Numeric.Interval.Internal:bisect from intervals-0.7.1, A

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Precision: binary64

Cost: 320

• Unsound rule application detected in e-graph. Results may not be sound.

Math

\[x + \frac{y - x}{2} \]

↓

\[0.5 \cdot \left(y + x\right) \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (- y x) 2.0)))

↓

(FPCore (x y) :precision binary64 (* 0.5 (+ y x)))

C

double code(double x, double y) {
	return x + ((y - x) / 2.0);
}

↓

double code(double x, double y) {
	return 0.5 * (y + x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((y - x) / 2.0d0)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 * (y + x)
end function

Java

public static double code(double x, double y) {
	return x + ((y - x) / 2.0);
}

↓

public static double code(double x, double y) {
	return 0.5 * (y + x);
}

Python

def code(x, y):
	return x + ((y - x) / 2.0)

↓

def code(x, y):
	return 0.5 * (y + x)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(y - x) / 2.0))
end

↓

function code(x, y)
	return Float64(0.5 * Float64(y + x))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((y - x) / 2.0);
end

↓

function tmp = code(x, y)
	tmp = 0.5 * (y + x);
end

Wolfram

code[x_, y_] := N[(x + N[(N[(y - x), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(0.5 * N[(y + x), $MachinePrecision]), $MachinePrecision]

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

\[0.5 \cdot \left(x + y\right) \]

Derivation

1. Initial program 100.0%

   \[x + \frac{y - x}{2} \]

2. Simplified 100.0%

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

   [Start] 100.0%	\[ x + \frac{y - x}{2} \]
   +-commutative [=>] 100.0%	\[ \color{blue}{\frac{y - x}{2} + x} \]
   div-sub [=>] 100.0%	\[ \color{blue}{\left(\frac{y}{2} - \frac{x}{2}\right)} + x \]
   sub-neg [=>] 100.0%	\[ \color{blue}{\left(\frac{y}{2} + \left(-\frac{x}{2}\right)\right)} + x \]
   associate-+l+ [=>] 100.0%	\[ \color{blue}{\frac{y}{2} + \left(\left(-\frac{x}{2}\right) + x\right)} \]
   *-lft-identity [<=] 100.0%	\[ \frac{\color{blue}{1 \cdot y}}{2} + \left(\left(-\frac{x}{2}\right) + x\right) \]
   metadata-eval [<=] 100.0%	\[ \frac{\color{blue}{\left(--1\right)} \cdot y}{2} + \left(\left(-\frac{x}{2}\right) + x\right) \]
   associate-/l* [=>] 99.8%	\[ \color{blue}{\frac{--1}{\frac{2}{y}}} + \left(\left(-\frac{x}{2}\right) + x\right) \]
   associate-/r/ [=>] 100.0%	\[ \color{blue}{\frac{--1}{2} \cdot y} + \left(\left(-\frac{x}{2}\right) + x\right) \]
   metadata-eval [=>] 100.0%	\[ \frac{\color{blue}{1}}{2} \cdot y + \left(\left(-\frac{x}{2}\right) + x\right) \]
   metadata-eval [=>] 100.0%	\[ \color{blue}{0.5} \cdot y + \left(\left(-\frac{x}{2}\right) + x\right) \]
   metadata-eval [<=] 100.0%	\[ \color{blue}{\left(1 + -0.5\right)} \cdot y + \left(\left(-\frac{x}{2}\right) + x\right) \]
   metadata-eval [<=] 100.0%	\[ \left(\color{blue}{\left(--1\right)} + -0.5\right) \cdot y + \left(\left(-\frac{x}{2}\right) + x\right) \]
   metadata-eval [<=] 100.0%	\[ \left(\left(--1\right) + \color{blue}{\frac{-1}{2}}\right) \cdot y + \left(\left(-\frac{x}{2}\right) + x\right) \]
   *-commutative [=>] 100.0%	\[ \color{blue}{y \cdot \left(\left(--1\right) + \frac{-1}{2}\right)} + \left(\left(-\frac{x}{2}\right) + x\right) \]
   +-commutative [=>] 100.0%	\[ y \cdot \left(\left(--1\right) + \frac{-1}{2}\right) + \color{blue}{\left(x + \left(-\frac{x}{2}\right)\right)} \]
   *-lft-identity [<=] 100.0%	\[ y \cdot \left(\left(--1\right) + \frac{-1}{2}\right) + \left(\color{blue}{1 \cdot x} + \left(-\frac{x}{2}\right)\right) \]
   metadata-eval [<=] 100.0%	\[ y \cdot \left(\left(--1\right) + \frac{-1}{2}\right) + \left(\color{blue}{\left(--1\right)} \cdot x + \left(-\frac{x}{2}\right)\right) \]
   distribute-neg-frac [=>] 100.0%	\[ y \cdot \left(\left(--1\right) + \frac{-1}{2}\right) + \left(\left(--1\right) \cdot x + \color{blue}{\frac{-x}{2}}\right) \]
   neg-mul-1 [=>] 100.0%	\[ y \cdot \left(\left(--1\right) + \frac{-1}{2}\right) + \left(\left(--1\right) \cdot x + \frac{\color{blue}{-1 \cdot x}}{2}\right) \]
   associate-/l* [=>] 99.9%	\[ y \cdot \left(\left(--1\right) + \frac{-1}{2}\right) + \left(\left(--1\right) \cdot x + \color{blue}{\frac{-1}{\frac{2}{x}}}\right) \]
   associate-/r/ [=>] 100.0%	\[ y \cdot \left(\left(--1\right) + \frac{-1}{2}\right) + \left(\left(--1\right) \cdot x + \color{blue}{\frac{-1}{2} \cdot x}\right) \]
   distribute-rgt-out [=>] 100.0%	\[ y \cdot \left(\left(--1\right) + \frac{-1}{2}\right) + \color{blue}{x \cdot \left(\left(--1\right) + \frac{-1}{2}\right)} \]
   distribute-rgt-in [<=] 100.0%	\[ \color{blue}{\left(\left(--1\right) + \frac{-1}{2}\right) \cdot \left(y + x\right)} \]
   metadata-eval [=>] 100.0%	\[ \left(\color{blue}{1} + \frac{-1}{2}\right) \cdot \left(y + x\right) \]
   metadata-eval [=>] 100.0%	\[ \left(1 + \color{blue}{-0.5}\right) \cdot \left(y + x\right) \]
   metadata-eval [=>] 100.0%	\[ \color{blue}{0.5} \cdot \left(y + x\right) \]

3. Final simplification 100.0%

   \[\leadsto 0.5 \cdot \left(y + x\right) \]

Reproduce

herbie shell --seed 2023272 
(FPCore (x y)
  :name "Numeric.Interval.Internal:bisect from intervals-0.7.1, A"
  :precision binary64

  :herbie-target
  (* 0.5 (+ x y))

  (+ x (/ (- y x) 2.0)))