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

Average Error: 14.4 → 0.5

Time: 3.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -6.0997181107985366 \cdot 10^{+171}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1.348124329597327 \cdot 10^{-160}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 1.924299285446891 \cdot 10^{-222}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 1.7647629544811067 \cdot 10^{+232}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{y}}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ y z) -6.0997181107985366e+171)
   (* (* y x) (/ 1.0 z))
   (if (<= (/ y z) -1.348124329597327e-160)
     (/ x (/ z y))
     (if (<= (/ y z) 1.924299285446891e-222)
       (* y (/ x z))
       (if (<= (/ y z) 1.7647629544811067e+232)
         (/ x (/ z y))
         (/ (/ x z) (/ 1.0 y)))))))

C

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y / z) <= -6.0997181107985366e+171) {
		tmp = (y * x) * (1.0 / z);
	} else if ((y / z) <= -1.348124329597327e-160) {
		tmp = x / (z / y);
	} else if ((y / z) <= 1.924299285446891e-222) {
		tmp = y * (x / z);
	} else if ((y / z) <= 1.7647629544811067e+232) {
		tmp = x / (z / y);
	} else {
		tmp = (x / z) / (1.0 / y);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.4
Target	1.3
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} < -1.20672205123045 \cdot 10^{+245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < -5.907522236933906 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < 5.658954423153415 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < 2.0087180502407133 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (/.f64 y z) < -6.09971811079853655e171

   1. Initial program 36.7

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

   2. Simplified 22.9

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied div-inv_binary64_18489 23.1

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

   5. Applied associate-*r*_binary64_18432 2.2

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

   if -6.09971811079853655e171 < (/.f64 y z) < -1.348124329597327e-160 or 1.9242992854468911e-222 < (/.f64 y z) < 1.76476295448110672e232

   1. Initial program 8.2

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied associate-*r/_binary64_18434 9.6

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]
   5. Using strategy rm

   6. Applied associate-/l*_binary64_18437 0.2

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

   if -1.348124329597327e-160 < (/.f64 y z) < 1.9242992854468911e-222

   1. Initial program 16.8

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

   2. Simplified 9.9

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied associate-*r/_binary64_18434 0.7

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]
   5. Using strategy rm

   6. Applied associate-/l*_binary64_18437 10.6

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

   8. Applied associate-/r/_binary64_18438 0.6

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y}\]

   if 1.76476295448110672e232 < (/.f64 y z)

   1. Initial program 47.2

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

   2. Simplified 32.4

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied associate-*r/_binary64_18434 0.4

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]
   5. Using strategy rm

   6. Applied associate-/l*_binary64_18437 30.0

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

   8. Applied div-inv_binary64_18489 30.0

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

   9. Applied associate-/r*_binary64_18436 0.6

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{1}{y}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -6.0997181107985366 \cdot 10^{+171}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1.348124329597327 \cdot 10^{-160}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 1.924299285446891 \cdot 10^{-222}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 1.7647629544811067 \cdot 10^{+232}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

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