3frac (problem 3.3.3)

Average Error: 9.8 → 0.1

Time: 9.8s

Precision: binary64

Cost: 576

Math

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

↓

\[\frac{\frac{-2}{x}}{1 - x \cdot x} \]

FPCore

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

↓

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

Target

Original	9.8
Target	0.3
Herbie	0.1

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

Derivation

1. Initial program 9.8

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

2. Simplified 9.8

   \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]
   Proof
   (-.f64 (/.f64 1 (+.f64 1 x)) (-.f64 (/.f64 2 x) (/.f64 1 (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) (-.f64 (/.f64 2 x) (/.f64 1 (+.f64 x -1)))): 0 points increase in error, 10 points decrease in error
   (-.f64 (/.f64 1 (+.f64 x 1)) (-.f64 (/.f64 2 x) (/.f64 1 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))))): 0 points increase in error, 10 points decrease in error
   (-.f64 (/.f64 1 (+.f64 x 1)) (-.f64 (/.f64 2 x) (/.f64 1 (Rewrite<= sub-neg_binary64 (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 1 (+.f64 x 1)) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 (/.f64 2 x) (/.f64 1 (-.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (/.f64 1 (+.f64 x 1)) (*.f64 (neg.f64 1) (-.f64 (/.f64 2 x) (/.f64 1 (-.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 x 1)) (*.f64 (Rewrite=> metadata-eval -1) (-.f64 (/.f64 2 x) (/.f64 1 (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 x 1)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (/.f64 2 x) (/.f64 1 (-.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 (+.f64 x 1)) (-.f64 (/.f64 2 x) (/.f64 1 (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 26.4

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

4. Applied egg-rr 25.5

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

5. Simplified 25.5

   \[\leadsto \color{blue}{\frac{\frac{\left(-1 + x \cdot x\right) + \left(\left(x + -2\right) \cdot \frac{-1 - x}{x}\right) \cdot \left(x + 1\right)}{-1 - x}}{1 - x \cdot x}} \]
   Proof
   (/.f64 (/.f64 (+.f64 (+.f64 -1 (*.f64 x x)) (*.f64 (*.f64 (+.f64 x -2) (/.f64 (-.f64 -1 x) x)) (+.f64 x 1))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (*.f64 x x)) (*.f64 (*.f64 (+.f64 x -2) (/.f64 (-.f64 -1 x) x)) (+.f64 x 1))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 0 points increase in error, 15 points decrease in error
   (/.f64 (/.f64 (+.f64 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 (*.f64 x x)))) (*.f64 (*.f64 (+.f64 x -2) (/.f64 (-.f64 -1 x) x)) (+.f64 x 1))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 0 points increase in error, 15 points decrease in error
   (/.f64 (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 1 (*.f64 x x)))) (*.f64 (*.f64 (+.f64 x -2) (/.f64 (-.f64 -1 x) x)) (+.f64 x 1))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 15 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (-.f64 1 (*.f64 x x))) (*.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 -2 x)) (/.f64 (-.f64 -1 x) x)) (+.f64 x 1))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 0 points increase in error, 15 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (-.f64 1 (*.f64 x x))) (*.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (+.f64 -2 x) 1)) (/.f64 (-.f64 -1 x) x)) (+.f64 x 1))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 15 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (-.f64 1 (*.f64 x x))) (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 -2 x) (-.f64 -1 x)) (*.f64 1 x))) (+.f64 x 1))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (-.f64 1 (*.f64 x x))) (*.f64 (/.f64 (*.f64 (+.f64 -2 x) (-.f64 -1 x)) (Rewrite=> *-lft-identity_binary64 x)) (+.f64 x 1))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 0 points increase in error, 15 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (-.f64 1 (*.f64 x x))) (*.f64 (/.f64 (*.f64 (+.f64 -2 x) (-.f64 -1 x)) x) (Rewrite=> +-commutative_binary64 (+.f64 1 x)))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 15 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (-.f64 1 (*.f64 x x))) (*.f64 (/.f64 (*.f64 (+.f64 -2 x) (-.f64 -1 x)) x) (+.f64 (Rewrite<= metadata-eval (-.f64 0 -1)) x))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 0 points increase in error, 15 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (-.f64 1 (*.f64 x x))) (*.f64 (/.f64 (*.f64 (+.f64 -2 x) (-.f64 -1 x)) x) (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 -1 x))))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 15 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (-.f64 1 (*.f64 x x))) (*.f64 (/.f64 (*.f64 (+.f64 -2 x) (-.f64 -1 x)) x) (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 -1 x))))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 15 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (-.f64 1 (*.f64 x x))) (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (-.f64 -1 x)) (/.f64 (*.f64 (+.f64 -2 x) (-.f64 -1 x)) x)))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (neg.f64 (-.f64 1 (*.f64 x x))) (*.f64 (-.f64 -1 x) (/.f64 (*.f64 (+.f64 -2 x) (-.f64 -1 x)) x)))) (-.f64 -1 x)) (-.f64 1 (*.f64 x x))): 0 points increase in error, 15 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 (-.f64 (neg.f64 (-.f64 1 (*.f64 x x))) (*.f64 (-.f64 -1 x) (/.f64 (*.f64 (+.f64 -2 x) (-.f64 -1 x)) x))) (*.f64 (-.f64 -1 x) (-.f64 1 (*.f64 x x))))): 15 points increase in error, 0 points decrease in error

6. Taylor expanded in x around inf 31.6

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

7. Taylor expanded in x around inf 0.1

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

8. Final simplification 0.1

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

Alternatives

Alternative 1
Error	11.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+51}:\\ \;\;\;\;1 + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Error	10.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	10.7
Cost	448

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

Alternative 4
Error	61.9
Cost	64

\[-2 \]

Alternative 5
Error	41.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022349 
(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))))