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

Percentage Accurate: 99.8% → 99.8%

Time: 7.6s

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 99.7%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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. Taylor expanded in x around inf 85.9%

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

   1. +-commutative 85.9%

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

   2. distribute-lft-in 85.9%

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

   3. associate-*l/ 88.2%

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

   4. associate-/l* 85.9%

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

   5. *-rgt-identity 85.9%

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

7. Applied egg-rr 85.9%

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

   1. associate-*r* 85.0%

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

   2. clear-num 85.0%

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

   3. un-div-inv 85.2%

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

   4. *-commutative 85.2%

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

9. Applied egg-rr 85.2%

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

    1. *-un-lft-identity 85.2%

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

    2. div-inv 85.2%

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

    3. metadata-eval 85.2%

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

    4. sqrt-div 85.3%

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

    5. times-frac 83.9%

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

    6. *-commutative 83.9%

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

    7. inv-pow 83.9%

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

    8. sqrt-pow1 83.9%

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

    9. metadata-eval 83.9%

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

11. Applied egg-rr 83.9%

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

    1. associate-*l/ 84.0%

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

    2. *-lft-identity 84.0%

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

    3. associate-/l* 88.9%

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

    4. associate-*l/ 99.8%

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

    5. *-inverses 99.8%

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

    6. *-lft-identity 99.8%

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

13. Simplified 99.8%

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

Alternative 2: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t\_0 \leq -20000000000000 \lor \neg \left(t\_0 \leq 100\right):\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -20000000000000.0) (not (<= t_0 100.0)))
     (* 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 <= -20000000000000.0) || !(t_0 <= 100.0)) {
		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 <= (-20000000000000.0d0)) .or. (.not. (t_0 <= 100.0d0))) 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 <= -20000000000000.0) || !(t_0 <= 100.0)) {
		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 <= -20000000000000.0) or not (t_0 <= 100.0):
		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 <= -20000000000000.0) || !(t_0 <= 100.0))
		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 <= -20000000000000.0) || ~((t_0 <= 100.0)))
		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, -20000000000000.0], N[Not[LessEqual[t$95$0, 100.0]], $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)) < -2e13 or 100 < (*.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. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. add-sqr-sqrt 99.1%

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

      4. associate-*l* 99.2%

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

      5. fma-define 99.2%

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

      6. pow1/2 99.2%

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

      7. sqrt-pow1 99.3%

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

      8. metadata-eval 99.3%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{\color{blue}{0.25}}, \sqrt{\sqrt{z}} \cdot y, x\right) \]

      9. pow1/2 99.3%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, \sqrt{\color{blue}{{z}^{0.5}}} \cdot y, x\right) \]

      10. sqrt-pow1 99.3%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, \color{blue}{{z}^{\left(\frac{0.5}{2}\right)}} \cdot y, x\right) \]

      11. metadata-eval 99.3%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, {z}^{\color{blue}{0.25}} \cdot y, x\right) \]

   6. Applied egg-rr 99.3%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left({z}^{0.25}, {z}^{0.25} \cdot y, x\right)} \]

   7. Taylor expanded in z around inf 80.5%

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

   if -2e13 < (*.f64 y (sqrt.f64 z)) < 100

   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. Add Preprocessing

   5. Taylor expanded in x around inf 81.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -20000000000000 \lor \neg \left(y \cdot \sqrt{z} \leq 100\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]
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 99.7%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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. 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 99.7%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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. Taylor expanded in x around inf 49.1%

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

Reproduce

herbie shell --seed 2024096 
(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)))))