3frac (problem 3.3.3)

Average Accuracy: 84.6% → 99.6%

Time: 8.8s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ -2.0 (* (+ x 1.0) (- x (* 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 / ((x + 1.0) * (x - (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) / ((x + 1.0d0) * (x - (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 / ((x + 1.0) * (x - (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 / ((x + 1.0) * (x - (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(-2.0 / Float64(Float64(x + 1.0) * Float64(x - Float64(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 / ((x + 1.0) * (x - (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[(-2.0 / N[(N[(x + 1.0), $MachinePrecision] * N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	84.6%
Target	99.6%
Herbie	99.6%

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

Derivation

1. Initial program 86.5%

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

2. Simplified 86.5%

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

3. Applied egg-rr 60.8%

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

4. Simplified 60.8%

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

5. Taylor expanded in x around 0 52.8%

   \[\leadsto \frac{1}{1 + x} - \frac{\color{blue}{-2}}{x \cdot x - x} \]

6. Applied egg-rr 60.4%

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

7. Taylor expanded in x around 0 99.6%

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

8. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	76.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} - x\\ \end{array} \]

Alternative 2
Accuracy	76.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x}\\ \end{array} \]

Alternative 3
Accuracy	83.3%
Cost	448

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

Alternative 4
Accuracy	51.6%
Cost	192

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

Alternative 5
Accuracy	3.3%
Cost	64

\[-1 \]

Alternative 6
Accuracy	3.2%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023157 -o generate:proofs
(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))))