Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3

Average Error: 8.63% → 0.21%

Time: 10.0s

Precision: binary64

Cost: 704

Math

\[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]

↓

\[\frac{1 - x}{y} \cdot \left(1 - \frac{x}{3}\right) \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))

↓

(FPCore (x y) :precision binary64 (* (/ (- 1.0 x) y) (- 1.0 (/ x 3.0))))

C

double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

↓

double code(double x, double y) {
	return ((1.0 - x) / y) * (1.0 - (x / 3.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) * (3.0d0 - x)) / (y * 3.0d0)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) / y) * (1.0d0 - (x / 3.0d0))
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

↓

public static double code(double x, double y) {
	return ((1.0 - x) / y) * (1.0 - (x / 3.0));
}

Python

def code(x, y):
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0)

↓

def code(x, y):
	return ((1.0 - x) / y) * (1.0 - (x / 3.0))

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) * Float64(3.0 - x)) / Float64(y * 3.0))
end

↓

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) / y) * Float64(1.0 - Float64(x / 3.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) * (3.0 - x)) / (y * 3.0);
end

↓

function tmp = code(x, y)
	tmp = ((1.0 - x) / y) * (1.0 - (x / 3.0));
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 - N[(x / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	8.63%
Target	0.21%
Herbie	0.21%

\[\frac{1 - x}{y} \cdot \frac{3 - x}{3} \]

Derivation

1. Initial program 8.63

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]

2. Simplified 0.21

   \[\leadsto \color{blue}{\frac{1 - x}{y} \cdot \left(1 - \frac{x}{3}\right)} \]
   Proof

   [Start] 8.63	\[ \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   times-frac [=>] 0.21	\[ \color{blue}{\frac{1 - x}{y} \cdot \frac{3 - x}{3}} \]
   div-sub [=>] 0.21	\[ \frac{1 - x}{y} \cdot \color{blue}{\left(\frac{3}{3} - \frac{x}{3}\right)} \]
   metadata-eval [=>] 0.21	\[ \frac{1 - x}{y} \cdot \left(\color{blue}{1} - \frac{x}{3}\right) \]

3. Final simplification 0.21

   \[\leadsto \frac{1 - x}{y} \cdot \left(1 - \frac{x}{3}\right) \]

Alternatives

Alternative 1
Error	1.33%
Cost	904

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;\frac{x}{y} \cdot \left(0.3333333333333333 \cdot \left(x + -4\right)\right)\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;\frac{1}{y} + \frac{x \cdot -1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{-3 \cdot \frac{y}{x + -4}}\\ \end{array} \]

Alternative 2
Error	1.34%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 1.75\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(0.3333333333333333 \cdot \left(x + -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \]

Alternative 3
Error	1.33%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;\frac{x}{y} \cdot \left(0.3333333333333333 \cdot \left(x + -4\right)\right)\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{3} \cdot \frac{x + -4}{y}\\ \end{array} \]

Alternative 4
Error	1.32%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;\frac{x}{y} \cdot \left(0.3333333333333333 \cdot \left(x + -4\right)\right)\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;\frac{1}{y} + \frac{x \cdot -1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{3} \cdot \frac{x + -4}{y}\\ \end{array} \]

Alternative 5
Error	2.82%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 5.1\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]

Alternative 6
Error	2.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 5.1\right):\\ \;\;\;\;x \cdot \frac{x}{\frac{y}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]

Alternative 7
Error	2.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;x \cdot \frac{x}{\frac{y}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \frac{1}{y}\\ \end{array} \]

Alternative 8
Error	2.04%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \lor \neg \left(x \leq 0.64\right):\\ \;\;\;\;x \cdot \frac{x}{\frac{y}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \]

Alternative 9
Error	2.82%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 5.1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 10
Error	0.64%
Cost	704

\[\left(1 - x\right) \cdot \left(\left(3 - x\right) \cdot \frac{0.3333333333333333}{y}\right) \]

Alternative 11
Error	0.25%
Cost	704

\[\left(1 - x\right) \cdot \frac{1 + x \cdot -0.3333333333333333}{y} \]

Alternative 12
Error	32.22%
Cost	192

\[\frac{1}{y} \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (* (/ (- 1.0 x) y) (/ (- 3.0 x) 3.0))

  (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))