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

Percentage Accurate: 99.8% → 99.8%

Time: 12.7s

Precision: binary64

Cost: 6912

Math

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

↓

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

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 (pow z -0.5)))))

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 / 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 = (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 / (z ** (-0.5d0))))
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.pow(z, -0.5)));
}

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.pow(z, -0.5)))

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 / (z ^ -0.5))))
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 / (z ^ -0.5)));
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[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Simplified 99.9%

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

   [Start] 99.9%	\[ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   metadata-eval [=>] 99.9%	\[ \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]

3. Applied egg-rr 99.1%

   \[\leadsto 0.5 \cdot \left(x + \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{z}}\right)}^{3}}\right) \]
   Step-by-step derivation

   [Start] 99.9%	\[ 0.5 \cdot \left(x + y \cdot \sqrt{z}\right) \]
   add-cube-cbrt [=>] 99.1%	\[ 0.5 \cdot \left(x + \color{blue}{\left(\sqrt[3]{y \cdot \sqrt{z}} \cdot \sqrt[3]{y \cdot \sqrt{z}}\right) \cdot \sqrt[3]{y \cdot \sqrt{z}}}\right) \]
   pow3 [=>] 99.1%	\[ 0.5 \cdot \left(x + \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{z}}\right)}^{3}}\right) \]

4. Applied egg-rr 99.5%

   \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left(y \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{\sqrt{z}}}\right) \]
   Step-by-step derivation

   [Start] 99.1%	\[ 0.5 \cdot \left(x + {\left(\sqrt[3]{y \cdot \sqrt{z}}\right)}^{3}\right) \]
   rem-cube-cbrt [=>] 99.9%	\[ 0.5 \cdot \left(x + \color{blue}{y \cdot \sqrt{z}}\right) \]
   add-cbrt-cube [=>] 79.1%	\[ 0.5 \cdot \left(x + y \cdot \color{blue}{\sqrt[3]{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \sqrt{z}}}\right) \]
   add-sqr-sqrt [<=] 79.1%	\[ 0.5 \cdot \left(x + y \cdot \sqrt[3]{\color{blue}{z} \cdot \sqrt{z}}\right) \]
   cbrt-prod [=>] 99.5%	\[ 0.5 \cdot \left(x + y \cdot \color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{\sqrt{z}}\right)}\right) \]
   associate-*r* [=>] 99.5%	\[ 0.5 \cdot \left(x + \color{blue}{\left(y \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{\sqrt{z}}}\right) \]

5. Applied egg-rr 99.9%

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

   [Start] 99.5%	\[ 0.5 \cdot \left(x + \left(y \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{\sqrt{z}}\right) \]
   associate-*l* [=>] 99.5%	\[ 0.5 \cdot \left(x + \color{blue}{y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{\sqrt{z}}\right)}\right) \]
   cbrt-prod [<=] 79.1%	\[ 0.5 \cdot \left(x + y \cdot \color{blue}{\sqrt[3]{z \cdot \sqrt{z}}}\right) \]
   add-sqr-sqrt [=>] 79.1%	\[ 0.5 \cdot \left(x + y \cdot \sqrt[3]{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)} \cdot \sqrt{z}}\right) \]
   add-cbrt-cube [<=] 99.9%	\[ 0.5 \cdot \left(x + y \cdot \color{blue}{\sqrt{z}}\right) \]
   remove-double-div [<=] 99.8%	\[ 0.5 \cdot \left(x + y \cdot \color{blue}{\frac{1}{\frac{1}{\sqrt{z}}}}\right) \]
   metadata-eval [<=] 99.8%	\[ 0.5 \cdot \left(x + y \cdot \frac{1}{\frac{\color{blue}{\sqrt{1}}}{\sqrt{z}}}\right) \]
   sqrt-div [<=] 99.8%	\[ 0.5 \cdot \left(x + y \cdot \frac{1}{\color{blue}{\sqrt{\frac{1}{z}}}}\right) \]
   div-inv [<=] 99.8%	\[ 0.5 \cdot \left(x + \color{blue}{\frac{y}{\sqrt{\frac{1}{z}}}}\right) \]
   inv-pow [=>] 99.8%	\[ 0.5 \cdot \left(x + \frac{y}{\sqrt{\color{blue}{{z}^{-1}}}}\right) \]
   sqrt-pow1 [=>] 99.9%	\[ 0.5 \cdot \left(x + \frac{y}{\color{blue}{{z}^{\left(\frac{-1}{2}\right)}}}\right) \]
   metadata-eval [=>] 99.9%	\[ 0.5 \cdot \left(x + \frac{y}{{z}^{\color{blue}{-0.5}}}\right) \]

6. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	6912

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

Alternative 2
Accuracy	73.6%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	6848

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

Alternative 4
Accuracy	50.8%
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2023271 
(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)))))