Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Average Error: 6.2 → 5.8

Time: 10.4s

Precision: 64

Math

\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.61197335195023666 \cdot 10^{60}:\\ \;\;\;\;\sqrt{\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)} \cdot \sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r475089 = x;
        double r475090 = 0.5;
        double r475091 = r475089 - r475090;
        double r475092 = log(r475089);
        double r475093 = r475091 * r475092;
        double r475094 = r475093 - r475089;
        double r475095 = 0.91893853320467;
        double r475096 = r475094 + r475095;
        double r475097 = y;
        double r475098 = 0.0007936500793651;
        double r475099 = r475097 + r475098;
        double r475100 = z;
        double r475101 = r475099 * r475100;
        double r475102 = 0.0027777777777778;
        double r475103 = r475101 - r475102;
        double r475104 = r475103 * r475100;
        double r475105 = 0.083333333333333;
        double r475106 = r475104 + r475105;
        double r475107 = r475106 / r475089;
        double r475108 = r475096 + r475107;
        return r475108;
}

↓

double f(double x, double y, double z) {
        double r475109 = x;
        double r475110 = 1.6119733519502367e+60;
        bool r475111 = r475109 <= r475110;
        double r475112 = cbrt(r475109);
        double r475113 = r475112 * r475112;
        double r475114 = log(r475113);
        double r475115 = 0.5;
        double r475116 = r475109 - r475115;
        double r475117 = r475114 * r475116;
        double r475118 = log(r475112);
        double r475119 = r475118 * r475116;
        double r475120 = r475119 - r475109;
        double r475121 = 0.91893853320467;
        double r475122 = r475120 + r475121;
        double r475123 = r475117 + r475122;
        double r475124 = sqrt(r475123);
        double r475125 = log(r475109);
        double r475126 = r475116 * r475125;
        double r475127 = r475126 - r475109;
        double r475128 = r475127 + r475121;
        double r475129 = sqrt(r475128);
        double r475130 = r475124 * r475129;
        double r475131 = y;
        double r475132 = 0.0007936500793651;
        double r475133 = r475131 + r475132;
        double r475134 = z;
        double r475135 = r475133 * r475134;
        double r475136 = 0.0027777777777778;
        double r475137 = r475135 - r475136;
        double r475138 = r475137 * r475134;
        double r475139 = 0.083333333333333;
        double r475140 = r475138 + r475139;
        double r475141 = r475140 / r475109;
        double r475142 = r475130 + r475141;
        double r475143 = 2.0;
        double r475144 = r475143 * r475118;
        double r475145 = r475116 * r475144;
        double r475146 = r475116 * r475118;
        double r475147 = r475145 + r475146;
        double r475148 = r475147 - r475109;
        double r475149 = r475148 + r475121;
        double r475150 = pow(r475134, r475143);
        double r475151 = r475150 / r475109;
        double r475152 = r475132 * r475151;
        double r475153 = 1.0;
        double r475154 = r475153 / r475109;
        double r475155 = r475139 * r475154;
        double r475156 = r475152 + r475155;
        double r475157 = r475134 / r475109;
        double r475158 = r475136 * r475157;
        double r475159 = r475156 - r475158;
        double r475160 = r475149 + r475159;
        double r475161 = r475111 ? r475142 : r475160;
        return r475161;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.2
Target	1.2
Herbie	5.8

\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

1. Split input into 2 regimes

2. if x < 1.6119733519502367e+60

   1. Initial program 0.7

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 0.8

      \[\leadsto \color{blue}{\sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} \cdot \sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001}} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 0.8

      \[\leadsto \sqrt{\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.91893853320467001} \cdot \sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   6. Applied log-prod 0.8

      \[\leadsto \sqrt{\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001} \cdot \sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   7. Applied distribute-rgt-in 0.8

      \[\leadsto \sqrt{\left(\color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right)} - x\right) + 0.91893853320467001} \cdot \sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   8. Applied associate--l+ 0.8

      \[\leadsto \sqrt{\color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right)} + 0.91893853320467001} \cdot \sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   9. Applied associate-+l+ 0.8

      \[\leadsto \sqrt{\color{blue}{\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)}} \cdot \sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   if 1.6119733519502367e+60 < x

   1. Initial program 12.1

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 12.1

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   4. Applied log-prod 12.1

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   5. Applied distribute-lft-in 12.1

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   6. Simplified 12.1

      \[\leadsto \left(\left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   7. Taylor expanded around 0 11.3

      \[\leadsto \left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \color{blue}{\left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.61197335195023666 \cdot 10^{60}:\\ \;\;\;\;\sqrt{\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)} \cdot \sqrt{\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))