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

Average Error: 5.7 → 0.1

Time: 8.5s

Precision: binary64

Cost: 704

Math

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

↓

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

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 (- 3.0 x)) 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 / (3.0 - x)) * 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 / (3.0d0 - x)) * 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 / (3.0 - x)) * 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 / (3.0 - x)) * 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(3.0 / Float64(3.0 - x)) * 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 / (3.0 - x)) * 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[(3.0 / N[(3.0 - x), $MachinePrecision]), $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

   \[\leadsto \color{blue}{\frac{1 - x}{y} \cdot \frac{3 - x}{3}} \]
   Proof
   (*.f64 (/.f64 (-.f64 1 x) y) (/.f64 (-.f64 3 x) 3)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 1 x) (-.f64 3 x)) (*.f64 y 3))): 70 points increase in error, 19 points decrease in error

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.1
Cost	840

\[\begin{array}{l} t_0 := \frac{x}{3 \cdot y} \cdot \left(x + -4\right)\\ \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.1
Cost	840

\[\begin{array}{l} t_0 := \frac{x}{3 \cdot y} \cdot \left(x + -4\right)\\ \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;\frac{1}{y} + \frac{x}{y} \cdot -1.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	2.0
Cost	712

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

Alternative 4
Error	2.0
Cost	712

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

Alternative 5
Error	2.0
Cost	712

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

Alternative 6
Error	2.0
Cost	712

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

Alternative 7
Error	2.0
Cost	712

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

Alternative 8
Error	2.0
Cost	712

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

Alternative 9
Error	1.6
Cost	712

\[\begin{array}{l} t_0 := x \cdot \frac{x}{3 \cdot y}\\ \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	0.4
Cost	704

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

Alternative 11
Error	0.1
Cost	704

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

Alternative 12
Error	21.5
Cost	192

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

Reproduce

herbie shell --seed 2022325 
(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)))