Average Error: 19.9 → 0.4
Time: 12.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.03500197122292579899748687600879258676 \cdot 10^{135} \lor \neg \left(z \le 1.221975873940990937748175085037372761265 \cdot 10^{-9}\right):\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\\ \end{array}\]
x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.03500197122292579899748687600879258676 \cdot 10^{135} \lor \neg \left(z \le 1.221975873940990937748175085037372761265 \cdot 10^{-9}\right):\\
\;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{\frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\\

\end{array}
double f(double x, double y, double z) {
        double r270531 = x;
        double r270532 = y;
        double r270533 = z;
        double r270534 = 0.0692910599291889;
        double r270535 = r270533 * r270534;
        double r270536 = 0.4917317610505968;
        double r270537 = r270535 + r270536;
        double r270538 = r270537 * r270533;
        double r270539 = 0.279195317918525;
        double r270540 = r270538 + r270539;
        double r270541 = r270532 * r270540;
        double r270542 = 6.012459259764103;
        double r270543 = r270533 + r270542;
        double r270544 = r270543 * r270533;
        double r270545 = 3.350343815022304;
        double r270546 = r270544 + r270545;
        double r270547 = r270541 / r270546;
        double r270548 = r270531 + r270547;
        return r270548;
}
↓
double f(double x, double y, double z) {
        double r270549 = z;
        double r270550 = -2.0350019712229258e+135;
        bool r270551 = r270549 <= r270550;
        double r270552 = 1.221975873940991e-09;
        bool r270553 = r270549 <= r270552;
        double r270554 = !r270553;
        bool r270555 = r270551 || r270554;
        double r270556 = x;
        double r270557 = 0.0692910599291889;
        double r270558 = y;
        double r270559 = r270557 * r270558;
        double r270560 = r270558 / r270549;
        double r270561 = 0.07512208616047561;
        double r270562 = 0.40462203869992125;
        double r270563 = r270562 / r270549;
        double r270564 = r270561 - r270563;
        double r270565 = r270560 * r270564;
        double r270566 = r270559 + r270565;
        double r270567 = r270556 + r270566;
        double r270568 = r270549 * r270557;
        double r270569 = 0.4917317610505968;
        double r270570 = r270568 + r270569;
        double r270571 = r270570 * r270549;
        double r270572 = 0.279195317918525;
        double r270573 = r270571 + r270572;
        double r270574 = 6.012459259764103;
        double r270575 = r270549 + r270574;
        double r270576 = r270575 * r270549;
        double r270577 = 3.350343815022304;
        double r270578 = r270576 + r270577;
        double r270579 = sqrt(r270578);
        double r270580 = r270573 / r270579;
        double r270581 = r270580 / r270579;
        double r270582 = r270558 * r270581;
        double r270583 = r270556 + r270582;
        double r270584 = r270555 ? r270567 : r270583;
        return r270584;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747248172760009765625:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737678336:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.0350019712229258e+135 or 1.221975873940991e-09 < z

    1. Initial program 46.4

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity46.4

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\color{blue}{1 \cdot \left(\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084\right)}}\]

    4. Applied times-frac40.9

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\]

    5. Simplified40.9

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

    6. Taylor expanded around inf 0.7

      \[\leadsto x + \color{blue}{\left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right)}\]

    7. Simplified0.7

      \[\leadsto x + \color{blue}{\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)}\]

    if -2.0350019712229258e+135 < z < 1.221975873940991e-09

    1. Initial program 2.9

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity2.9

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\color{blue}{1 \cdot \left(\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084\right)}}\]

    4. Applied times-frac0.1

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\]

    5. Simplified0.1

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt0.5

      \[\leadsto x + y \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\color{blue}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084} \cdot \sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}\]

    8. Applied associate-/r*0.2

      \[\leadsto x + y \cdot \color{blue}{\frac{\frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.03500197122292579899748687600879258676 \cdot 10^{135} \lor \neg \left(z \le 1.221975873940990937748175085037372761265 \cdot 10^{-9}\right):\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))