Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A

Average Error: 7.9 → 0.5

Time: 5.7s

Precision: binary64

Cost: 1928

Math

\[\frac{x + y}{1 - \frac{y}{z}}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -9.090678777324014 \cdot 10^{-235} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= (/ (+ x y) (- 1.0 (/ y z))) -9.090678777324014e-235)
         (not (<= (/ (+ x y) (- 1.0 (/ y z))) 0.0)))
   (/ (+ x y) (- 1.0 (/ y z)))
   (* z (- -1.0 (/ x y)))))

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

↓

double code(double x, double y, double z) {
	double tmp;
	if ((((x + y) / (1.0 - (y / z))) <= -9.090678777324014e-235) || !(((x + y) / (1.0 - (y / z))) <= 0.0)) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.9
Target	4.1
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Alternatives

Alternative 1
Error	15.5
Cost	2374

\[\begin{array}{l} \mathbf{if}\;y \leq -5.763062329060862 \cdot 10^{+24}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -6.773913547581241 \cdot 10^{-233}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.1838034291801365 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.6830616281677957 \cdot 10^{-118}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.3034857445990075 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 8.025233456175185 \cdot 10^{+20}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array}\]

Alternative 2
Error	15.4
Cost	776

\[\begin{array}{l} \mathbf{if}\;y \leq -6.84198291251251 \cdot 10^{+24} \lor \neg \left(y \leq 4.7945480054321624 \cdot 10^{+23}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array}\]

Alternative 3
Error	20.3
Cost	520

\[\begin{array}{l} \mathbf{if}\;y \leq -8.180434561156343 \cdot 10^{+98} \lor \neg \left(y \leq 9.679364348904078 \cdot 10^{+114}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array}\]

Alternative 4
Error	26.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2705230329553015 \cdot 10^{+25} \lor \neg \left(y \leq 2.5204102414744295 \cdot 10^{+25}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 5
Error	41.7
Cost	64

\[x\]

Alternative 6
Error	61.6
Cost	64

\[-1\]

Alternative 7
Error	61.7
Cost	64

\[0\]

Alternative 8
Error	61.6
Cost	64

\[1\]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -9.09067877732401422e-235 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 0.1

      \[\frac{x + y}{1 - \frac{y}{z}}\]

   2. Simplified 0.1

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

   if -9.09067877732401422e-235 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 52.0

      \[\frac{x + y}{1 - \frac{y}{z}}\]

   2. Taylor expanded around 0 52.0

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

   3. Simplified 52.0

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

   4. Taylor expanded around 0 3.1

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

   5. Simplified 3.1

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

   6. Simplified 3.1

      \[\leadsto \color{blue}{z \cdot \left(-1 - \frac{x}{y}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -9.090678777324014 \cdot 10^{-235} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1.0 (/ y z))))