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

Average Accuracy: 90.8% → 99.8%

Time: 10.8s

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	90.8%
Target	99.8%
Herbie	99.8%

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

Derivation

1. Initial program 90.8%

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

2. Simplified 99.5%

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

   [Start] 90.8	\[ \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   associate-/l* [=>] 99.5	\[ \color{blue}{\frac{1 - x}{\frac{y \cdot 3}{3 - x}}} \]
   *-commutative [=>] 99.5	\[ \frac{1 - x}{\frac{\color{blue}{3 \cdot y}}{3 - x}} \]

3. Applied egg-rr 99.8%

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

   [Start] 99.5	\[ \frac{1 - x}{\frac{3 \cdot y}{3 - x}} \]
   associate-/l* [=>] 99.7	\[ \frac{1 - x}{\color{blue}{\frac{3}{\frac{3 - x}{y}}}} \]
   associate-/r/ [=>] 99.8	\[ \frac{1 - x}{\color{blue}{\frac{3}{3 - x} \cdot y}} \]

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	97.7%
Cost	841

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

Alternative 2
Accuracy	97.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{3}\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1}{y} + -1.3333333333333333 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{x}{3}}}\\ \end{array} \]

Alternative 3
Accuracy	97.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{1 - x}{-3 \cdot \frac{y}{x}}\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1}{y} + -1.3333333333333333 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{x}{3}}}\\ \end{array} \]

Alternative 4
Accuracy	96.9%
Cost	713

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

Alternative 5
Accuracy	97.0%
Cost	713

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

Alternative 6
Accuracy	97.0%
Cost	712

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

Alternative 7
Accuracy	97.0%
Cost	712

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

Alternative 8
Accuracy	97.0%
Cost	712

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

Alternative 9
Accuracy	97.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{3}\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{x}{3}}}\\ \end{array} \]

Alternative 10
Accuracy	99.7%
Cost	704

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

Alternative 11
Accuracy	99.8%
Cost	704

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

Alternative 12
Accuracy	66.8%
Cost	192

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

Reproduce

herbie shell --seed 2023151 
(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)))