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

Percentage Accurate: 99.8% → 99.8%

Time: 9.1s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(x + 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%

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

   2. +-commutative 99.8%

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

   3. fma-define 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 81.0%

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

   1. pow1 81.0%

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

   2. fma-define 81.1%

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

   3. inv-pow 81.1%

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

   4. sqrt-pow1 81.1%

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

   5. metadata-eval 81.1%

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

7. Applied egg-rr 81.1%

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

   1. unpow1 81.1%

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

   2. fma-undefine 81.0%

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

   3. +-commutative 81.0%

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

   4. distribute-lft-in 81.0%

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

   5. associate-*r/ 77.6%

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

   6. *-rgt-identity 77.6%

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

   7. times-frac 93.5%

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

   8. *-inverses 93.5%

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

   9. /-rgt-identity 93.5%

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

   10. *-lft-identity 93.5%

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

9. Simplified 93.5%

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

    1. *-commutative 93.5%

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

    2. metadata-eval 93.5%

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

    3. pow-flip 93.4%

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

    4. pow1/2 93.4%

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

    5. associate-/r/ 93.4%

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

    6. un-div-inv 94.1%

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

11. Applied egg-rr 94.1%

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

    1. associate-/r/ 99.8%

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

13. Simplified 99.8%

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

    1. clear-num 99.7%

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

    2. clear-num 99.7%

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

    3. pow1 99.7%

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

    4. pow1/2 99.7%

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

    5. pow-div 99.7%

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

    6. metadata-eval 99.7%

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

    7. pow1/2 99.7%

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

    8. associate-*l/ 99.8%

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

    9. *-un-lft-identity 99.8%

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

    10. pow1/2 99.8%

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

    11. pow-flip 99.9%

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

    12. metadata-eval 99.9%

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

15. Applied egg-rr 99.9%

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

Alternative 2: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-18} \lor \neg \left(y \leq 1.6 \cdot 10^{-28}\right):\\ \;\;\;\;\sqrt{z} \cdot \left(0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.55e-18) (not (<= y 1.6e-28)))
   (* (sqrt z) (* 0.5 y))
   (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.55e-18) || !(y <= 1.6e-28)) {
		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 <= (-1.55d-18)) .or. (.not. (y <= 1.6d-28))) 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 <= -1.55e-18) || !(y <= 1.6e-28)) {
		tmp = Math.sqrt(z) * (0.5 * y);
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.55e-18) or not (y <= 1.6e-28):
		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 <= -1.55e-18) || !(y <= 1.6e-28))
		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 <= -1.55e-18) || ~((y <= 1.6e-28)))
		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, -1.55e-18], N[Not[LessEqual[y, 1.6e-28]], $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 < -1.55000000000000003e-18 or 1.59999999999999991e-28 < y

   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%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 70.7%

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

      1. associate-*r* 70.7%

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

   7. Simplified 70.7%

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

   if -1.55000000000000003e-18 < y < 1.59999999999999991e-28

   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%

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

      2. +-commutative 99.9%

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

      3. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 78.0%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 78.0%

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

   7. Simplified 78.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-18} \lor \neg \left(y \leq 1.6 \cdot 10^{-28}\right):\\ \;\;\;\;\sqrt{z} \cdot \left(0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]
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: 50.7% 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%

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

   2. +-commutative 99.8%

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

   3. fma-define 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 52.9%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. Step-by-step derivation

   1. *-commutative 52.9%

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

7. Simplified 52.9%

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

8. Final simplification 52.9%

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

Reproduce

herbie shell --seed 2024111 
(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)))))