3frac (problem 3.3.3)

Average Error: 10.0 → 0.1

Time: 7.3s

Precision: binary64

Cost: 704

Math

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

↓

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

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 (* x x))) (- -1.0 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 - (x * x))) / (-1.0 - 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 - (x * x))) / ((-1.0d0) - 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 - (x * x))) / (-1.0 - 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 - (x * x))) / (-1.0 - 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(x - Float64(x * x))) / Float64(-1.0 - 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 - (x * x))) / (-1.0 - 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[(x - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]

Target

Original	10.0
Target	0.3
Herbie	0.1

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

Derivation

1. Initial program 10.0

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

2. Simplified 10.0

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)} \]
   Proof
   (+.f64 (/.f64 1 (+.f64 1 x)) (+.f64 (/.f64 1 (+.f64 x -1)) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) (+.f64 (/.f64 1 (+.f64 x -1)) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (-.f64 x 1)) (/.f64 (Rewrite<= metadata-eval (neg.f64 2)) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (-.f64 x 1)) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2 x))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 1 (+.f64 x 1)) (Rewrite<= +-commutative_binary64 (+.f64 (neg.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)) (neg.f64 (/.f64 2 x))) (/.f64 1 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= sub-neg_binary64 (-.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 26.2

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

4. Simplified 26.2

   \[\leadsto \frac{1}{1 + x} + \color{blue}{\frac{\left(-2 + x \cdot 2\right) - x}{x - x \cdot x}} \]
   Proof
   (/.f64 (-.f64 (+.f64 -2 (*.f64 x 2)) x) (-.f64 x (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 -1 2)) (*.f64 x 2)) x) (-.f64 x (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 2 (+.f64 -1 x))) x) (-.f64 x (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 x -1))) x) (-.f64 x (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 x -1) 2)) x) (-.f64 x (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (+.f64 x -1) 2) 0)) x) (-.f64 x (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x -1) 2) 0) x) (-.f64 x (Rewrite<= unpow2_binary64 (pow.f64 x 2)))): 1 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x -1) 2) 0) x) (Rewrite<= unsub-neg_binary64 (+.f64 x (neg.f64 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x -1) 2) 0) x) (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 x))) (neg.f64 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x -1) 2) 0) x) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 x) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x -1) 2) 0) x) (neg.f64 (Rewrite=> +-commutative_binary64 (+.f64 (pow.f64 x 2) (neg.f64 x))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x -1) 2) 0) x) (neg.f64 (+.f64 (Rewrite=> unpow2_binary64 (*.f64 x x)) (neg.f64 x)))): 0 points increase in error, 1 points decrease in error
   (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x -1) 2) 0) x) (neg.f64 (+.f64 (*.f64 x x) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x -1) 2) 0) x) (neg.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 x (+.f64 x -1))))): 3 points increase in error, 0 points decrease in error
   (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (+.f64 (*.f64 (+.f64 x -1) 2) 0) (neg.f64 (*.f64 x (+.f64 x -1)))) (/.f64 x (neg.f64 (*.f64 x (+.f64 x -1)))))): 5 points increase in error, 3 points decrease in error
   (-.f64 (/.f64 (Rewrite=> +-rgt-identity_binary64 (*.f64 (+.f64 x -1) 2)) (neg.f64 (*.f64 x (+.f64 x -1)))) (/.f64 x (neg.f64 (*.f64 x (+.f64 x -1))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 (+.f64 x -1) 2) x) (neg.f64 (*.f64 x (+.f64 x -1))))): 3 points increase in error, 5 points decrease in error
   (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 (+.f64 x -1) 2) (neg.f64 x))) (neg.f64 (*.f64 x (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 x) (*.f64 (+.f64 x -1) 2))) (neg.f64 (*.f64 x (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x)) (*.f64 (+.f64 x -1) 2)) (neg.f64 (*.f64 x (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 x -1)) (*.f64 (+.f64 x -1) 2)) (neg.f64 (*.f64 x (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= fma-udef_binary64 (fma.f64 x -1 (*.f64 (+.f64 x -1) 2))) (neg.f64 (*.f64 x (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 x -1 (*.f64 (+.f64 x -1) 2)) (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 x) (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 x -1 (*.f64 (+.f64 x -1) 2)) (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 x -1) (neg.f64 x)))): 0 points increase in error, 0 points decrease in error

