3frac (problem 3.3.3)

Average Error: 10.0 → 0.2

Time: 12.2s

Precision: binary64

Cost: 16456

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x, x, -x\right)\\ t_1 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ t_2 := \left(1 + x\right) \cdot \left(\left(-2 - x\right) - x \cdot -2\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-19}:\\ \;\;\;\;\frac{x \cdot x - \left(x + t_2\right)}{\left(1 + x\right) \cdot t_0}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 - t_0}{t_0 \cdot \left(-1 - x\right)}\\ \end{array} \]

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 (+ (+ (/ 1.0 (+ 1.0 x)) (/ -2.0 x)) (/ 1.0 (+ x -1.0))))
        (t_2 (* (+ 1.0 x) (- (- -2.0 x) (* x -2.0)))))
   (if (<= t_1 -1e-19)
     (/ (- (* x x) (+ x t_2)) (* (+ 1.0 x) t_0))
     (if (<= t_1 5e-324)
       (* 2.0 (pow x -3.0))
       (/ (- t_2 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 = ((1.0 / (1.0 + x)) + (-2.0 / x)) + (1.0 / (x + -1.0));
	double t_2 = (1.0 + x) * ((-2.0 - x) - (x * -2.0));
	double tmp;
	if (t_1 <= -1e-19) {
		tmp = ((x * x) - (x + t_2)) / ((1.0 + x) * t_0);
	} else if (t_1 <= 5e-324) {
		tmp = 2.0 * pow(x, -3.0);
	} else {
		tmp = (t_2 - 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(1.0 / Float64(1.0 + x)) + Float64(-2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	t_2 = Float64(Float64(1.0 + x) * Float64(Float64(-2.0 - x) - Float64(x * -2.0)))
	tmp = 0.0
	if (t_1 <= -1e-19)
		tmp = Float64(Float64(Float64(x * x) - Float64(x + t_2)) / Float64(Float64(1.0 + x) * t_0));
	elseif (t_1 <= 5e-324)
		tmp = Float64(2.0 * (x ^ -3.0));
	else
		tmp = Float64(Float64(t_2 - 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[(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$2 = N[(N[(1.0 + x), $MachinePrecision] * N[(N[(-2.0 - x), $MachinePrecision] - N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-19], N[(N[(N[(x * x), $MachinePrecision] - N[(x + t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-324], N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 - t$95$0), $MachinePrecision] / N[(t$95$0 * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	10.0
Target	0.3
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -9.9999999999999998e-20

   1. Initial program 0.5

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

   2. Simplified 0.5

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

      [Start] 0.5	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 0.5	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.5	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 0.5	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 0.5	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 0.5	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 0.5	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 0.5	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 0.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 0.1

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

   4. Simplified 0.1

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

      [Start] 0.1	\[ \frac{x \cdot x - \left(x + \left(1 + x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 0.1	\[ \frac{x \cdot x - \left(x + \color{blue}{\left(x + 1\right)} \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 0.1	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \color{blue}{\left(\left(2 \cdot x - x\right) + -2\right)}\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      associate-+l- [=>] 0.1	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \color{blue}{\left(2 \cdot x - \left(x - -2\right)\right)}\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      *-commutative [=>] 0.1	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(\color{blue}{x \cdot 2} - \left(x - -2\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 0.1	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(x \cdot 2 - \left(x - -2\right)\right)\right)}{\color{blue}{\left(x + 1\right)} \cdot \mathsf{fma}\left(x, x, -x\right)} \]

   if -9.9999999999999998e-20 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 4.94066e-324

   1. Initial program 19.5

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

   2. Simplified 19.5

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

      [Start] 19.5	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 19.5	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.5	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 19.5	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 19.5	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 19.5	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 19.5	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 19.5	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 19.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 0.6

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

   4. Applied egg-rr 0.0

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

   if 4.94066e-324 < (+.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 0.8

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

   4. Simplified 0.8

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

      [Start] 0.8	\[ \frac{-1 \cdot \mathsf{fma}\left(x, x, -x\right) - \left(-1 - x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      mul-1-neg [=>] 0.8	\[ \frac{\color{blue}{\left(-\mathsf{fma}\left(x, x, -x\right)\right)} - \left(-1 - x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      *-commutative [=>] 0.8	\[ \frac{\left(-\mathsf{fma}\left(x, x, -x\right)\right) - \color{blue}{\left(-2 + \left(2 \cdot x - x\right)\right) \cdot \left(-1 - x\right)}}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 0.8	\[ \frac{\left(-\mathsf{fma}\left(x, x, -x\right)\right) - \color{blue}{\left(\left(2 \cdot x - x\right) + -2\right)} \cdot \left(-1 - x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      associate-+l- [=>] 0.8	\[ \frac{\left(-\mathsf{fma}\left(x, x, -x\right)\right) - \color{blue}{\left(2 \cdot x - \left(x - -2\right)\right)} \cdot \left(-1 - x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      *-commutative [=>] 0.8	\[ \frac{\left(-\mathsf{fma}\left(x, x, -x\right)\right) - \left(\color{blue}{x \cdot 2} - \left(x - -2\right)\right) \cdot \left(-1 - x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      *-commutative [=>] 0.8	\[ \frac{\left(-\mathsf{fma}\left(x, x, -x\right)\right) - \left(x \cdot 2 - \left(x - -2\right)\right) \cdot \left(-1 - x\right)}{\color{blue}{\mathsf{fma}\left(x, x, -x\right) \cdot \left(-1 - x\right)}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-19}:\\ \;\;\;\;\frac{x \cdot x - \left(x + \left(1 + x\right) \cdot \left(\left(-2 - x\right) - x \cdot -2\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-324}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot \left(\left(-2 - x\right) - x \cdot -2\right) - \mathsf{fma}\left(x, x, -x\right)}{\mathsf{fma}\left(x, x, -x\right) \cdot \left(-1 - x\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	9092

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-19}:\\ \;\;\;\;\frac{x \cdot x - \left(x + \left(1 + x\right) \cdot \left(\left(-2 - x\right) - x \cdot -2\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.8
Cost	8712

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -1000000:\\ \;\;\;\;x \cdot -2 + \frac{-2}{x}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	10.0
Cost	960

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

Alternative 4
Error	10.9
Cost	448

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

Alternative 5
Error	10.9
Cost	448

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

Alternative 6
Error	30.9
Cost	192

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

Reproduce

herbie shell --seed 2022356 
(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))))