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

Average Error: 14.8 → 1.5

Time: 4.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \frac{\frac{y}{z} \cdot t}{t} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot \frac{\frac{y}{z} \cdot t}{t} \leq -9.579476509853023 \cdot 10^{-296}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x (/ (* (/ y z) t) t)) (- INFINITY))
   (* y (/ x z))
   (if (<= (* x (/ (* (/ y z) t) t)) -9.579476509853023e-296)
     (* x (/ y z))
     (*
      (* x (/ (* (cbrt y) (cbrt y)) (* (cbrt z) (cbrt z))))
      (/ (cbrt y) (cbrt 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 ((x * (((y / z) * t) / t)) <= -((double) INFINITY)) {
		tmp = y * (x / z);
	} else if ((x * (((y / z) * t) / t)) <= -9.579476509853023e-296) {
		tmp = x * (y / z);
	} else {
		tmp = (x * ((cbrt(y) * cbrt(y)) / (cbrt(z) * cbrt(z)))) * (cbrt(y) / cbrt(z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.8
Target	1.6
Herbie	1.5

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (/.f64 (*.f64 (/.f64 y z) t) t)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 29.4

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

   4. Applied add-cube-cbrt_binary64_14435 30.0

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

   5. Applied associate-*l*_binary64_14341 30.0

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

   6. Simplified 30.0

      \[\leadsto \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \color{blue}{\left(\frac{y}{z} \cdot \sqrt[3]{x}\right)}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity_binary64_14400 30.0

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

   9. Applied add-cube-cbrt_binary64_14435 30.2

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

   10. Applied times-frac_binary64_14406 30.2

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

   11. Applied associate-*l*_binary64_14341 19.3

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

   12. Simplified 19.3

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

   13. Taylor expanded around 0 3.4

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

   14. Simplified 2.6

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

   if -inf.0 < (*.f64 x (/.f64 (*.f64 (/.f64 y z) t) t)) < -9.5794765098530229e-296

   1. Initial program 0.8

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

   2. Simplified 0.4

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

   if -9.5794765098530229e-296 < (*.f64 x (/.f64 (*.f64 (/.f64 y z) t) t))

   1. Initial program 16.6

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

   2. Simplified 7.2

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

   4. Applied add-cube-cbrt_binary64_14435 7.9

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

   5. Applied add-cube-cbrt_binary64_14435 8.1

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

   6. Applied times-frac_binary64_14406 8.1

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

   7. Applied associate-*r*_binary64_14340 1.9

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\frac{y}{z} \cdot t}{t} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot \frac{\frac{y}{z} \cdot t}{t} \leq -9.579476509853023 \cdot 10^{-296}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(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)))