?

Average Accuracy: 75.4% → 99.9%
Time: 11.8s
Precision: binary64
Cost: 46852

?

\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.005:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))
↓
(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.005)
   (+
    (* 0.125 (pow x 2.0))
    (+ (* 0.0673828125 (pow x 6.0)) (* -0.0859375 (pow x 4.0))))
   (/
    (/
     (- 0.125 (/ 0.125 (pow (hypot 1.0 x) 3.0)))
     (+ 0.25 (+ (/ 0.25 (fma x x 1.0)) (/ 0.25 (hypot 1.0 x)))))
    (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x))))))))
double code(double x) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
}
↓
double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.005) {
		tmp = (0.125 * pow(x, 2.0)) + ((0.0673828125 * pow(x, 6.0)) + (-0.0859375 * pow(x, 4.0)));
	} else {
		tmp = ((0.125 - (0.125 / pow(hypot(1.0, x), 3.0))) / (0.25 + ((0.25 / fma(x, x, 1.0)) + (0.25 / hypot(1.0, x))))) / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x)))));
	}
	return tmp;
}

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))))
end
↓
function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.005)
		tmp = Float64(Float64(0.125 * (x ^ 2.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(-0.0859375 * (x ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(0.125 - Float64(0.125 / (hypot(1.0, x) ^ 3.0))) / Float64(0.25 + Float64(Float64(0.25 / fma(x, x, 1.0)) + Float64(0.25 / hypot(1.0, x))))) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x))))));
	end
	return tmp
end

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.005], N[(N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.125 - N[(0.125 / N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.25 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.005:\\
\;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\


\end{array}

Error?

Derivation?

  1. Split input into 2 regimes

  2. if (hypot.f64 1 x) < 1.0049999999999999

    1. Initial program 53.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

    2. Simplified53.2%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      Proof

      [Start]53.2

      \[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

      distribute-lft-in [=>]53.2

      \[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]

      metadata-eval [=>]53.2

      \[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]

      associate-*r/ [=>]53.2

      \[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]

      metadata-eval [=>]53.2

      \[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

    3. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)} \]

    if 1.0049999999999999 < (hypot.f64 1 x)

    1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

    2. Simplified98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      Proof

      [Start]98.5

      \[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

      distribute-lft-in [=>]98.5

      \[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]

      metadata-eval [=>]98.5

      \[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]

      associate-*r/ [=>]98.5

      \[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]

      metadata-eval [=>]98.5

      \[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

    3. Applied egg-rr100.0%

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      Proof

      [Start]98.5

      \[ 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \]

      flip-- [=>]98.5

      \[ \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

      metadata-eval [=>]98.5

      \[ \frac{\color{blue}{1} - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      add-sqr-sqrt [<=]99.9

      \[ \frac{1 - \color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      associate--r+ [=>]100.0

      \[ \frac{\color{blue}{\left(1 - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      metadata-eval [=>]100.0

      \[ \frac{\color{blue}{0.5} - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

    4. Applied egg-rr99.9%

      \[\leadsto \frac{\color{blue}{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      Proof

      [Start]100.0

      \[ \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      flip3-- [=>]99.9

      \[ \frac{\color{blue}{\frac{{0.5}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.5 \cdot 0.5 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      metadata-eval [=>]99.9

      \[ \frac{\frac{\color{blue}{0.125} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.5 \cdot 0.5 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      cube-div [=>]99.9

      \[ \frac{\frac{0.125 - \color{blue}{\frac{{0.5}^{3}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}}{0.5 \cdot 0.5 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      metadata-eval [=>]99.9

      \[ \frac{\frac{0.125 - \frac{\color{blue}{0.125}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.5 \cdot 0.5 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      metadata-eval [=>]99.9

      \[ \frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{\color{blue}{0.25} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      frac-times [=>]99.9

      \[ \frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\color{blue}{\frac{0.5 \cdot 0.5}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      metadata-eval [=>]99.9

      \[ \frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{\color{blue}{0.25}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      hypot-udef [=>]99.9

      \[ \frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{0.25}{\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      hypot-udef [=>]99.9

      \[ \frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{0.25}{\sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      add-sqr-sqrt [<=]99.9

      \[ \frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{0.25}{\color{blue}{1 \cdot 1 + x \cdot x}} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      metadata-eval [=>]99.9

      \[ \frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{0.25}{\color{blue}{1} + x \cdot x} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      +-commutative [=>]99.9

      \[ \frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{0.25}{\color{blue}{x \cdot x + 1}} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      fma-def [=>]99.9

      \[ \frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{0.25}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}} + 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      associate-*r/ [=>]99.9

      \[ \frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + \color{blue}{\frac{0.5 \cdot 0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      metadata-eval [=>]99.9

      \[ \frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + \frac{\color{blue}{0.25}}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification99.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.005:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.125 - \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy99.9%
Cost26756
\[\begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.005:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t_0}{1 + \sqrt{0.5 + t_0}}\\ \end{array} \]
Alternative 2
Accuracy99.0%
Cost20360
\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{elif}\;x \leq 0.0125:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]
Alternative 3
Accuracy98.9%
Cost13576
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{elif}\;x \leq 0.0029:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]
Alternative 4
Accuracy99.0%
Cost7624
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 - \frac{0.5}{x}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]
Alternative 5
Accuracy99.0%
Cost7496
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]
Alternative 6
Accuracy98.6%
Cost7305
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.1\right):\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot \left(x \cdot x\right)\\ \end{array} \]
Alternative 7
Accuracy98.3%
Cost6985
\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \end{array} \]
Alternative 8
Accuracy97.6%
Cost6857
\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \end{array} \]
Alternative 9
Accuracy61.6%
Cost712
\[\begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.8:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{-0.5}{x}}{2}\\ \end{array} \]
Alternative 10
Accuracy61.6%
Cost712
\[\begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{2}\\ \mathbf{elif}\;x \leq 1.8:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{-0.5}{x}}{2}\\ \end{array} \]
Alternative 11
Accuracy61.6%
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]
Alternative 12
Accuracy61.6%
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]
Alternative 13
Accuracy13.3%
Cost64
\[0.25 \]

Error

Reproduce?

herbie shell --seed 2023151 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))