Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B

Average Error: 20.0 → 0.2

Time: 1.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -6134214662947.4755859375 \lor \neg \left(z \le 0.02992000358306671353725292306080518756062\right):\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + 0.07512208616047560960637952121032867580652 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{z \cdot \left(0.06929105992918889456166908757950295694172 \cdot z + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736}{3.350343815022303939343828460550867021084 + z \cdot \left(z + 6.012459259764103336465268512256443500519\right)}\right)}^{3}} \cdot y + x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r297455 = x;
        double r297456 = y;
        double r297457 = z;
        double r297458 = 0.0692910599291889;
        double r297459 = r297457 * r297458;
        double r297460 = 0.4917317610505968;
        double r297461 = r297459 + r297460;
        double r297462 = r297461 * r297457;
        double r297463 = 0.279195317918525;
        double r297464 = r297462 + r297463;
        double r297465 = r297456 * r297464;
        double r297466 = 6.012459259764103;
        double r297467 = r297457 + r297466;
        double r297468 = r297467 * r297457;
        double r297469 = 3.350343815022304;
        double r297470 = r297468 + r297469;
        double r297471 = r297465 / r297470;
        double r297472 = r297455 + r297471;
        return r297472;
}

↓

double f(double x, double y, double z) {
        double r297473 = z;
        double r297474 = -6134214662947.476;
        bool r297475 = r297473 <= r297474;
        double r297476 = 0.029920003583066714;
        bool r297477 = r297473 <= r297476;
        double r297478 = !r297477;
        bool r297479 = r297475 || r297478;
        double r297480 = x;
        double r297481 = 0.0692910599291889;
        double r297482 = y;
        double r297483 = r297481 * r297482;
        double r297484 = 0.07512208616047561;
        double r297485 = r297482 / r297473;
        double r297486 = r297484 * r297485;
        double r297487 = r297483 + r297486;
        double r297488 = r297480 + r297487;
        double r297489 = r297481 * r297473;
        double r297490 = 0.4917317610505968;
        double r297491 = r297489 + r297490;
        double r297492 = r297473 * r297491;
        double r297493 = 0.279195317918525;
        double r297494 = r297492 + r297493;
        double r297495 = 3.350343815022304;
        double r297496 = 6.012459259764103;
        double r297497 = r297473 + r297496;
        double r297498 = r297473 * r297497;
        double r297499 = r297495 + r297498;
        double r297500 = r297494 / r297499;
        double r297501 = 3.0;
        double r297502 = pow(r297500, r297501);
        double r297503 = cbrt(r297502);
        double r297504 = r297503 * r297482;
        double r297505 = r297504 + r297480;
        double r297506 = r297479 ? r297488 : r297505;
        return r297506;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.0
Target	0.2
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if z < -6134214662947.476 or 0.029920003583066714 < z

   1. Initial program 40.8

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

   2. Simplified 32.4

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

   3. Taylor expanded around inf 0.3

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

   4. Simplified 0.3

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

   if -6134214662947.476 < z < 0.029920003583066714

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   4. Applied add-cbrt-cube 0.1

      \[\leadsto \frac{\left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z + 0.2791953179185249767080279070796677842736}{\color{blue}{\sqrt[3]{\left(\left(z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084\right) \cdot \left(z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084\right)\right) \cdot \left(z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084\right)}}} \cdot y + x\]

   5. Applied add-cbrt-cube 0.1

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z + 0.2791953179185249767080279070796677842736\right) \cdot \left(\left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \left(\left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}}}{\sqrt[3]{\left(\left(z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084\right) \cdot \left(z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084\right)\right) \cdot \left(z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084\right)}} \cdot y + x\]

   6. Applied cbrt-undiv 0.1

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(\left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z + 0.2791953179185249767080279070796677842736\right) \cdot \left(\left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \left(\left(0.4917317610505967939715787906607147306204 + z \cdot 0.06929105992918889456166908757950295694172\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(\left(z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084\right) \cdot \left(z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084\right)\right) \cdot \left(z \cdot \left(z + 6.012459259764103336465268512256443500519\right) + 3.350343815022303939343828460550867021084\right)}}} \cdot y + x\]

   7. Simplified 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6134214662947.4755859375 \lor \neg \left(z \le 0.02992000358306671353725292306080518756062\right):\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + 0.07512208616047560960637952121032867580652 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{z \cdot \left(0.06929105992918889456166908757950295694172 \cdot z + 0.4917317610505967939715787906607147306204\right) + 0.2791953179185249767080279070796677842736}{3.350343815022303939343828460550867021084 + z \cdot \left(z + 6.012459259764103336465268512256443500519\right)}\right)}^{3}} \cdot y + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1.0 (+ (* (+ 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))))