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

Average Error: 24.2 → 9.4

Time: 5.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(y - x\right) \cdot \frac{1}{a - t}, z - t, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -5.11466826064621464 \cdot 10^{-284}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;y\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 2.46784281805568474 \cdot 10^{285}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - x\right) \cdot \frac{1}{a - t}, z - t, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r714934 = x;
        double r714935 = y;
        double r714936 = r714935 - r714934;
        double r714937 = z;
        double r714938 = t;
        double r714939 = r714937 - r714938;
        double r714940 = r714936 * r714939;
        double r714941 = a;
        double r714942 = r714941 - r714938;
        double r714943 = r714940 / r714942;
        double r714944 = r714934 + r714943;
        return r714944;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r714945 = x;
        double r714946 = y;
        double r714947 = r714946 - r714945;
        double r714948 = z;
        double r714949 = t;
        double r714950 = r714948 - r714949;
        double r714951 = r714947 * r714950;
        double r714952 = a;
        double r714953 = r714952 - r714949;
        double r714954 = r714951 / r714953;
        double r714955 = r714945 + r714954;
        double r714956 = -inf.0;
        bool r714957 = r714955 <= r714956;
        double r714958 = 1.0;
        double r714959 = r714958 / r714953;
        double r714960 = r714947 * r714959;
        double r714961 = fma(r714960, r714950, r714945);
        double r714962 = -5.1146682606462146e-284;
        bool r714963 = r714955 <= r714962;
        double r714964 = 0.0;
        bool r714965 = r714955 <= r714964;
        double r714966 = 2.467842818055685e+285;
        bool r714967 = r714955 <= r714966;
        double r714968 = r714967 ? r714955 : r714961;
        double r714969 = r714965 ? r714946 : r714968;
        double r714970 = r714963 ? r714955 : r714969;
        double r714971 = r714957 ? r714961 : r714970;
        return r714971;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.2
Target	9.5
Herbie	9.4

\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -inf.0 or 2.467842818055685e+285 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 62.2

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

   2. Simplified 17.3

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

   4. Applied div-inv 17.4

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

   if -inf.0 < (+ x (/ (* (- y x) (- z t)) (- a t))) < -5.1146682606462146e-284 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 2.467842818055685e+285

   1. Initial program 2.0

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

   if -5.1146682606462146e-284 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 59.5

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

   2. Simplified 59.6

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

   3. Taylor expanded around 0 36.5

      \[\leadsto \color{blue}{y}\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(y - x\right) \cdot \frac{1}{a - t}, z - t, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -5.11466826064621464 \cdot 10^{-284}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;y\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 2.46784281805568474 \cdot 10^{285}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - x\right) \cdot \frac{1}{a - t}, z - t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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))))