3frac (problem 3.3.3)

Average Error: 9.9 → 0.1

Time: 1.4min

Precision: binary64

Cost: 704

Math

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

↓

\[\frac{\frac{-2}{-1 - x}}{\left(-1 + x\right) \cdot x} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

↓

(FPCore (x) :precision binary64 (/ (/ -2.0 (- -1.0 x)) (* (+ -1.0 x) x)))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

double code(double x) {
	return (-2.0 / (-1.0 - x)) / ((-1.0 + x) * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-2.0d0) / ((-1.0d0) - x)) / (((-1.0d0) + x) * x)
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

public static double code(double x) {
	return (-2.0 / (-1.0 - x)) / ((-1.0 + x) * x);
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

↓

def code(x):
	return (-2.0 / (-1.0 - x)) / ((-1.0 + x) * x)

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

↓

function code(x)
	return Float64(Float64(-2.0 / Float64(-1.0 - x)) / Float64(Float64(-1.0 + x) * x))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

↓

function tmp = code(x)
	tmp = (-2.0 / (-1.0 - x)) / ((-1.0 + x) * x);
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(-2.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Target

Original	9.9
Target	0.2
Herbie	0.1

\[\frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

1. Initial program 9.9

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

2. Simplified 9.9

   \[\leadsto \color{blue}{\frac{-1}{-1 - x} + \left(\frac{-2}{x} + \frac{-1}{1 - x}\right)} \]
   Proof

3. Applied egg-rr 25.8

   \[\leadsto \color{blue}{\frac{\left(\left(-1 + x\right) \cdot 2 - x\right) \cdot \left(-1 - x\right) + \left(-1 + x\right) \cdot x}{\left(\left(1 - x\right) \cdot x\right) \cdot \left(-1 - x\right)}} \]

4. Taylor expanded in x around 0 0.2

   \[\leadsto \frac{\color{blue}{2}}{\left(\left(1 - x\right) \cdot x\right) \cdot \left(-1 - x\right)} \]

5. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{\frac{-2}{-1 - x}}{\left(-1 + x\right) \cdot x}} \]

Alternatives

Alternative 1
Error	0.9
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{2}{x}}{\left(-1 + x\right) \cdot x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;\frac{-2}{x} - \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.2
Cost	704

\[\frac{2}{\left(\left(1 - x\right) \cdot x\right) \cdot \left(-1 - x\right)} \]

Alternative 3
Error	10.7
Cost	448

\[1 + \left(\frac{-2}{x} + -1\right) \]

Alternative 4
Error	30.8
Cost	192

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

Alternative 5
Error	61.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))