3frac (problem 3.3.3)

Average Error: 9.9 → 0.7

Time: 9.5s

Precision: binary64

Cost: 8712

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \frac{1}{x + -1}\\ t_2 := \left(t_0 + \frac{-2}{x}\right) + t_1\\ \mathbf{if}\;t_2 \leq -5:\\ \;\;\;\;t_0 + \frac{\left(1 - x\right) - x \cdot -0.5}{\left(1 - x\right) \cdot \left(x \cdot -0.5\right)}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(t_1 + \frac{-2}{x}\right)\\ \end{array} \]

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 (/ 1.0 (+ 1.0 x)))
        (t_1 (/ 1.0 (+ x -1.0)))
        (t_2 (+ (+ t_0 (/ -2.0 x)) t_1)))
   (if (<= t_2 -5.0)
     (+ t_0 (/ (- (- 1.0 x) (* x -0.5)) (* (- 1.0 x) (* x -0.5))))
     (if (<= t_2 5e-27) (/ 2.0 (pow x 3.0)) (+ t_0 (+ t_1 (/ -2.0 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 = 1.0 / (1.0 + x);
	double t_1 = 1.0 / (x + -1.0);
	double t_2 = (t_0 + (-2.0 / x)) + t_1;
	double tmp;
	if (t_2 <= -5.0) {
		tmp = t_0 + (((1.0 - x) - (x * -0.5)) / ((1.0 - x) * (x * -0.5)));
	} else if (t_2 <= 5e-27) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = t_0 + (t_1 + (-2.0 / 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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + x)
    t_1 = 1.0d0 / (x + (-1.0d0))
    t_2 = (t_0 + ((-2.0d0) / x)) + t_1
    if (t_2 <= (-5.0d0)) then
        tmp = t_0 + (((1.0d0 - x) - (x * (-0.5d0))) / ((1.0d0 - x) * (x * (-0.5d0))))
    else if (t_2 <= 5d-27) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else
        tmp = t_0 + (t_1 + ((-2.0d0) / 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 = 1.0 / (1.0 + x);
	double t_1 = 1.0 / (x + -1.0);
	double t_2 = (t_0 + (-2.0 / x)) + t_1;
	double tmp;
	if (t_2 <= -5.0) {
		tmp = t_0 + (((1.0 - x) - (x * -0.5)) / ((1.0 - x) * (x * -0.5)));
	} else if (t_2 <= 5e-27) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else {
		tmp = t_0 + (t_1 + (-2.0 / 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 = 1.0 / (1.0 + x)
	t_1 = 1.0 / (x + -1.0)
	t_2 = (t_0 + (-2.0 / x)) + t_1
	tmp = 0
	if t_2 <= -5.0:
		tmp = t_0 + (((1.0 - x) - (x * -0.5)) / ((1.0 - x) * (x * -0.5)))
	elif t_2 <= 5e-27:
		tmp = 2.0 / math.pow(x, 3.0)
	else:
		tmp = t_0 + (t_1 + (-2.0 / 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(1.0 / Float64(1.0 + x))
	t_1 = Float64(1.0 / Float64(x + -1.0))
	t_2 = Float64(Float64(t_0 + Float64(-2.0 / x)) + t_1)
	tmp = 0.0
	if (t_2 <= -5.0)
		tmp = Float64(t_0 + Float64(Float64(Float64(1.0 - x) - Float64(x * -0.5)) / Float64(Float64(1.0 - x) * Float64(x * -0.5))));
	elseif (t_2 <= 5e-27)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(t_0 + Float64(t_1 + Float64(-2.0 / 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 = 1.0 / (1.0 + x);
	t_1 = 1.0 / (x + -1.0);
	t_2 = (t_0 + (-2.0 / x)) + t_1;
	tmp = 0.0;
	if (t_2 <= -5.0)
		tmp = t_0 + (((1.0 - x) - (x * -0.5)) / ((1.0 - x) * (x * -0.5)));
	elseif (t_2 <= 5e-27)
		tmp = 2.0 / (x ^ 3.0);
	else
		tmp = t_0 + (t_1 + (-2.0 / 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[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -5.0], N[(t$95$0 + N[(N[(N[(1.0 - x), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - x), $MachinePrecision] * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-27], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(t$95$1 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	9.9
Target	0.3
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -5

   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. Applied egg-rr 0.0

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

   4. Applied egg-rr 0.0

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

   if -5 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 5.0000000000000002e-27

   1. Initial program 19.6

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

   2. Simplified 19.6

      \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)} \]
      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 53.4

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

   4. Applied egg-rr 61.9

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

   5. Applied egg-rr 61.9

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

   6. Taylor expanded in x around inf 1.0

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

   if 5.0000000000000002e-27 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 0.8

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

   2. Simplified 0.8

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq -5:\\ \;\;\;\;\frac{1}{1 + x} + \frac{\left(1 - x\right) - x \cdot -0.5}{\left(1 - x\right) \cdot \left(x \cdot -0.5\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	9.9
Cost	960

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

Alternative 2
Error	15.6
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{-2}{x}}{x}\\ \mathbf{if}\;x \leq -505410841.0814072:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.011003540048445906:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	15.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -505410841.0814072:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.011003540048445906:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \end{array} \]

Alternative 4
Error	15.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -505410841.0814072:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.011003540048445906:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \end{array} \]

Alternative 5
Error	10.8
Cost	448

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

Alternative 6
Error	30.9
Cost	192

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

Alternative 7
Error	61.9
Cost	64

\[2 \]

Alternative 8
Error	61.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022301 
(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))))