Toniolo and Linder, Equation (7)

Average Accuracy: 32.9% → 86.7%

Time: 22.9s

Precision: binary64

Cost: 13900.00

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{-51}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-148}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{t \cdot t}{\mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -1.05e-51)
     (- t_1)
     (if (<= t 3.6e-148)
       (/ t (hypot t (/ l (sqrt x))))
       (if (<= t 2.05e+15) (sqrt (/ (* t t) (fma t t (* l (/ l x))))) t_1)))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

↓

double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1.05e-51) {
		tmp = -t_1;
	} else if (t <= 3.6e-148) {
		tmp = t / hypot(t, (l / sqrt(x)));
	} else if (t <= 2.05e+15) {
		tmp = sqrt(((t * t) / fma(t, t, (l * (l / x)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

↓

function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1.05e-51)
		tmp = Float64(-t_1);
	elseif (t <= 3.6e-148)
		tmp = Float64(t / hypot(t, Float64(l / sqrt(x))));
	elseif (t <= 2.05e+15)
		tmp = sqrt(Float64(Float64(t * t) / fma(t, t, Float64(l * Float64(l / x)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.05e-51], (-t$95$1), If[LessEqual[t, 3.6e-148], N[(t / N[Sqrt[t ^ 2 + N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.05e+15], N[Sqrt[N[(N[(t * t), $MachinePrecision] / N[(t * t + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.05000000000000001e-51

   1. Initial program 38.0%

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

   2. Taylor expanded in l around 0 28.0%

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

   3. Simplified 45.8%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\frac{-1 + x}{x + 1}}}}} \]
      Proof

      [Start] 28.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}} \]
      *-commutative [=>] 28.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{\color{blue}{{t}^{2} \cdot \left(1 + x\right)}}{x - 1}}} \]
      associate-/l* [=>] 45.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \color{blue}{\frac{{t}^{2}}{\frac{x - 1}{1 + x}}}}} \]
      unpow2 [=>] 45.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{\color{blue}{t \cdot t}}{\frac{x - 1}{1 + x}}}} \]
      sub-neg [=>] 45.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{t \cdot t}{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}}} \]
      metadata-eval [=>] 45.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{t \cdot t}{\frac{x + \color{blue}{-1}}{1 + x}}}} \]
      +-commutative [=>] 45.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{t \cdot t}{\frac{\color{blue}{-1 + x}}{1 + x}}}} \]
      +-commutative [=>] 45.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{t \cdot t}{\frac{-1 + x}{\color{blue}{x + 1}}}}} \]

   4. Taylor expanded in t around -inf 89.9%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]

   5. Simplified 89.9%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]
      Proof

      [Start] 89.9	\[ -1 \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      mul-1-neg [=>] 89.9	\[ \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]
      sub-neg [=>] 89.9	\[ -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]
      metadata-eval [=>] 89.9	\[ -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   if -1.05000000000000001e-51 < t < 3.5999999999999998e-148

   1. Initial program 15.6%

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

   2. Simplified 8.8%

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

      [Start] 15.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 15.6	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 17.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 7.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 7.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 7.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 7.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 7.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 7.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 8.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around -inf 58.4%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}}} \]

   4. Simplified 58.4%

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

      [Start] 58.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 58.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 58.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 58.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 58.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 58.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 58.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   5. Taylor expanded in t around 0 58.3%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\color{blue}{\frac{{\ell}^{2}}{x}} + t \cdot t\right)}} \]

   6. Simplified 58.3%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\color{blue}{\frac{\ell \cdot \ell}{x}} + t \cdot t\right)}} \]
      Proof

      [Start] 58.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + t \cdot t\right)}} \]
      unpow2 [=>] 58.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + t \cdot t\right)}} \]

   7. Applied egg-rr 39.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\right)} - 1} \]

   8. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}} \]
      Proof

      [Start] 39.4	\[ e^{\mathsf{log1p}\left(\frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\right)} - 1 \]
      expm1-def [=>] 75.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\right)\right)} \]
      expm1-log1p [=>] 76.9	\[ \color{blue}{\frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}} \]
      associate-/l* [=>] 77.0	\[ \frac{\color{blue}{\frac{t}{\frac{\sqrt{2}}{\sqrt{2}}}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)} \]
      associate-/l/ [=>] 77.0	\[ \color{blue}{\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right) \cdot \frac{\sqrt{2}}{\sqrt{2}}}} \]
      *-inverses [=>] 77.0	\[ \frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right) \cdot \color{blue}{1}} \]
      *-rgt-identity [=>] 77.0	\[ \frac{t}{\color{blue}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}} \]

   if 3.5999999999999998e-148 < t < 2.05e15

   1. Initial program 52.2%

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

   2. Simplified 35.0%

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

      [Start] 52.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 52.1	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 52.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 34.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 34.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 34.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 34.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 34.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 34.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 35.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around -inf 85.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}}} \]

   4. Simplified 85.2%

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

      [Start] 85.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 85.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 85.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 85.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 85.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 85.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 85.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   5. Taylor expanded in t around 0 84.4%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\color{blue}{\frac{{\ell}^{2}}{x}} + t \cdot t\right)}} \]

   6. Simplified 84.4%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\color{blue}{\frac{\ell \cdot \ell}{x}} + t \cdot t\right)}} \]
      Proof

