3frac (problem 3.3.3)

Average Error: 9.9 → 0.6

Time: 13.8s

Precision: binary64

Cost: 8712

Math

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

↓

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ t_1 := x \cdot x - x\\ \mathbf{if}\;t_0 \leq -1000:\\ \;\;\;\;\frac{x \cdot -2}{1 - x \cdot x} + \frac{-2}{x}\\ \mathbf{elif}\;t_0 \leq 10^{-25}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \left(-1 - x\right) \cdot \left(x + -2\right)}{\left(1 + x\right) \cdot t_1}\\ \end{array} \]

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))))
        (t_1 (- (* x x) x)))
   (if (<= t_0 -1000.0)
     (+ (/ (* x -2.0) (- 1.0 (* x x))) (/ -2.0 x))
     (if (<= t_0 1e-25)
       (/ 2.0 (pow x 3.0))
       (/ (+ t_1 (* (- -1.0 x) (+ x -2.0))) (* (+ 1.0 x) t_1))))))

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 t_1 = (x * x) - x;
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = ((x * -2.0) / (1.0 - (x * x))) + (-2.0 / x);
	} else if (t_0 <= 1e-25) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = (t_1 + ((-1.0 - x) * (x + -2.0))) / ((1.0 + x) * t_1);
	}
	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 = ((1.0d0 / (1.0d0 + x)) + ((-2.0d0) / x)) + (1.0d0 / (x + (-1.0d0)))
    t_1 = (x * x) - x
    if (t_0 <= (-1000.0d0)) then
        tmp = ((x * (-2.0d0)) / (1.0d0 - (x * x))) + ((-2.0d0) / x)
    else if (t_0 <= 1d-25) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else
        tmp = (t_1 + (((-1.0d0) - x) * (x + (-2.0d0)))) / ((1.0d0 + x) * t_1)
    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 t_1 = (x * x) - x;
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = ((x * -2.0) / (1.0 - (x * x))) + (-2.0 / x);
	} else if (t_0 <= 1e-25) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else {
		tmp = (t_1 + ((-1.0 - x) * (x + -2.0))) / ((1.0 + x) * t_1);
	}
	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))
	t_1 = (x * x) - x
	tmp = 0
	if t_0 <= -1000.0:
		tmp = ((x * -2.0) / (1.0 - (x * x))) + (-2.0 / x)
	elif t_0 <= 1e-25:
		tmp = 2.0 / math.pow(x, 3.0)
	else:
		tmp = (t_1 + ((-1.0 - x) * (x + -2.0))) / ((1.0 + x) * t_1)
	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)))
	t_1 = Float64(Float64(x * x) - x)
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = Float64(Float64(Float64(x * -2.0) / Float64(1.0 - Float64(x * x))) + Float64(-2.0 / x));
	elseif (t_0 <= 1e-25)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(Float64(t_1 + Float64(Float64(-1.0 - x) * Float64(x + -2.0))) / Float64(Float64(1.0 + x) * t_1));
	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));
	t_1 = (x * x) - x;
	tmp = 0.0;
	if (t_0 <= -1000.0)
		tmp = ((x * -2.0) / (1.0 - (x * x))) + (-2.0 / x);
	elseif (t_0 <= 1e-25)
		tmp = 2.0 / (x ^ 3.0);
	else
		tmp = (t_1 + ((-1.0 - x) * (x + -2.0))) / ((1.0 + x) * t_1);
	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]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], N[(N[(N[(x * -2.0), $MachinePrecision] / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-25], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[(N[(-1.0 - x), $MachinePrecision] * N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	9.9
Target	0.2
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -1e3

   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

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

   4. Applied egg-rr 0.0

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

   5. Simplified 0.0

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

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

   if -1e3 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1.00000000000000004e-25

   1. Initial program 19.8

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

   2. Simplified 19.8

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

      [Start] 19.8	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 19.8	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.8	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 19.8	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 19.8	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 19.8	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 19.8	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 19.8	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.8	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 19.8	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 1.0

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

   if 1.00000000000000004e-25 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 0.7

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

   2. Simplified 0.7

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

      [Start] 0.7	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 0.7	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.7	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 0.7	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 0.7	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 0.7	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 0.7	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 0.7	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.7	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 0.7	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 0.7

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

   4. Applied egg-rr 0.7

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

   5. Applied egg-rr 0.7

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

   6. Simplified 0.7

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

      [Start] 0.7	\[ \frac{1}{1 + x} - \frac{2 \cdot \left(-1 + x\right) - x}{x \cdot x - x} \]
      distribute-rgt-in [=>] 0.7	\[ \frac{1}{1 + x} - \frac{\color{blue}{\left(-1 \cdot 2 + x \cdot 2\right)} - x}{x \cdot x - x} \]
      metadata-eval [=>] 0.7	\[ \frac{1}{1 + x} - \frac{\left(\color{blue}{-2} + x \cdot 2\right) - x}{x \cdot x - x} \]

   7. Applied egg-rr 0.3

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq -1000:\\ \;\;\;\;\frac{x \cdot -2}{1 - x \cdot x} + \frac{-2}{x}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq 10^{-25}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(-1 - x\right) \cdot \left(x + -2\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.9
Cost	960

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

Alternative 2
Error	15.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.39\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} - x\\ \end{array} \]

Alternative 3
Error	10.7
Cost	448

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

Alternative 4
Error	30.4
Cost	192

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

Alternative 5
Error	61.9
Cost	64

\[-1 \]

Alternative 6
Error	61.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023017 
(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))))