Average Error: 43.0 → 9.2
Time: 47.2s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.651945504637402590186377945564960446005 \cdot 10^{98}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} - \frac{\frac{t}{\sqrt{2}}}{x}\right) - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2 - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 1.423524572675994608344388557654474711065 \cdot 10^{99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}} \cdot \sqrt{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{\frac{t}{\sqrt{2}}}{x} - \frac{t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)}\right) \cdot 2 + \frac{2}{x} \cdot \frac{\frac{t}{\sqrt{2}}}{x}\right) + \sqrt{2} \cdot t}\\ \end{array}\]
\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}
\mathbf{if}\;t \le -3.651945504637402590186377945564960446005 \cdot 10^{98}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} - \frac{\frac{t}{\sqrt{2}}}{x}\right) - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2 - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le 1.423524572675994608344388557654474711065 \cdot 10^{99}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}} \cdot \sqrt{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{\frac{t}{\sqrt{2}}}{x} - \frac{t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)}\right) \cdot 2 + \frac{2}{x} \cdot \frac{\frac{t}{\sqrt{2}}}{x}\right) + \sqrt{2} \cdot t}\\

\end{array}
double f(double x, double l, double t) {
        double r1727087 = 2.0;
        double r1727088 = sqrt(r1727087);
        double r1727089 = t;
        double r1727090 = r1727088 * r1727089;
        double r1727091 = x;
        double r1727092 = 1.0;
        double r1727093 = r1727091 + r1727092;
        double r1727094 = r1727091 - r1727092;
        double r1727095 = r1727093 / r1727094;
        double r1727096 = l;
        double r1727097 = r1727096 * r1727096;
        double r1727098 = r1727089 * r1727089;
        double r1727099 = r1727087 * r1727098;
        double r1727100 = r1727097 + r1727099;
        double r1727101 = r1727095 * r1727100;
        double r1727102 = r1727101 - r1727097;
        double r1727103 = sqrt(r1727102);
        double r1727104 = r1727090 / r1727103;
        return r1727104;
}
↓
double f(double x, double l, double t) {
        double r1727105 = t;
        double r1727106 = -3.6519455046374026e+98;
        bool r1727107 = r1727105 <= r1727106;
        double r1727108 = 2.0;
        double r1727109 = sqrt(r1727108);
        double r1727110 = r1727109 * r1727105;
        double r1727111 = x;
        double r1727112 = r1727111 * r1727111;
        double r1727113 = r1727108 * r1727109;
        double r1727114 = r1727112 * r1727113;
        double r1727115 = r1727105 / r1727114;
        double r1727116 = r1727105 / r1727109;
        double r1727117 = r1727116 / r1727111;
        double r1727118 = r1727115 - r1727117;
        double r1727119 = r1727116 / r1727112;
        double r1727120 = r1727118 - r1727119;
        double r1727121 = r1727120 * r1727108;
        double r1727122 = r1727121 - r1727110;
        double r1727123 = r1727110 / r1727122;
        double r1727124 = 1.4235245726759946e+99;
        bool r1727125 = r1727105 <= r1727124;
        double r1727126 = l;
        double r1727127 = cbrt(r1727111);
        double r1727128 = r1727126 / r1727127;
        double r1727129 = r1727127 * r1727127;
        double r1727130 = r1727126 / r1727129;
        double r1727131 = r1727128 * r1727130;
        double r1727132 = r1727105 * r1727105;
        double r1727133 = r1727131 + r1727132;
        double r1727134 = r1727108 * r1727133;
        double r1727135 = 4.0;
        double r1727136 = r1727111 / r1727132;
        double r1727137 = r1727135 / r1727136;
        double r1727138 = r1727134 + r1727137;
        double r1727139 = sqrt(r1727138);
        double r1727140 = sqrt(r1727139);
        double r1727141 = r1727140 * r1727140;
        double r1727142 = r1727110 / r1727141;
        double r1727143 = r1727117 - r1727115;
        double r1727144 = r1727143 * r1727108;
        double r1727145 = r1727108 / r1727111;
        double r1727146 = r1727145 * r1727117;
        double r1727147 = r1727144 + r1727146;
        double r1727148 = r1727147 + r1727110;
        double r1727149 = r1727110 / r1727148;
        double r1727150 = r1727125 ? r1727142 : r1727149;
        double r1727151 = r1727107 ? r1727123 : r1727150;
        return r1727151;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -3.6519455046374026e+98

    1. Initial program 49.0

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

    2. Taylor expanded around -inf 2.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \left(\sqrt{2} \cdot t + 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right)\right)}}\]

    3. Simplified2.5

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

    if -3.6519455046374026e+98 < t < 1.4235245726759946e+99

    1. Initial program 38.7

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

    2. Taylor expanded around inf 17.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified17.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right) + \frac{4}{\frac{x}{t \cdot t}}}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt17.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right) + \frac{4}{\frac{x}{t \cdot t}}}}\]

    6. Applied times-frac13.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \color{blue}{\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}}\right) + \frac{4}{\frac{x}{t \cdot t}}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt13.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right) + \frac{4}{\frac{x}{t \cdot t}}} \cdot \sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right) + \frac{4}{\frac{x}{t \cdot t}}}}}}\]

    9. Applied sqrt-prod13.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right) + \frac{4}{\frac{x}{t \cdot t}}}} \cdot \sqrt{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right) + \frac{4}{\frac{x}{t \cdot t}}}}}}\]

    if 1.4235245726759946e+99 < t

    1. Initial program 49.5

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

    2. Taylor expanded around inf 3.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \left(\sqrt{2} \cdot t + 2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}}\right)\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]

    3. Simplified3.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2} + \left(\frac{2}{x} \cdot \frac{\frac{t}{\sqrt{2}}}{x} + 2 \cdot \left(\frac{\frac{t}{\sqrt{2}}}{x} - \frac{t}{\left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot 2\right)}\right)\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.651945504637402590186377945564960446005 \cdot 10^{98}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)} - \frac{\frac{t}{\sqrt{2}}}{x}\right) - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2 - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 1.423524572675994608344388557654474711065 \cdot 10^{99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}} \cdot \sqrt{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} + t \cdot t\right) + \frac{4}{\frac{x}{t \cdot t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{\frac{t}{\sqrt{2}}}{x} - \frac{t}{\left(x \cdot x\right) \cdot \left(2 \cdot \sqrt{2}\right)}\right) \cdot 2 + \frac{2}{x} \cdot \frac{\frac{t}{\sqrt{2}}}{x}\right) + \sqrt{2} \cdot t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))