Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Average Error: 7.6 → 4.5

Time: 9.5s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le -2.72059074699743426 \cdot 10^{236} \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 2.40235735934087887 \cdot 10^{45}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r4014 = x;
        double r4015 = y;
        double r4016 = z;
        double r4017 = r4015 * r4016;
        double r4018 = r4017 - r4014;
        double r4019 = t;
        double r4020 = r4019 * r4016;
        double r4021 = r4020 - r4014;
        double r4022 = r4018 / r4021;
        double r4023 = r4014 + r4022;
        double r4024 = 1.0;
        double r4025 = r4014 + r4024;
        double r4026 = r4023 / r4025;
        return r4026;
}

↓

double f(double x, double y, double z, double t) {
        double r4027 = x;
        double r4028 = y;
        double r4029 = z;
        double r4030 = r4028 * r4029;
        double r4031 = r4030 - r4027;
        double r4032 = t;
        double r4033 = r4032 * r4029;
        double r4034 = r4033 - r4027;
        double r4035 = r4031 / r4034;
        double r4036 = r4027 + r4035;
        double r4037 = 1.0;
        double r4038 = r4027 + r4037;
        double r4039 = r4036 / r4038;
        double r4040 = -2.7205907469974343e+236;
        bool r4041 = r4039 <= r4040;
        double r4042 = 2.402357359340879e+45;
        bool r4043 = r4039 <= r4042;
        double r4044 = !r4043;
        bool r4045 = r4041 || r4044;
        double r4046 = r4028 / r4032;
        double r4047 = r4027 + r4046;
        double r4048 = r4047 / r4038;
        double r4049 = r4045 ? r4048 : r4039;
        return r4049;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.6
Target	0.4
Herbie	4.5

\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

1. Split input into 2 regimes

2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < -2.7205907469974343e+236 or 2.402357359340879e+45 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

   1. Initial program 37.4

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

   2. Taylor expanded around inf 20.6

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]

   if -2.7205907469974343e+236 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 2.402357359340879e+45

   1. Initial program 0.8

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le -2.72059074699743426 \cdot 10^{236} \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 2.40235735934087887 \cdot 10^{45}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))