+30) || (!(x <= -1.4e-8) && ((x <= -5.2e-46) || !(x <= 8.4e+64))))
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(sqrt(z) * Float64(0.5 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.02e+30) || (~((x <= -1.4e-8)) && ((x <= -5.2e-46) || ~((x <= 8.4e+64)))))
		tmp = 0.5 * x;
	else
		tmp = sqrt(z) * (0.5 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.02e+30], And[N[Not[LessEqual[x, -1.4e-8]], $MachinePrecision], Or[LessEqual[x, -5.2e-46], N[Not[LessEqual[x, 8.4e+64]], $MachinePrecision]]]], N[(0.5 * x), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(0.5 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.02e30 or -1.4e-8 < x < -5.2000000000000004e-46 or 8.4000000000000001e64 < x

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. add-sqr-sqrt 52.0%

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

      3. sqrt-unprod 86.2%

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

      4. sqrt-prod 83.9%

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

      5. pow2 83.9%

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

   6. Applied egg-rr 83.9%

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

   7. Taylor expanded in x around inf 83.7%

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

   if -1.02e30 < x < -1.4e-8 or -5.2000000000000004e-46 < x < 8.4000000000000001e64

   1. Initial program 99.7%

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

      1. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. add-sqr-sqrt 48.6%

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

      3. sqrt-unprod 54.5%

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

      4. sqrt-prod 50.5%

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

      5. pow2 50.5%

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

   6. Applied egg-rr 50.5%

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

      1. flip-+ 37.9%

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

      2. add-sqr-sqrt 37.8%

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

      3. div-sub 37.8%

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

      4. pow2 37.8%

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

      5. *-commutative 37.8%

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

      6. sqrt-prod 37.8%

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

      7. unpow2 37.8%

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

      8. sqrt-prod 27.7%

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

      9. add-sqr-sqrt 38.6%

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

      10. *-commutative 38.6%

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

      11. sqrt-prod 38.8%

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

      12. unpow2 38.8%

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

      13. sqrt-prod 28.2%

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

      14. add-sqr-sqrt 52.1%

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

   8. Applied egg-rr 52.1%

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

      1. div-sub 52.1%

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

   10. Simplified 52.1%

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

   11. Taylor expanded in x around 0 35.9%

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

       1. associate-*r* 35.9%

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

       2. neg-mul-1 35.9%

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

       3. *-commutative 35.9%

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

   13. Simplified 35.9%

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

   14. Taylor expanded in y around inf 73.8%

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

       1. associate-*r* 73.8%

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

   16. Simplified 73.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+30} \lor \neg \left(x \leq -1.4 \cdot 10^{-8}\right) \land \left(x \leq -5.2 \cdot 10^{-46} \lor \neg \left(x \leq 8.4 \cdot 10^{+64}\right)\right):\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(0.5 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.6% accurate, 36.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. add-sqr-sqrt 50.2%

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

   3. sqrt-unprod 69.2%

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

   4. sqrt-prod 66.0%

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

   5. pow2 66.0%

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

6. Applied egg-rr 66.0%

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

7. Taylor expanded in x around inf 54.0%

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

8. Final simplification 54.0%

   \[\leadsto 0.5 \cdot x \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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)))))