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

Percentage Accurate: 99.8% → 99.8%

Time: 6.9s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(N[(y / N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision] + x), $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 83.4%

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

   1. distribute-rgt-in 83.5%

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

   2. associate-*l* 87.6%

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

   3. fma-define 87.6%

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

   4. *-commutative 87.6%

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

   5. associate-*l/ 86.8%

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

   6. associate-/l* 99.7%

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

   7. *-inverses 99.7%

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

   8. *-rgt-identity 99.7%

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

7. Simplified 99.7%

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

   1. fma-undefine 99.8%

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

   2. *-commutative 99.8%

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

   3. sqrt-div 99.7%

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

   4. metadata-eval 99.7%

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

   5. un-div-inv 99.8%

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

9. Applied egg-rr 99.8%

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

    1. clear-num 99.7%

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

    2. pow1/2 99.7%

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

    3. pow1 99.7%

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

    4. pow-div 99.8%

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

    5. metadata-eval 99.8%

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

    6. metadata-eval 99.8%

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

    7. sqrt-pow1 99.7%

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

    8. inv-pow 99.7%

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

    9. associate-*l/ 99.8%

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

    10. *-un-lft-identity 99.8%

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

    11. inv-pow 99.8%

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

    12. sqrt-pow1 99.8%

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

    13. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

Alternative 2: 72.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-77} \lor \neg \left(x \leq 3.4 \cdot 10^{-87}\right):\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.8e-77) (not (<= x 3.4e-87)))
   (* 0.5 x)
   (* 0.5 (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e-77) || !(x <= 3.4e-87)) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (y * sqrt(z));
	}
	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 <= (-6.8d-77)) .or. (.not. (x <= 3.4d-87))) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * (y * sqrt(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e-77) || !(x <= 3.4e-87)) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (y * Math.sqrt(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6.8e-77) or not (x <= 3.4e-87):
		tmp = 0.5 * x
	else:
		tmp = 0.5 * (y * math.sqrt(z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.8e-77) || !(x <= 3.4e-87))
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(y * sqrt(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -6.8e-77) || ~((x <= 3.4e-87)))
		tmp = 0.5 * x;
	else
		tmp = 0.5 * (y * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.8e-77], N[Not[LessEqual[x, 3.4e-87]], $MachinePrecision]], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.79999999999999966e-77 or 3.3999999999999999e-87 < 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%

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

      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 76.4%

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

   if -6.79999999999999966e-77 < x < 3.3999999999999999e-87

   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%

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

      2. +-commutative 99.6%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around inf 89.7%

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

      1. distribute-rgt-in 89.7%

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

      2. associate-*l* 98.5%

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

      3. fma-define 98.5%

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

      4. *-commutative 98.5%

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

      5. associate-*l/ 89.8%

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

      6. associate-/l* 99.5%

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

      7. *-inverses 99.5%

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

      8. *-rgt-identity 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in y around inf 83.3%

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

      1. *-commutative 83.3%

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

   10. Simplified 83.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-77} \lor \neg \left(x \leq 3.4 \cdot 10^{-87}\right):\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \end{array} \]
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: 50.5% 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 56.0%

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

Reproduce

herbie shell --seed 2024106 
(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)))))