Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Average Error: 11.3 → 1.0

Time: 7.8s

Precision: binary64

Cost: 3217

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -\infty \lor \neg \left(\frac{x \cdot \left(y - z\right)}{t - z} \leq -6.9544927341437 \cdot 10^{-318} \lor \neg \left(\frac{x \cdot \left(y - z\right)}{t - z} \leq 0\right) \land \frac{x \cdot \left(y - z\right)}{t - z} \leq 2.751194885489083 \cdot 10^{-79}\right):\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \end{array}\]

FPCore

(FPCore (x y z t) :precision binary64 (/ (* x (- y z)) (- t z)))

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ (* x (- y z)) (- t z)) (- INFINITY))
         (not
          (or (<= (/ (* x (- y z)) (- t z)) -6.9544927341437e-318)
              (and (not (<= (/ (* x (- y z)) (- t z)) 0.0))
                   (<= (/ (* x (- y z)) (- t z)) 2.751194885489083e-79)))))
   (* (- y z) (/ x (- t z)))
   (/ (* x (- y z)) (- t z))))

C

double code(double x, double y, double z, double t) {
	return (x * (y - z)) / (t - z);
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((x * (y - z)) / (t - z)) <= -((double) INFINITY)) || !((((x * (y - z)) / (t - z)) <= -6.9544927341437e-318) || (!(((x * (y - z)) / (t - z)) <= 0.0) && (((x * (y - z)) / (t - z)) <= 2.751194885489083e-79)))) {
		tmp = (y - z) * (x / (t - z));
	} else {
		tmp = (x * (y - z)) / (t - z);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	11.3
Target	2.1
Herbie	1.0

\[\frac{x}{\frac{t - z}{y - z}}\]

Alternatives

Alternative 1
Error	2.1
Cost	576

\[\frac{x}{\frac{t - z}{y - z}}\]

Alternative 2
Error	7.0
Cost	1860

\[\begin{array}{l} \mathbf{if}\;z \leq -5.687623818202684 \cdot 10^{+193}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{elif}\;z \leq -1.6110760326216112 \cdot 10^{-265}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{elif}\;z \leq 3.6274392266717513 \cdot 10^{-236}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{elif}\;z \leq 4.3843029754894383 \cdot 10^{+160}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \end{array}\]

Alternative 3
Error	16.0
Cost	1090

\[\begin{array}{l} \mathbf{if}\;z \leq -8.672423740642967 \cdot 10^{-28}:\\ \;\;\;\;\frac{-x}{\frac{z}{y - z}}\\ \mathbf{elif}\;z \leq 2.727684865905723 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \end{array}\]

Alternative 4
Error	16.8
Cost	2374

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7351672597883488 \cdot 10^{+193}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{elif}\;z \leq -6.142361359959382 \cdot 10^{+141}:\\ \;\;\;\;\frac{x \cdot \left(z - y\right)}{z}\\ \mathbf{elif}\;z \leq -4.4556909337014073 \cdot 10^{+80}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{elif}\;z \leq -1.3958838192896972 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{elif}\;z \leq -2.311832737277049 \cdot 10^{-06}:\\ \;\;\;\;-z \cdot \frac{x}{t - z}\\ \mathbf{elif}\;z \leq 2.9433372621969594 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \end{array}\]

Alternative 5
Error	16.7
Cost	1683

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7351672597883488 \cdot 10^{+193}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{elif}\;z \leq -1.0623796772080383 \cdot 10^{+141}:\\ \;\;\;\;\frac{x \cdot \left(z - y\right)}{z}\\ \mathbf{elif}\;z \leq -4.4556909337014073 \cdot 10^{+80} \lor \neg \left(z \leq -8.180405541503082 \cdot 10^{+20} \lor \neg \left(z \leq -8.918392719515627 \cdot 10^{-07}\right) \land z \leq 2.5120324696144867 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \end{array}\]

Alternative 6
Error	16.0
Cost	1041

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4681140235601384 \cdot 10^{+80} \lor \neg \left(z \leq -4.803985098633738 \cdot 10^{+18} \lor \neg \left(z \leq -0.00010068718543739433\right) \land z \leq 3.5902944510706688 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \end{array}\]

Alternative 7
Error	21.0
Cost	1090

\[\begin{array}{l} \mathbf{if}\;z \leq -2.212728744097138 \cdot 10^{+80}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.189567283923334 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 8
Error	24.9
Cost	962

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2500329658531297 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.8355110640513414 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 9
Error	38.3
Cost	706

\[\begin{array}{l} \mathbf{if}\;z \leq -3.714802340550633 \cdot 10^{-67}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.2813127836209075 \cdot 10^{-295}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 10
Error	56.1
Cost	64

\[0\]

Alternative 11
Error	61.7
Cost	64

\[1\]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -inf.0 or -6.95449273e-318 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 0.0 or 2.7511948854890829e-79 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 23.9

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

   3. Applied associate-/l*_binary64_16732 1.2

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

   5. Applied associate-/r/_binary64_16733 1.7

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

   6. Simplified 1.7

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

   if -inf.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -6.95449273e-318 or 0.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 2.7511948854890829e-79

   1. Initial program 0.3

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

   2. Simplified 0.3

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -\infty \lor \neg \left(\frac{x \cdot \left(y - z\right)}{t - z} \leq -6.9544927341437 \cdot 10^{-318} \lor \neg \left(\frac{x \cdot \left(y - z\right)}{t - z} \leq 0\right) \land \frac{x \cdot \left(y - z\right)}{t - z} \leq 2.751194885489083 \cdot 10^{-79}\right):\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (/ x (/ (- t z) (- y z)))

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