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

Average Error: 20.2 → 0.1

Time: 16.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -65097058713919798902784 \lor \neg \left(z \le 203454031.5286219120025634765625\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r229565 = x;
        double r229566 = y;
        double r229567 = z;
        double r229568 = 0.0692910599291889;
        double r229569 = r229567 * r229568;
        double r229570 = 0.4917317610505968;
        double r229571 = r229569 + r229570;
        double r229572 = r229571 * r229567;
        double r229573 = 0.279195317918525;
        double r229574 = r229572 + r229573;
        double r229575 = r229566 * r229574;
        double r229576 = 6.012459259764103;
        double r229577 = r229567 + r229576;
        double r229578 = r229577 * r229567;
        double r229579 = 3.350343815022304;
        double r229580 = r229578 + r229579;
        double r229581 = r229575 / r229580;
        double r229582 = r229565 + r229581;
        return r229582;
}

↓

double f(double x, double y, double z) {
        double r229583 = z;
        double r229584 = -6.50970587139198e+22;
        bool r229585 = r229583 <= r229584;
        double r229586 = 203454031.5286219;
        bool r229587 = r229583 <= r229586;
        double r229588 = !r229587;
        bool r229589 = r229585 || r229588;
        double r229590 = 0.07512208616047561;
        double r229591 = y;
        double r229592 = r229591 / r229583;
        double r229593 = 0.0692910599291889;
        double r229594 = x;
        double r229595 = fma(r229593, r229591, r229594);
        double r229596 = fma(r229590, r229592, r229595);
        double r229597 = 6.012459259764103;
        double r229598 = r229583 + r229597;
        double r229599 = 3.350343815022304;
        double r229600 = fma(r229598, r229583, r229599);
        double r229601 = r229591 / r229600;
        double r229602 = 0.4917317610505968;
        double r229603 = fma(r229583, r229593, r229602);
        double r229604 = 0.279195317918525;
        double r229605 = fma(r229603, r229583, r229604);
        double r229606 = fma(r229601, r229605, r229594);
        double r229607 = r229589 ? r229596 : r229606;
        return r229607;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.2
Target	0.2
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if z < -6.50970587139198e+22 or 203454031.5286219 < z

   1. Initial program 41.8

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

   2. Simplified 34.4

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

   4. Applied clear-num 34.5

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}{y}}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]
   5. Using strategy rm

   6. Applied div-inv 34.5

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

   7. Applied add-cube-cbrt 34.5

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right) \cdot \frac{1}{y}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]

   8. Applied times-frac 34.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)} \cdot \frac{\sqrt[3]{1}}{\frac{1}{y}}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]

   9. Simplified 34.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{y}}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]

   10. Simplified 34.4

       \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)} \cdot \color{blue}{y}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]
   11. Using strategy rm

   12. Applied add-exp-log 35.6

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

   13. Applied rec-exp 35.6

       \[\leadsto \mathsf{fma}\left(\color{blue}{e^{-\log \left(\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)\right)}} \cdot y, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\]

   14. Taylor expanded around inf 0.0

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

   15. Simplified 0.0

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)}\]

   if -6.50970587139198e+22 < z < 203454031.5286219

   1. Initial program 0.2

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

   2. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -65097058713919798902784 \lor \neg \left(z \le 203454031.5286219120025634765625\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047560960637952121032867580652, \frac{y}{z}, \mathsf{fma}\left(0.06929105992918889456166908757950295694172, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z + 6.012459259764103336465268512256443500519, z, 3.350343815022303939343828460550867021084\right)}, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.06929105992918889456166908757950295694172, 0.4917317610505967939715787906607147306204\right), z, 0.2791953179185249767080279070796677842736\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.6524566747) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291888946) y) (- (/ (* 0.404622038699921249 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291888946) 0.49173176105059679) z) 0.279195317918524977)) (/ 1 (+ (* (+ z 6.0124592597641033) z) 3.35034381502230394)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291888946) y) (- (/ (* 0.404622038699921249 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291888946) 0.49173176105059679) z) 0.279195317918524977)) (+ (* (+ z 6.0124592597641033) z) 3.35034381502230394))))