3frac (problem 3.3.3)

Average Error: 9.7 → 0.1

Time: 9.2s

Precision: binary64

Cost: 704

Math

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

↓

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

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

Target

Original	9.7
Target	0.3
Herbie	0.1

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

Derivation

1. Initial program 9.7

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

2. Simplified 9.7

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)} \]
   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 25.9

   \[\leadsto \frac{1}{1 + x} + \color{blue}{\frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)}} \]

4. Applied egg-rr 9.7

   \[\leadsto \color{blue}{\frac{\left(1 - x\right) + \left(1 + x\right) \cdot \frac{\mathsf{fma}\left(x, 0.5, 1 - x\right)}{x \cdot -0.5}}{\left(1 + x\right) \cdot \left(1 - x\right)}} \]

5. Taylor expanded in x around 0 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.7
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{-2}{x}}{x \cdot \left(-x\right)}\\ \mathbf{if}\;x \leq -50.943474180845776:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0057372745632231605:\\ \;\;\;\;-2 \cdot x + -2 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.8
Cost	776

\[\begin{array}{l} t_0 := \frac{\frac{-2}{x}}{x \cdot \left(-x\right)}\\ \mathbf{if}\;x \leq -50.943474180845776:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0057372745632231605:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	15.9
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{-2}{x}}{x}\\ \mathbf{if}\;x \leq -50.943474180845776:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0057372745632231605:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	10.7
Cost	448

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

Alternative 5
Error	31.0
Cost	192

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

Alternative 6
Error	61.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022306 
(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))))