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

Average Error: 24.8 → 11.0

Time: 4.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(a - t\right) \cdot \frac{1}{y - x}}, z - t, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.1217753974023432 \cdot 10^{-287}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\frac{a - t}{y - x}} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r587294 = x;
        double r587295 = y;
        double r587296 = r587295 - r587294;
        double r587297 = z;
        double r587298 = t;
        double r587299 = r587297 - r587298;
        double r587300 = r587296 * r587299;
        double r587301 = a;
        double r587302 = r587301 - r587298;
        double r587303 = r587300 / r587302;
        double r587304 = r587294 + r587303;
        return r587304;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r587305 = x;
        double r587306 = y;
        double r587307 = r587306 - r587305;
        double r587308 = z;
        double r587309 = t;
        double r587310 = r587308 - r587309;
        double r587311 = r587307 * r587310;
        double r587312 = a;
        double r587313 = r587312 - r587309;
        double r587314 = r587311 / r587313;
        double r587315 = r587305 + r587314;
        double r587316 = -inf.0;
        bool r587317 = r587315 <= r587316;
        double r587318 = 1.0;
        double r587319 = r587318 / r587307;
        double r587320 = r587313 * r587319;
        double r587321 = r587318 / r587320;
        double r587322 = fma(r587321, r587310, r587305);
        double r587323 = -1.1217753974023432e-287;
        bool r587324 = r587315 <= r587323;
        double r587325 = 0.0;
        bool r587326 = r587315 <= r587325;
        double r587327 = r587313 / r587307;
        double r587328 = r587310 / r587327;
        double r587329 = r587328 + r587305;
        double r587330 = r587326 ? r587306 : r587329;
        double r587331 = r587324 ? r587315 : r587330;
        double r587332 = r587317 ? r587322 : r587331;
        return r587332;
}

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	9.4
Herbie	11.0

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 64.0

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

   2. Simplified 16.3

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

   4. Applied clear-num 16.4

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

   6. Applied div-inv 16.4

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

   if -inf.0 < (+ x (/ (* (- y x) (- z t)) (- a t))) < -1.1217753974023432e-287

   1. Initial program 2.1

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

   if -1.1217753974023432e-287 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 60.1

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

   2. Simplified 60.2

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

   3. Taylor expanded around 0 37.1

      \[\leadsto \color{blue}{y}\]

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

   1. Initial program 21.7

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

   2. Simplified 10.6

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

   4. Applied clear-num 10.8

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

   6. Applied fma-udef 10.9

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

   7. Simplified 10.6

      \[\leadsto \color{blue}{\frac{z - t}{\frac{a - t}{y - x}}} + x\]
3. Recombined 4 regimes into one program.

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(a - t\right) \cdot \frac{1}{y - x}}, z - t, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.1217753974023432 \cdot 10^{-287}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\frac{a - t}{y - x}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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))))