3frac (problem 3.3.3)

Average Error: 10.1 → 0.9

Time: 15.4s

Precision: binary64

Cost: 10120

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -50000:\\ \;\;\;\;t_0 - \left(1 + \left(x + \left(\frac{2}{x} + x \cdot x\right)\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - \left(x + \left(1 + x\right) \cdot \left(x \cdot 2 + \left(-2 - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)}\\ \end{array} \]

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 (+ (+ t_0 (/ -2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -50000.0)
     (- t_0 (+ 1.0 (+ x (+ (/ 2.0 x) (* x x)))))
     (if (<= t_1 0.0)
       (/ 2.0 (pow x 3.0))
       (/
        (- (* x x) (+ x (* (+ 1.0 x) (+ (* x 2.0) (- -2.0 x)))))
        (* (+ 1.0 x) (fma x x (- 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 = (t_0 + (-2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = t_0 - (1.0 + (x + ((2.0 / x) + (x * x))));
	} else if (t_1 <= 0.0) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = ((x * x) - (x + ((1.0 + x) * ((x * 2.0) + (-2.0 - x))))) / ((1.0 + x) * fma(x, x, -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(Float64(t_0 + Float64(-2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -50000.0)
		tmp = Float64(t_0 - Float64(1.0 + Float64(x + Float64(Float64(2.0 / x) + Float64(x * x)))));
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(Float64(Float64(x * x) - Float64(x + Float64(Float64(1.0 + x) * Float64(Float64(x * 2.0) + Float64(-2.0 - x))))) / Float64(Float64(1.0 + x) * fma(x, x, Float64(-x))));
	end
	return 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[(N[(t$95$0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000.0], N[(t$95$0 - N[(1.0 + N[(x + N[(N[(2.0 / x), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] - N[(x + N[(N[(1.0 + x), $MachinePrecision] * N[(N[(x * 2.0), $MachinePrecision] + N[(-2.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * N[(x * x + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	10.1
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))) < -5e4

   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. Taylor expanded in x around 0 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \frac{1}{1 + x} - \left(1 + \left({x}^{2} + \left(2 \cdot \frac{1}{x} + x\right)\right)\right) \]
      associate-+r+ [=>] 0.0	\[ \frac{1}{1 + x} - \left(1 + \color{blue}{\left(\left({x}^{2} + 2 \cdot \frac{1}{x}\right) + x\right)}\right) \]
      unpow2 [=>] 0.0	\[ \frac{1}{1 + x} - \left(1 + \left(\left(\color{blue}{x \cdot x} + 2 \cdot \frac{1}{x}\right) + x\right)\right) \]
      associate-*r/ [=>] 0.0	\[ \frac{1}{1 + x} - \left(1 + \left(\left(x \cdot x + \color{blue}{\frac{2 \cdot 1}{x}}\right) + x\right)\right) \]
      metadata-eval [=>] 0.0	\[ \frac{1}{1 + x} - \left(1 + \left(\left(x \cdot x + \frac{\color{blue}{2}}{x}\right) + x\right)\right) \]

   if -5e4 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 19.7

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

   2. Simplified 19.7

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

      [Start] 19.7	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 19.7	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.7	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 19.7	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 19.7	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 19.7	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 19.7	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 19.7	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.7	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 19.7	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 1.3

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

   if 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 1.5

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

   2. Simplified 1.5

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

      [Start] 1.5	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 1.5	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 1.5	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 1.5	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 1.5	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 1.5	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 1.5	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 1.5	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 1.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 1.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 1.2

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

   4. Simplified 1.2

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

      [Start] 1.2	\[ \frac{x \cdot x - \left(x + \left(1 + x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 1.2	\[ \frac{x \cdot x - \left(x + \color{blue}{\left(x + 1\right)} \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 1.2	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \color{blue}{\left(\left(2 \cdot x - x\right) + -2\right)}\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      associate-+l- [=>] 1.2	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \color{blue}{\left(2 \cdot x - \left(x - -2\right)\right)}\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      *-commutative [=>] 1.2	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(\color{blue}{x \cdot 2} - \left(x - -2\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 1.2	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(x \cdot 2 - \left(x - -2\right)\right)\right)}{\color{blue}{\left(x + 1\right)} \cdot \mathsf{fma}\left(x, x, -x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq -50000:\\ \;\;\;\;\frac{1}{1 + x} - \left(1 + \left(x + \left(\frac{2}{x} + x \cdot x\right)\right)\right)\\ \mathbf{elif}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq 0:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - \left(x + \left(1 + x\right) \cdot \left(x \cdot 2 + \left(-2 - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	8712

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -50000:\\ \;\;\;\;t_0 - \left(1 + \left(x + \left(\frac{2}{x} + x \cdot x\right)\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{x + -2}{x \cdot \left(x + -1\right)}\\ \end{array} \]

Alternative 2
Error	10.1
Cost	960

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

Alternative 3
Error	15.4
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.52\right):\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 4
Error	15.2
Cost	584

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

Alternative 5
Error	10.9
Cost	448

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

Alternative 6
Error	30.6
Cost	192

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

Alternative 7
Error	61.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023066 
(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))))