3frac (problem 3.3.3)

Average Error: 9.6 → 0.9

Time: 7.1s

Precision: binary64

Cost: 8712

Math

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

↓

\[\begin{array}{l} t_0 := \left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{1}{x + -1}\\ t_1 := x \cdot x - x\\ \mathbf{if}\;t_0 \leq -200000000000:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \left(1 + x\right) \cdot \left(x + -2 \cdot \left(x + -1\right)\right)}{t_1 \cdot \left(1 + x\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (+ (/ -2.0 x) (/ 1.0 (+ 1.0 x))) (/ 1.0 (+ x -1.0))))
        (t_1 (- (* x x) x)))
   (if (<= t_0 -200000000000.0)
     (/ -2.0 x)
     (if (<= t_0 4e-28)
       (* 2.0 (pow x -3.0))
       (/
        (+ t_1 (* (+ 1.0 x) (+ x (* -2.0 (+ x -1.0)))))
        (* t_1 (+ 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) {
	double t_0 = ((-2.0 / x) + (1.0 / (1.0 + x))) + (1.0 / (x + -1.0));
	double t_1 = (x * x) - x;
	double tmp;
	if (t_0 <= -200000000000.0) {
		tmp = -2.0 / x;
	} else if (t_0 <= 4e-28) {
		tmp = 2.0 * pow(x, -3.0);
	} else {
		tmp = (t_1 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / (t_1 * (1.0 + x));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((-2.0d0) / x) + (1.0d0 / (1.0d0 + x))) + (1.0d0 / (x + (-1.0d0)))
    t_1 = (x * x) - x
    if (t_0 <= (-200000000000.0d0)) then
        tmp = (-2.0d0) / x
    else if (t_0 <= 4d-28) then
        tmp = 2.0d0 * (x ** (-3.0d0))
    else
        tmp = (t_1 + ((1.0d0 + x) * (x + ((-2.0d0) * (x + (-1.0d0)))))) / (t_1 * (1.0d0 + x))
    end if
    code = tmp
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) {
	double t_0 = ((-2.0 / x) + (1.0 / (1.0 + x))) + (1.0 / (x + -1.0));
	double t_1 = (x * x) - x;
	double tmp;
	if (t_0 <= -200000000000.0) {
		tmp = -2.0 / x;
	} else if (t_0 <= 4e-28) {
		tmp = 2.0 * Math.pow(x, -3.0);
	} else {
		tmp = (t_1 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / (t_1 * (1.0 + x));
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = ((-2.0 / x) + (1.0 / (1.0 + x))) + (1.0 / (x + -1.0))
	t_1 = (x * x) - x
	tmp = 0
	if t_0 <= -200000000000.0:
		tmp = -2.0 / x
	elif t_0 <= 4e-28:
		tmp = 2.0 * math.pow(x, -3.0)
	else:
		tmp = (t_1 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / (t_1 * (1.0 + x))
	return tmp

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)
	t_0 = Float64(Float64(Float64(-2.0 / x) + Float64(1.0 / Float64(1.0 + x))) + Float64(1.0 / Float64(x + -1.0)))
	t_1 = Float64(Float64(x * x) - x)
	tmp = 0.0
	if (t_0 <= -200000000000.0)
		tmp = Float64(-2.0 / x);
	elseif (t_0 <= 4e-28)
		tmp = Float64(2.0 * (x ^ -3.0));
	else
		tmp = Float64(Float64(t_1 + Float64(Float64(1.0 + x) * Float64(x + Float64(-2.0 * Float64(x + -1.0))))) / Float64(t_1 * Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = ((-2.0 / x) + (1.0 / (1.0 + x))) + (1.0 / (x + -1.0));
	t_1 = (x * x) - x;
	tmp = 0.0;
	if (t_0 <= -200000000000.0)
		tmp = -2.0 / x;
	elseif (t_0 <= 4e-28)
		tmp = 2.0 * (x ^ -3.0);
	else
		tmp = (t_1 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / (t_1 * (1.0 + x));
	end
	tmp_2 = tmp;
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_] := Block[{t$95$0 = N[(N[(N[(-2.0 / x), $MachinePrecision] + N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -200000000000.0], N[(-2.0 / x), $MachinePrecision], If[LessEqual[t$95$0, 4e-28], N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[(N[(1.0 + x), $MachinePrecision] * N[(x + N[(-2.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	9.6
Target	0.3
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -2e11

   1. Initial program 0.0

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)} \]
      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. Taylor expanded in x around 0 0

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

   if -2e11 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 3.99999999999999988e-28

   1. Initial program 18.3

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

   2. Simplified 18.3

      \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)} \]
      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. Taylor expanded in x around inf 2.1

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]

   4. Applied egg-rr 1.6

      \[\leadsto \color{blue}{{x}^{-3} \cdot 2} \]

   if 3.99999999999999988e-28 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 0.9

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

   2. Simplified 0.9

      \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)} \]
      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 0.9

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

   4. Applied egg-rr 0.9

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

   5. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{\left(x \cdot x - x\right) + \left(1 + x\right) \cdot \left(x + \left(x + -1\right) \cdot -2\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{1}{x + -1} \leq -200000000000:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{1}{x + -1} \leq 4 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(1 + x\right) \cdot \left(x + -2 \cdot \left(x + -1\right)\right)}{\left(x \cdot x - x\right) \cdot \left(1 + x\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.6
Cost	960

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

Alternative 2
Error	15.6
Cost	712

\[\begin{array}{l} t_0 := \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -5533963371.988736:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.669966069998323 \cdot 10^{-12}:\\ \;\;\;\;\frac{-2}{x} + x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	15.7
Cost	584

\[\begin{array}{l} t_0 := \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -5533963371.988736:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.669966069998323 \cdot 10^{-12}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	15.7
Cost	584

\[\begin{array}{l} t_0 := \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -5533963371.988736:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.669966069998323 \cdot 10^{-12}:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	10.4
Cost	448

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

Alternative 6
Error	56.4
Cost	192

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

Alternative 7
Error	31.2
Cost	192

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

Alternative 8
Error	61.9
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022291 
(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))))