Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Average Error: 16.5 → 9.4

Time: 7.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -2.922927865251897 \cdot 10^{-288}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}, \frac{\sqrt[3]{y}}{a - t} \cdot \left(t - z\right), x + y\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.71333 \cdot 10^{-166}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right) \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(t - z\right), x + y\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double temp;
	if ((((x + y) - (((z - t) * y) / (a - t))) <= -2.922927865251897e-288)) {
		temp = fma(((cbrt(y) * cbrt(y)) / 1.0), ((cbrt(y) / (a - t)) * (t - z)), (x + y));
	} else {
		double temp_1;
		if ((((x + y) - (((z - t) * y) / (a - t))) <= 2.7133285516175262e-166)) {
			temp_1 = x;
		} else {
			temp_1 = fma(((cbrt(y) * cbrt(y)) / ((cbrt((cbrt((a - t)) * cbrt((a - t)))) * cbrt(cbrt((a - t)))) * cbrt((a - t)))), ((cbrt(y) / cbrt((a - t))) * (t - z)), (x + y));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.5
Target	8.7
Herbie	9.4

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -2.922927865251897e-288

   1. Initial program 12.9

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

   2. Simplified 7.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a - t}, t - z, x + y\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 7.4

      \[\leadsto \color{blue}{\frac{y}{a - t} \cdot \left(t - z\right) + \left(x + y\right)}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 7.4

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

   7. Applied add-cube-cbrt 7.7

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

   8. Applied times-frac 7.7

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

   9. Applied associate-*l* 7.1

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

   10. Applied fma-def 7.1

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}, \frac{\sqrt[3]{y}}{a - t} \cdot \left(t - z\right), x + y\right)}\]

   if -2.922927865251897e-288 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 2.7133285516175262e-166

   1. Initial program 50.3

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

   2. Simplified 50.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a - t}, t - z, x + y\right)}\]

   3. Taylor expanded around 0 33.1

      \[\leadsto \color{blue}{x}\]

   if 2.7133285516175262e-166 < (- (+ x y) (/ (* (- z t) y) (- a t)))

   1. Initial program 12.6

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

   2. Simplified 7.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a - t}, t - z, x + y\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 7.3

      \[\leadsto \color{blue}{\frac{y}{a - t} \cdot \left(t - z\right) + \left(x + y\right)}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 7.6

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

   7. Applied add-cube-cbrt 7.6

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

   8. Applied times-frac 7.6

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

   9. Applied associate-*l* 6.3

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

   10. Applied fma-def 6.3

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(t - z\right), x + y\right)}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 6.3

       \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(t - z\right), x + y\right)\]

   13. Applied cbrt-prod 6.3

       \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right)} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(t - z\right), x + y\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -2.922927865251897 \cdot 10^{-288}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}, \frac{\sqrt[3]{y}}{a - t} \cdot \left(t - z\right), x + y\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.71333 \cdot 10^{-166}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right) \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(t - z\right), x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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