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

Average Error: 14.5 → 2.3

Time: 3.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -4.259602136673019 \cdot 10^{+88}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1.5121500752879295 \cdot 10^{-132}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z} \cdot t}{t}\\ \mathbf{elif}\;\frac{y}{z} \leq 8.489377968209908 \cdot 10^{-168} \lor \neg \left(\frac{y}{z} \leq 1.3753328211343246 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{y}{z}} \cdot \left(x \cdot \sqrt{\frac{y}{z}}\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ y z) -4.259602136673019e+88)
   (* y (/ x z))
   (if (<= (/ y z) -1.5121500752879295e-132)
     (* x (/ (* (/ y z) t) t))
     (if (or (<= (/ y z) 8.489377968209908e-168)
             (not (<= (/ y z) 1.3753328211343246e+162)))
       (/ (* y x) z)
       (* (sqrt (/ y z)) (* x (sqrt (/ y z))))))))

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) <= -4.259602136673019e+88) {
		tmp = y * (x / z);
	} else if ((y / z) <= -1.5121500752879295e-132) {
		tmp = x * (((y / z) * t) / t);
	} else if (((y / z) <= 8.489377968209908e-168) || !((y / z) <= 1.3753328211343246e+162)) {
		tmp = (y * x) / z;
	} else {
		tmp = sqrt(y / z) * (x * sqrt(y / z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.5
Target	1.6
Herbie	2.3

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 y z) < -4.259602136673019e88

   1. Initial program 27.5

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

   2. Simplified 11.9

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

   4. Applied add-cube-cbrt_binary64 12.8

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

   5. Applied *-un-lft-identity_binary64 12.8

      \[\leadsto x \cdot \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

   6. Applied times-frac_binary64 12.8

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right)}\]

   7. Applied associate-*r*_binary64 4.8

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{y}{\sqrt[3]{z}}}\]

   8. Simplified 4.9

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\sqrt[3]{z}}\]

   9. Taylor expanded around 0 6.0

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

   10. Simplified 3.9

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

   if -4.259602136673019e88 < (/.f64 y z) < -1.51215007528793e-132

   1. Initial program 5.2

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

   if -1.51215007528793e-132 < (/.f64 y z) < 8.48937796820990803e-168 or 1.37533282113432464e162 < (/.f64 y z)

   1. Initial program 19.5

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

   2. Simplified 10.9

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

   4. Applied associate-*r/_binary64 1.5

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

   if 8.48937796820990803e-168 < (/.f64 y z) < 1.37533282113432464e162

   1. Initial program 6.6

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

   2. Simplified 0.2

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

   4. Applied add-sqr-sqrt_binary64 0.6

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

   5. Applied associate-*r*_binary64 0.6

      \[\leadsto \color{blue}{\left(x \cdot \sqrt{\frac{y}{z}}\right) \cdot \sqrt{\frac{y}{z}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -4.259602136673019 \cdot 10^{+88}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1.5121500752879295 \cdot 10^{-132}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z} \cdot t}{t}\\ \mathbf{elif}\;\frac{y}{z} \leq 8.489377968209908 \cdot 10^{-168} \lor \neg \left(\frac{y}{z} \leq 1.3753328211343246 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{y}{z}} \cdot \left(x \cdot \sqrt{\frac{y}{z}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(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)))