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

Average Error: 0.1 → 0.1

Time: 7.3s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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 = (1.0d0 / 2.0d0) * (x + (y * sqrt(z)))
end function

↓

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 (1.0 / 2.0) * (x + (y * Math.sqrt(z)));
}

↓

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 (1.0 / 2.0) * (x + (y * math.sqrt(z)))

↓

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

↓

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

MATLAB

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

↓

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

↓

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   metadata-eval [=>] 0.1	\[ \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	17.1
Cost	33425

\[\begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;t_0 \leq 10^{-64} \lor \neg \left(t_0 \leq 2 \cdot 10^{-47}\right) \land t_0 \leq 10^{+64}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{{z}^{-0.5}}\\ \end{array} \]

Alternative 2
Error	17.1
Cost	33424

\[\begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;t_0 \leq 10^{-64}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-47}:\\ \;\;\;\;0.5 \cdot \sqrt{y \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;t_0 \leq 10^{+64}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{{z}^{-0.5}}\\ \end{array} \]

Alternative 3
Error	17.1
Cost	33362

\[\begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-29} \lor \neg \left(t_0 \leq 10^{-64}\right) \land \left(t_0 \leq 2 \cdot 10^{-47} \lor \neg \left(t_0 \leq 10^{+64}\right)\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 4
Error	28.6
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2023016 
(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)))))