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

Average Error: 5.7 → 0.1

Time: 10.1s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

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) * (((3.0 - x) / 3.0) / y);
}

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) * (((3.0d0 - x) / 3.0d0) / y)
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) * (((3.0 - x) / 3.0) / y);
}

Python

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

↓

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

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(1.0 - x) * Float64(Float64(Float64(3.0 - x) / 3.0) / y))
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) * (((3.0 - x) / 3.0) / y);
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[(1.0 - x), $MachinePrecision] * N[(N[(N[(3.0 - x), $MachinePrecision] / 3.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Target

Original	5.7
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 5.7

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

2. Simplified 0.1

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

   [Start] 5.7	\[ \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   rational.json-simplify-2 [=>] 5.7	\[ \frac{\color{blue}{\left(3 - x\right) \cdot \left(1 - x\right)}}{y \cdot 3} \]
   rational.json-simplify-49 [=>] 0.4	\[ \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]
   rational.json-simplify-2 [=>] 0.4	\[ \left(1 - x\right) \cdot \frac{3 - x}{\color{blue}{3 \cdot y}} \]
   rational.json-simplify-46 [=>] 0.1	\[ \left(1 - x\right) \cdot \color{blue}{\frac{\frac{3 - x}{3}}{y}} \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.6
Cost	840

\[\begin{array}{l} t_0 := x \cdot \left(\left(3 - x\right) \cdot \frac{-0.3333333333333333}{y}\right)\\ \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{1 - x \cdot 1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;\left(3 - x\right) \cdot \left(-0.3333333333333333 \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{1 - x \cdot 1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(3 - x\right) \cdot \frac{-0.3333333333333333}{y}\right)\\ \end{array} \]

Alternative 3
Error	1.5
Cost	840

\[\begin{array}{l} t_0 := \left(3 - x\right) \cdot \frac{\frac{x}{-3}}{y}\\ \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{1 - x \cdot 1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;\left(3 - x\right) \cdot \frac{\frac{x}{-3}}{y}\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{1 - x \cdot 1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(3 - x\right) \cdot \frac{\frac{x}{y}}{-3}\\ \end{array} \]

Alternative 5
Error	1.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;\left(3 - x\right) \cdot \frac{\frac{x}{-3}}{y}\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{1}{y} + \frac{x}{y} \cdot -1.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(3 - x\right) \cdot \frac{\frac{x}{y}}{-3}\\ \end{array} \]

Alternative 6
Error	1.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;\left(3 - x\right) \cdot \frac{\frac{x}{-3}}{y}\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{1}{y} + \frac{x}{y} \cdot -1.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{3 - x}{-3 \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 7
Error	2.1
Cost	712

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot \left(x \cdot 0.3333333333333333\right)\\ \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	1.6
Cost	712

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot \left(x \cdot 0.3333333333333333\right)\\ \mathbf{if}\;x \leq -4.7:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1 - x \cdot 1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	21.1
Cost	192

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

Reproduce

herbie shell --seed 2023075 
(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)))