      [Start] 84.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + t \cdot t\right)}} \]
      unpow2 [=>] 84.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + t \cdot t\right)}} \]

   7. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \frac{t \cdot t}{2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)}}} \]

   8. Simplified 86.8%

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

      [Start] 86.8	\[ \sqrt{2 \cdot \frac{t \cdot t}{2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)}} \]
      associate-*r/ [=>] 86.8	\[ \sqrt{\color{blue}{\frac{2 \cdot \left(t \cdot t\right)}{2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)}}} \]
      unpow2 [<=] 86.8	\[ \sqrt{\frac{2 \cdot \color{blue}{{t}^{2}}}{2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)}} \]
      times-frac [=>] 86.8	\[ \sqrt{\color{blue}{\frac{2}{2} \cdot \frac{{t}^{2}}{\mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)}}} \]
      metadata-eval [=>] 86.8	\[ \sqrt{\color{blue}{1} \cdot \frac{{t}^{2}}{\mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)}} \]
      unpow2 [=>] 86.8	\[ \sqrt{1 \cdot \frac{\color{blue}{t \cdot t}}{\mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)}} \]
      associate-/r/ [=>] 86.8	\[ \sqrt{1 \cdot \frac{t \cdot t}{\mathsf{fma}\left(t, t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      *-commutative [=>] 86.8	\[ \sqrt{1 \cdot \frac{t \cdot t}{\mathsf{fma}\left(t, t, \color{blue}{\ell \cdot \frac{\ell}{x}}\right)}} \]

   if 2.05e15 < t

   1. Initial program 34.9%

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

   2. Taylor expanded in l around 0 19.2%

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

   3. Simplified 40.8%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\frac{-1 + x}{x + 1}}}}} \]
      Proof

      [Start] 19.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}} \]
      *-commutative [=>] 19.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{\color{blue}{{t}^{2} \cdot \left(1 + x\right)}}{x - 1}}} \]
      associate-/l* [=>] 40.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \color{blue}{\frac{{t}^{2}}{\frac{x - 1}{1 + x}}}}} \]
      unpow2 [=>] 40.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{\color{blue}{t \cdot t}}{\frac{x - 1}{1 + x}}}} \]
      sub-neg [=>] 40.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{t \cdot t}{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}}} \]
      metadata-eval [=>] 40.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{t \cdot t}{\frac{x + \color{blue}{-1}}{1 + x}}}} \]
      +-commutative [=>] 40.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{t \cdot t}{\frac{\color{blue}{-1 + x}}{1 + x}}}} \]
      +-commutative [=>] 40.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{t \cdot t}{\frac{-1 + x}{\color{blue}{x + 1}}}}} \]

   4. Taylor expanded in t around 0 92.4%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-51}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-148}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{t \cdot t}{\mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	79.9%
Cost	13580.00

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+109}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{+82}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -500000:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \end{array} \]

Alternative 2
Accuracy	76.8%
Cost	7112.00

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-221}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-190}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 3
Accuracy	77.5%
Cost	7112.00

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-220}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-190}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	76.5%
Cost	7048.00

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-221}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-191}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{0.5 + \frac{0.5 + \frac{0.5}{x}}{x}}{x} + \frac{\frac{-1}{x}}{x}\right)}{1 + \left(x + \frac{1}{x}\right)}\\ \end{array} \]

Alternative 5
Accuracy	76.6%
Cost	7048.00

\[\begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-221}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-191}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{0.5 + \frac{0.5 + \frac{0.5}{x}}{x}}{x} + \frac{\frac{-1}{x}}{x}\right)}{1 + \left(x + \frac{1}{x}\right)}\\ \end{array} \]

Alternative 6
Accuracy	76.3%
Cost	6984.00

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-221}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{0.5 + \frac{0.5 + \frac{0.5}{x}}{x}}{x} + \frac{\frac{-1}{x}}{x}\right)}{1 + \left(x + \frac{1}{x}\right)}\\ \end{array} \]

Alternative 7
Accuracy	75.8%
Cost	6984.00

\[\begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-162}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-308}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{0.5 + \frac{0.5 + \frac{0.5}{x}}{x}}{x} + \frac{\frac{-1}{x}}{x}\right)}{1 + \left(x + \frac{1}{x}\right)}\\ \end{array} \]

Alternative 8
Accuracy	75.7%
Cost	1860.00

\[\begin{array}{l} \mathbf{if}\;t \leq -7.9 \cdot 10^{-299}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{0.5 + \frac{0.5 + \frac{0.5}{x}}{x}}{x} + \frac{\frac{-1}{x}}{x}\right)}{1 + \left(x + \frac{1}{x}\right)}\\ \end{array} \]

Alternative 9
Accuracy	75.7%
Cost	836.00

\[\begin{array}{l} \mathbf{if}\;t \leq -7.9 \cdot 10^{-299}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x \cdot x} + \left(1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 10
Accuracy	75.5%
Cost	452.00

\[\begin{array}{l} \mathbf{if}\;t \leq -7.9 \cdot 10^{-299}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 11
Accuracy	75.2%
Cost	196.00

\[\begin{array}{l} \mathbf{if}\;t \leq -7.9 \cdot 10^{-299}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Accuracy	39.1%
Cost	64.00

\[-1 \]

Reproduce

herbie shell --seed 2023096 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))