Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Average Error: 25.0 → 7.5

Time: 6.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \left(\frac{t}{\frac{a - t}{y - x}} - x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -6.156727181680058362159255815336745142488 \cdot 10^{-230}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 1.508571022062013089185934498746472468243 \cdot 10^{205}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \left(\frac{t}{\frac{a - t}{y - x}} - x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r612673 = x;
        double r612674 = y;
        double r612675 = r612674 - r612673;
        double r612676 = z;
        double r612677 = t;
        double r612678 = r612676 - r612677;
        double r612679 = r612675 * r612678;
        double r612680 = a;
        double r612681 = r612680 - r612677;
        double r612682 = r612679 / r612681;
        double r612683 = r612673 + r612682;
        return r612683;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r612684 = x;
        double r612685 = y;
        double r612686 = r612685 - r612684;
        double r612687 = z;
        double r612688 = t;
        double r612689 = r612687 - r612688;
        double r612690 = r612686 * r612689;
        double r612691 = a;
        double r612692 = r612691 - r612688;
        double r612693 = r612690 / r612692;
        double r612694 = r612684 + r612693;
        double r612695 = -inf.0;
        bool r612696 = r612694 <= r612695;
        double r612697 = r612692 / r612686;
        double r612698 = r612687 / r612697;
        double r612699 = r612688 / r612697;
        double r612700 = r612699 - r612684;
        double r612701 = r612698 - r612700;
        double r612702 = -6.156727181680058e-230;
        bool r612703 = r612694 <= r612702;
        double r612704 = 0.0;
        bool r612705 = r612694 <= r612704;
        double r612706 = r612684 / r612688;
        double r612707 = r612687 * r612685;
        double r612708 = r612707 / r612688;
        double r612709 = r612685 - r612708;
        double r612710 = fma(r612706, r612687, r612709);
        double r612711 = 1.508571022062013e+205;
        bool r612712 = r612694 <= r612711;
        double r612713 = r612712 ? r612694 : r612701;
        double r612714 = r612705 ? r612710 : r612713;
        double r612715 = r612703 ? r612694 : r612714;
        double r612716 = r612696 ? r612701 : r612715;
        return r612716;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.0
Target	9.6
Herbie	7.5

\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -inf.0 or 1.508571022062013e+205 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 55.2

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

   2. Simplified 16.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a - t}, z - t, x\right)}\]
   3. Using strategy rm

   4. Applied clear-num 16.5

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y - x}}}, z - t, x\right)\]
   5. Using strategy rm

   6. Applied fma-udef 16.6

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

   7. Simplified 16.4

      \[\leadsto \color{blue}{\frac{z - t}{\frac{a - t}{y - x}}} + x\]
   8. Using strategy rm

   9. Applied div-sub 16.4

      \[\leadsto \color{blue}{\left(\frac{z}{\frac{a - t}{y - x}} - \frac{t}{\frac{a - t}{y - x}}\right)} + x\]

   10. Applied associate-+l- 12.2

       \[\leadsto \color{blue}{\frac{z}{\frac{a - t}{y - x}} - \left(\frac{t}{\frac{a - t}{y - x}} - x\right)}\]

   if -inf.0 < (+ x (/ (* (- y x) (- z t)) (- a t))) < -6.156727181680058e-230 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 1.508571022062013e+205

   1. Initial program 1.9

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

   if -6.156727181680058e-230 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 54.4

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

   2. Simplified 55.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{a - t}, z - t, x\right)}\]
   3. Using strategy rm

   4. Applied clear-num 55.1

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y - x}}}, z - t, x\right)\]
   5. Using strategy rm

   6. Applied fma-udef 55.3

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

   7. Simplified 55.3

      \[\leadsto \color{blue}{\frac{z - t}{\frac{a - t}{y - x}}} + x\]

   8. Taylor expanded around inf 22.4

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

   9. Simplified 24.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \left(\frac{t}{\frac{a - t}{y - x}} - x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -6.156727181680058362159255815336745142488 \cdot 10^{-230}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 1.508571022062013089185934498746472468243 \cdot 10^{205}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \left(\frac{t}{\frac{a - t}{y - x}} - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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