3frac (problem 3.3.3)

Average Error: 10.2 → 0.7

Time: 13.7s

Precision: binary64

Cost: 16456

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x, x, -x\right)\\ t_1 := \left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -10:\\ \;\;\;\;\frac{2}{\frac{-1 + x \cdot x}{x}} + \frac{-2}{x}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot 2 + \left(-2 - x\right)\right) \cdot \left(1 + x\right) - t_0}{t_0 \cdot \left(-1 - x\right)}\\ \end{array} \]

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 (fma x x (- x)))
        (t_1 (+ (+ (/ -2.0 x) (/ 1.0 (+ 1.0 x))) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -10.0)
     (+ (/ 2.0 (/ (+ -1.0 (* x x)) x)) (/ -2.0 x))
     (if (<= t_1 0.0)
       (/ 2.0 (pow x 3.0))
       (/
        (- (* (+ (* x 2.0) (- -2.0 x)) (+ 1.0 x)) t_0)
        (* t_0 (- -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 = fma(x, x, -x);
	double t_1 = ((-2.0 / x) + (1.0 / (1.0 + x))) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -10.0) {
		tmp = (2.0 / ((-1.0 + (x * x)) / x)) + (-2.0 / x);
	} else if (t_1 <= 0.0) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = ((((x * 2.0) + (-2.0 - x)) * (1.0 + x)) - t_0) / (t_0 * (-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 = fma(x, x, Float64(-x))
	t_1 = Float64(Float64(Float64(-2.0 / x) + Float64(1.0 / Float64(1.0 + x))) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = Float64(Float64(2.0 / Float64(Float64(-1.0 + Float64(x * x)) / x)) + Float64(-2.0 / x));
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * 2.0) + Float64(-2.0 - x)) * Float64(1.0 + x)) - t_0) / Float64(t_0 * Float64(-1.0 - 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[(x * x + (-x)), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -10.0], N[(N[(2.0 / N[(N[(-1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * 2.0), $MachinePrecision] + N[(-2.0 - x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	10.2
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))) < -10

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

   6. Applied egg-rr 0.0

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

   7. Simplified 0.0

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

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

   if -10 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

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

   1. Initial program 1.3

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

   2. Simplified 1.3

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

      [Start] 1.3	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 1.3	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 1.3	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 1.3	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 1.3	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 1.3	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 1.3	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 1.3	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 1.3	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 1.3	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 0.7

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

   4. Simplified 0.7

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

      [Start] 0.7	\[ \frac{-1 \cdot \mathsf{fma}\left(x, x, -x\right) - \left(-1 - x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      mul-1-neg [=>] 0.7	\[ \frac{\color{blue}{\left(-\mathsf{fma}\left(x, x, -x\right)\right)} - \left(-1 - x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      *-commutative [=>] 0.7	\[ \frac{\left(-\mathsf{fma}\left(x, x, -x\right)\right) - \color{blue}{\left(-2 + \left(2 \cdot x - x\right)\right) \cdot \left(-1 - x\right)}}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 0.7	\[ \frac{\left(-\mathsf{fma}\left(x, x, -x\right)\right) - \color{blue}{\left(\left(2 \cdot x - x\right) + -2\right)} \cdot \left(-1 - x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      associate-+l- [=>] 0.7	\[ \frac{\left(-\mathsf{fma}\left(x, x, -x\right)\right) - \color{blue}{\left(2 \cdot x - \left(x - -2\right)\right)} \cdot \left(-1 - x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      *-commutative [=>] 0.7	\[ \frac{\left(-\mathsf{fma}\left(x, x, -x\right)\right) - \left(\color{blue}{x \cdot 2} - \left(x - -2\right)\right) \cdot \left(-1 - x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      *-commutative [=>] 0.7	\[ \frac{\left(-\mathsf{fma}\left(x, x, -x\right)\right) - \left(x \cdot 2 - \left(x - -2\right)\right) \cdot \left(-1 - x\right)}{\color{blue}{\mathsf{fma}\left(x, x, -x\right) \cdot \left(-1 - x\right)}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{1}{x + -1} \leq -10:\\ \;\;\;\;\frac{2}{\frac{-1 + x \cdot x}{x}} + \frac{-2}{x}\\ \mathbf{elif}\;\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{1}{x + -1} \leq 0:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot 2 + \left(-2 - x\right)\right) \cdot \left(1 + x\right) - \mathsf{fma}\left(x, x, -x\right)}{\mathsf{fma}\left(x, x, -x\right) \cdot \left(-1 - x\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	8713

\[\begin{array}{l} t_0 := \left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -10 \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\frac{2}{\frac{-1 + x \cdot x}{x}} + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \end{array} \]

Alternative 2
Error	10.2
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -102000000:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;\frac{2}{\frac{-1 + x \cdot x}{x}} + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x} + \frac{-2}{x}\\ \end{array} \]

Alternative 3
Error	10.2
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -96000000:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+42}:\\ \;\;\;\;\frac{-2}{x} + \frac{x \cdot -2}{1 - x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x} + \frac{-2}{x}\\ \end{array} \]

Alternative 4
Error	10.2
Cost	1088

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

Alternative 5
Error	10.2
Cost	960

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

Alternative 6
Error	10.2
Cost	960

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

Alternative 7
Error	11.1
Cost	448

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

Alternative 8
Error	31.1
Cost	192

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

Alternative 9
Error	61.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023055 
(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))))