5. Applied egg-rr 25.9

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

6. Simplified 25.9

   \[\leadsto \color{blue}{\frac{\left(2 - x\right) \cdot \left(x + 1\right) - \left(x - x \cdot x\right)}{\left(x - x \cdot x\right) \cdot \left(-1 - x\right)}} \]
   Proof
   (/.f64 (-.f64 (*.f64 (-.f64 2 x) (+.f64 x 1)) (-.f64 x (*.f64 x x))) (*.f64 (-.f64 x (*.f64 x x)) (-.f64 -1 x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (*.f64 (Rewrite<= unsub-neg_binary64 (+.f64 2 (neg.f64 x))) (+.f64 x 1)) (-.f64 x (*.f64 x x))) (*.f64 (-.f64 x (*.f64 x x)) (-.f64 -1 x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (*.f64 (+.f64 (Rewrite<= metadata-eval (neg.f64 -2)) (neg.f64 x)) (+.f64 x 1)) (-.f64 x (*.f64 x x))) (*.f64 (-.f64 x (*.f64 x x)) (-.f64 -1 x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (*.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 -2 x))) (+.f64 x 1)) (-.f64 x (*.f64 x x))) (*.f64 (-.f64 x (*.f64 x x)) (-.f64 -1 x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (+.f64 -2 x) (+.f64 x 1)))) (-.f64 x (*.f64 x x))) (*.f64 (-.f64 x (*.f64 x x)) (-.f64 -1 x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (*.f64 (+.f64 -2 x) (+.f64 x 1))) (neg.f64 (-.f64 x (*.f64 x x))))) (*.f64 (-.f64 x (*.f64 x x)) (-.f64 -1 x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (*.f64 (+.f64 -2 x) (+.f64 x 1)) (-.f64 x (*.f64 x x))))) (*.f64 (-.f64 x (*.f64 x x)) (-.f64 -1 x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 x (*.f64 x x)) (*.f64 (+.f64 -2 x) (+.f64 x 1))))) (*.f64 (-.f64 x (*.f64 x x)) (-.f64 -1 x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (neg.f64 (+.f64 (-.f64 x (*.f64 x x)) (*.f64 (+.f64 -2 x) (+.f64 x 1)))) (*.f64 (-.f64 x (*.f64 x x)) (Rewrite<= unsub-neg_binary64 (+.f64 -1 (neg.f64 x))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (neg.f64 (+.f64 (-.f64 x (*.f64 x x)) (*.f64 (+.f64 -2 x) (+.f64 x 1)))) (*.f64 (-.f64 x (*.f64 x x)) (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 x) -1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (neg.f64 (+.f64 (-.f64 x (*.f64 x x)) (*.f64 (+.f64 -2 x) (+.f64 x 1)))) (*.f64 (-.f64 x (*.f64 x x)) (+.f64 (neg.f64 x) (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (neg.f64 (+.f64 (-.f64 x (*.f64 x x)) (*.f64 (+.f64 -2 x) (+.f64 x 1)))) (*.f64 (-.f64 x (*.f64 x x)) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 x 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (neg.f64 (+.f64 (-.f64 x (*.f64 x x)) (*.f64 (+.f64 -2 x) (+.f64 x 1)))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (-.f64 x (*.f64 x x)) (+.f64 x 1))))): 0 points increase in error, 0 points decrease in error

7. Taylor expanded in x around 0 0.3

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

8. Applied egg-rr 0.1

   \[\leadsto \color{blue}{0 + \frac{\frac{2}{x - x \cdot x}}{-1 - x}} \]

9. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.0
Cost	840

\[\begin{array}{l} t_0 := \frac{2}{\left(x \cdot x\right) \cdot \left(x + 1\right)}\\ \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot -2 + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.7
Cost	712

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

Alternative 3
Error	10.5
Cost	712

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

Alternative 4
Error	10.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;\frac{1}{x + 1} + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot -2 + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x} + \frac{-2}{x}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	704

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

Alternative 6
Error	0.1
Cost	704

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

Alternative 7
Error	15.2
Cost	584

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

Alternative 8
Error	30.8
Cost	192

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

Reproduce

herbie shell --seed 2022330 
(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))))