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

Average Error: 6.0 → 5.1

Time: 8.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 2.84987569520092838 \cdot 10^{298}:\\ \;\;\;\;\left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{1}{\frac{x}{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{1}{\left(0.400000000000006406 \cdot \left(x \cdot z\right) + 12.000000000000048 \cdot x\right) - 0.100952278095241613 \cdot \left(x \cdot {z}^{2}\right)}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r483087 = x;
        double r483088 = 0.5;
        double r483089 = r483087 - r483088;
        double r483090 = log(r483087);
        double r483091 = r483089 * r483090;
        double r483092 = r483091 - r483087;
        double r483093 = 0.91893853320467;
        double r483094 = r483092 + r483093;
        double r483095 = y;
        double r483096 = 0.0007936500793651;
        double r483097 = r483095 + r483096;
        double r483098 = z;
        double r483099 = r483097 * r483098;
        double r483100 = 0.0027777777777778;
        double r483101 = r483099 - r483100;
        double r483102 = r483101 * r483098;
        double r483103 = 0.083333333333333;
        double r483104 = r483102 + r483103;
        double r483105 = r483104 / r483087;
        double r483106 = r483094 + r483105;
        return r483106;
}

↓

double f(double x, double y, double z) {
        double r483107 = y;
        double r483108 = 0.0007936500793651;
        double r483109 = r483107 + r483108;
        double r483110 = z;
        double r483111 = r483109 * r483110;
        double r483112 = 0.0027777777777778;
        double r483113 = r483111 - r483112;
        double r483114 = r483113 * r483110;
        double r483115 = 2.8498756952009284e+298;
        bool r483116 = r483114 <= r483115;
        double r483117 = x;
        double r483118 = cbrt(r483117);
        double r483119 = r483118 * r483118;
        double r483120 = log(r483119);
        double r483121 = 0.5;
        double r483122 = r483117 - r483121;
        double r483123 = r483120 * r483122;
        double r483124 = 1.0;
        double r483125 = r483124 / r483117;
        double r483126 = -0.3333333333333333;
        double r483127 = pow(r483125, r483126);
        double r483128 = log(r483127);
        double r483129 = r483128 * r483122;
        double r483130 = r483129 - r483117;
        double r483131 = r483123 + r483130;
        double r483132 = 0.91893853320467;
        double r483133 = r483131 + r483132;
        double r483134 = 0.083333333333333;
        double r483135 = r483114 + r483134;
        double r483136 = r483117 / r483135;
        double r483137 = r483124 / r483136;
        double r483138 = r483133 + r483137;
        double r483139 = 0.4000000000000064;
        double r483140 = r483117 * r483110;
        double r483141 = r483139 * r483140;
        double r483142 = 12.000000000000048;
        double r483143 = r483142 * r483117;
        double r483144 = r483141 + r483143;
        double r483145 = 0.10095227809524161;
        double r483146 = 2.0;
        double r483147 = pow(r483110, r483146);
        double r483148 = r483117 * r483147;
        double r483149 = r483145 * r483148;
        double r483150 = r483144 - r483149;
        double r483151 = r483124 / r483150;
        double r483152 = r483133 + r483151;
        double r483153 = r483116 ? r483138 : r483152;
        return r483153;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	1.3
Herbie	5.1

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

Derivation

1. Split input into 2 regimes

2. if (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < 2.8498756952009284e+298

   1. Initial program 1.9

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

   3. Applied add-cube-cbrt 1.9

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

   4. Applied log-prod 1.9

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

   5. Applied distribute-rgt-in 1.9

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

   6. Applied associate--l+ 1.9

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

   7. Taylor expanded around inf 1.9

      \[\leadsto \left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \color{blue}{\left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)} \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
   8. Using strategy rm

   9. Applied clear-num 2.0

      \[\leadsto \left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \color{blue}{\frac{1}{\frac{x}{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}}}\]

   if 2.8498756952009284e+298 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)

   1. Initial program 59.9

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

   3. Applied add-cube-cbrt 59.9

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

   4. Applied log-prod 59.9

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

   5. Applied distribute-rgt-in 59.9

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

   6. Applied associate--l+ 60.0

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

   7. Taylor expanded around inf 59.9

      \[\leadsto \left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \color{blue}{\left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)} \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
   8. Using strategy rm

   9. Applied clear-num 60.0

      \[\leadsto \left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \color{blue}{\frac{1}{\frac{x}{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}}}\]

   10. Taylor expanded around 0 47.0

       \[\leadsto \left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{1}{\color{blue}{\left(0.400000000000006406 \cdot \left(x \cdot z\right) + 12.000000000000048 \cdot x\right) - 0.100952278095241613 \cdot \left(x \cdot {z}^{2}\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 2.84987569520092838 \cdot 10^{298}:\\ \;\;\;\;\left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{1}{\frac{x}{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{1}{\left(0.400000000000006406 \cdot \left(x \cdot z\right) + 12.000000000000048 \cdot x\right) - 0.100952278095241613 \cdot \left(x \cdot {z}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))