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

Average Error: 24.3 → 8.3

Time: 21.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.828093467919815950939216553514049710649 \cdot 10^{-286}:\\ \;\;\;\;x + \left(y - x\right) \cdot \left(\left(z - t\right) \cdot \frac{1}{a - t}\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 7.220777503459318304277048324694986155104 \cdot 10^{-276}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t} - \frac{t}{z - t}}{y - x}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r41996928 = x;
        double r41996929 = y;
        double r41996930 = r41996929 - r41996928;
        double r41996931 = z;
        double r41996932 = t;
        double r41996933 = r41996931 - r41996932;
        double r41996934 = r41996930 * r41996933;
        double r41996935 = a;
        double r41996936 = r41996935 - r41996932;
        double r41996937 = r41996934 / r41996936;
        double r41996938 = r41996928 + r41996937;
        return r41996938;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r41996939 = x;
        double r41996940 = y;
        double r41996941 = r41996940 - r41996939;
        double r41996942 = z;
        double r41996943 = t;
        double r41996944 = r41996942 - r41996943;
        double r41996945 = r41996941 * r41996944;
        double r41996946 = a;
        double r41996947 = r41996946 - r41996943;
        double r41996948 = r41996945 / r41996947;
        double r41996949 = r41996939 + r41996948;
        double r41996950 = -1.828093467919816e-286;
        bool r41996951 = r41996949 <= r41996950;
        double r41996952 = 1.0;
        double r41996953 = r41996952 / r41996947;
        double r41996954 = r41996944 * r41996953;
        double r41996955 = r41996941 * r41996954;
        double r41996956 = r41996939 + r41996955;
        double r41996957 = 7.220777503459318e-276;
        bool r41996958 = r41996949 <= r41996957;
        double r41996959 = r41996939 * r41996942;
        double r41996960 = r41996959 / r41996943;
        double r41996961 = r41996940 + r41996960;
        double r41996962 = r41996942 * r41996940;
        double r41996963 = r41996962 / r41996943;
        double r41996964 = r41996961 - r41996963;
        double r41996965 = r41996946 / r41996944;
        double r41996966 = r41996943 / r41996944;
        double r41996967 = r41996965 - r41996966;
        double r41996968 = r41996967 / r41996941;
        double r41996969 = r41996952 / r41996968;
        double r41996970 = r41996939 + r41996969;
        double r41996971 = r41996958 ? r41996964 : r41996970;
        double r41996972 = r41996951 ? r41996956 : r41996971;
        return r41996972;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.3
Target	9.2
Herbie	8.3

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

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -1.828093467919816e-286

   1. Initial program 21.8

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

   3. Applied *-un-lft-identity 21.8

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

   4. Applied times-frac 7.4

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

   5. Simplified 7.4

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

   7. Applied div-inv 7.5

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

   if -1.828093467919816e-286 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 7.220777503459318e-276

   1. Initial program 58.0

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

   2. Taylor expanded around inf 20.8

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

   if 7.220777503459318e-276 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 20.6

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

   3. Applied associate-/l* 6.8

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

   5. Applied div-sub 6.8

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

   7. Applied clear-num 6.9

      \[\leadsto x + \color{blue}{\frac{1}{\frac{\frac{a}{z - t} - \frac{t}{z - t}}{y - x}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.828093467919815950939216553514049710649 \cdot 10^{-286}:\\ \;\;\;\;x + \left(y - x\right) \cdot \left(\left(z - t\right) \cdot \frac{1}{a - t}\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 7.220777503459318304277048324694986155104 \cdot 10^{-276}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t} - \frac{t}{z - t}}{y - x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))))