3frac (problem 3.3.3)

Average Error: 9.6 → 1.6

Time: 12.5s

Precision: binary64

Cost: 8712

Math

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

↓

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -2000000:\\ \;\;\;\;x \cdot -2 + \frac{-2}{x}\\ \mathbf{elif}\;t_0 \leq 10^{-31}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

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)) (/ -2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_0 -2000000.0)
     (+ (* x -2.0) (/ -2.0 x))
     (if (<= t_0 1e-31) (/ 2.0 (pow x 3.0)) (/ -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)) + (-2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = (x * -2.0) + (-2.0 / x);
	} else if (t_0 <= 1e-31) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = -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) :: tmp
    t_0 = ((1.0d0 / (1.0d0 + x)) + ((-2.0d0) / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_0 <= (-2000000.0d0)) then
        tmp = (x * (-2.0d0)) + ((-2.0d0) / x)
    else if (t_0 <= 1d-31) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else
        tmp = (-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)) + (-2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = (x * -2.0) + (-2.0 / x);
	} else if (t_0 <= 1e-31) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else {
		tmp = -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)) + (-2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_0 <= -2000000.0:
		tmp = (x * -2.0) + (-2.0 / x)
	elif t_0 <= 1e-31:
		tmp = 2.0 / math.pow(x, 3.0)
	else:
		tmp = -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(Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= -2000000.0)
		tmp = Float64(Float64(x * -2.0) + Float64(-2.0 / x));
	elseif (t_0 <= 1e-31)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = 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)) + (-2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= -2000000.0)
		tmp = (x * -2.0) + (-2.0 / x);
	elseif (t_0 <= 1e-31)
		tmp = 2.0 / (x ^ 3.0);
	else
		tmp = -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[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000.0], N[(N[(x * -2.0), $MachinePrecision] + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-31], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(-2.0 / x), $MachinePrecision]]]]

Target

Original	9.6
Target	0.2
Herbie	1.6

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -2e6

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 0.0	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.0	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 0.0	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 0.0	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 0.0	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 0.0	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 0.0	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 0.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right) \]
      sub-neg [=>] 0.0	\[ \color{blue}{\left(\frac{1}{1 + x} + \left(-\frac{2}{x}\right)\right)} + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right) \]
      mul-1-neg [<=] 0.0	\[ \left(\frac{1}{1 + x} + \color{blue}{-1 \cdot \frac{2}{x}}\right) + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right) \]
      associate-+l+ [=>] 0.0	\[ \color{blue}{\frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right)} \]
      mul-1-neg [<=] 0.0	\[ \frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\mathsf{fma}\left(\color{blue}{-1 \cdot \frac{2}{x}}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      fma-udef [=>] 0.0	\[ \frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\color{blue}{\left(\left(-1 \cdot \frac{2}{x}\right) \cdot 1 + \frac{2}{x}\right)} - \frac{1}{1 - x}\right)\right) \]
      associate-*r/ [=>] 0.0	\[ \frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\left(\color{blue}{\frac{-1 \cdot 2}{x}} \cdot 1 + \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      metadata-eval [=>] 0.0	\[ \frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\left(\frac{\color{blue}{-2}}{x} \cdot 1 + \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      associate-*l/ [=>] 0.0	\[ \frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\left(\color{blue}{\frac{-2 \cdot 1}{x}} + \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      metadata-eval [=>] 0.0	\[ \frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\left(\frac{\color{blue}{-2}}{x} + \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      metadata-eval [<=] 0.0	\[ \frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\left(\frac{\color{blue}{-1 \cdot 2}}{x} + \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      associate-*l/ [<=] 0.0	\[ \frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\left(\color{blue}{\frac{-1}{x} \cdot 2} + \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      metadata-eval [<=] 0.0	\[ \frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\left(\frac{\color{blue}{-1}}{x} \cdot 2 + \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      distribute-neg-frac [<=] 0.0	\[ \frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\left(\color{blue}{\left(-\frac{1}{x}\right)} \cdot 2 + \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]

   5. Taylor expanded in x around 0 0.0

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

   6. Simplified 0.0

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

      [Start] 0.0	\[ -2 \cdot x - 2 \cdot \frac{1}{x} \]
      associate-*r/ [=>] 0.0	\[ -2 \cdot x - \color{blue}{\frac{2 \cdot 1}{x}} \]
      metadata-eval [=>] 0.0	\[ -2 \cdot x - \frac{\color{blue}{2}}{x} \]

   if -2e6 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1e-31

   1. Initial program 19.2

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

   2. Simplified 19.2

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

      [Start] 19.2	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 19.2	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.2	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 19.2	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 19.2	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 19.2	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 19.2	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 19.2	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.2	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 19.2	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 1.4

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

   if 1e-31 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 1.4

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

   2. Simplified 1.4

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

      [Start] 1.4	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 1.4	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 1.4	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 1.4	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 1.4	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 1.4	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 1.4	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 1.4	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 1.4	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 1.4	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around 0 3.3

      \[\leadsto \color{blue}{\frac{-2}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq -2000000:\\ \;\;\;\;x \cdot -2 + \frac{-2}{x}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq 10^{-31}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.6
Cost	960

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

Alternative 2
Error	10.5
Cost	584

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

Alternative 3
Error	10.5
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 4
Error	10.5
Cost	448

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

Alternative 5
Error	61.9
Cost	64

\[1 \]

Alternative 6
Error	42.2
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023027 
(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))))