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

Average Error: 24.8 → 8.2

Time: 45.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.533543267934166183549967405153551686617 \cdot 10^{-301}:\\ \;\;\;\;\frac{y - x}{\frac{a - t}{z - t}} + x\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\left(\frac{z \cdot x}{t} + y\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{\frac{a - t}{z - t}} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r29664086 = x;
        double r29664087 = y;
        double r29664088 = r29664087 - r29664086;
        double r29664089 = z;
        double r29664090 = t;
        double r29664091 = r29664089 - r29664090;
        double r29664092 = r29664088 * r29664091;
        double r29664093 = a;
        double r29664094 = r29664093 - r29664090;
        double r29664095 = r29664092 / r29664094;
        double r29664096 = r29664086 + r29664095;
        return r29664096;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r29664097 = x;
        double r29664098 = y;
        double r29664099 = r29664098 - r29664097;
        double r29664100 = z;
        double r29664101 = t;
        double r29664102 = r29664100 - r29664101;
        double r29664103 = r29664099 * r29664102;
        double r29664104 = a;
        double r29664105 = r29664104 - r29664101;
        double r29664106 = r29664103 / r29664105;
        double r29664107 = r29664097 + r29664106;
        double r29664108 = -1.5335432679341662e-301;
        bool r29664109 = r29664107 <= r29664108;
        double r29664110 = r29664105 / r29664102;
        double r29664111 = r29664099 / r29664110;
        double r29664112 = r29664111 + r29664097;
        double r29664113 = 0.0;
        bool r29664114 = r29664107 <= r29664113;
        double r29664115 = r29664100 * r29664097;
        double r29664116 = r29664115 / r29664101;
        double r29664117 = r29664116 + r29664098;
        double r29664118 = r29664100 * r29664098;
        double r29664119 = r29664118 / r29664101;
        double r29664120 = r29664117 - r29664119;
        double r29664121 = r29664114 ? r29664120 : r29664112;
        double r29664122 = r29664109 ? r29664112 : r29664121;
        return r29664122;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.8
Target	9.3
Herbie	8.2

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -1.5335432679341662e-301 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 21.5

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
   2. Using strategy rm

   3. Applied associate-/l* 7.3

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

   if -1.5335432679341662e-301 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 60.8

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

   2. Taylor expanded around inf 17.6

      \[\leadsto \color{blue}{\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.533543267934166183549967405153551686617 \cdot 10^{-301}:\\ \;\;\;\;\frac{y - x}{\frac{a - t}{z - t}} + x\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\left(\frac{z \cdot x}{t} + y\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{\frac{a - t}{z - t}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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