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

Average Error: 16.0 → 9.5

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.331687176622029 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}}, x + y\right)\\ \mathbf{elif}\;a \le 1.92285616397135727 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a - t}, y, x\right) + y\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((a <= -1.331687176622029e-240)) {
		VAR = ((double) fma(((double) (((double) (((double) cbrt(((double) (t - z)))) * ((double) cbrt(((double) (t - z)))))) / ((double) (((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))) / ((double) (((double) cbrt(y)) * ((double) cbrt(y)))))))), ((double) (((double) cbrt(((double) (t - z)))) / ((double) (((double) cbrt(((double) (a - t)))) / ((double) (((double) cbrt(((double) (((double) cbrt(y)) * ((double) cbrt(y)))))) * ((double) cbrt(((double) cbrt(y)))))))))), ((double) (x + y))));
	} else {
		double VAR_1;
		if ((a <= 1.9228561639713573e-07)) {
			VAR_1 = ((double) fma(((double) (z / t)), y, x));
		} else {
			VAR_1 = ((double) (((double) fma(((double) (((double) (t - z)) / ((double) (a - t)))), y, x)) + y));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.0
Target	8.4
Herbie	9.5

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

Derivation

1. Split input into 3 regimes

2. if a < -1.331687176622029e-240

   1. Initial program 15.8

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

   2. Simplified 11.0

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

   4. Applied clear-num 11.1

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y}}}, t - z, x + y\right)\]
   5. Using strategy rm

   6. Applied fma-udef 11.1

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

   7. Simplified 11.0

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

   9. Applied add-cube-cbrt 11.2

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

   10. Applied add-cube-cbrt 11.2

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

   11. Applied times-frac 11.2

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

   12. Applied add-cube-cbrt 11.2

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

   13. Applied times-frac 10.1

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

   14. Applied fma-def 10.0

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

   16. Applied add-cube-cbrt 10.1

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

   17. Applied cbrt-prod 10.1

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

   if -1.331687176622029e-240 < a < 1.9228561639713573e-07

   1. Initial program 19.4

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

   2. Simplified 17.9

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

   4. Applied clear-num 18.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y}}}, t - z, x + y\right)\]
   5. Using strategy rm

   6. Applied fma-udef 18.0

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

   7. Simplified 17.8

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

   8. Taylor expanded around inf 13.8

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

   9. Simplified 12.1

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

   if 1.9228561639713573e-07 < a

   1. Initial program 12.7

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

   2. Simplified 6.6

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

   4. Applied clear-num 6.7

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y}}}, t - z, x + y\right)\]
   5. Using strategy rm

   6. Applied fma-udef 6.7

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

   7. Simplified 6.7

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

   9. Applied associate-+r+ 6.7

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

   10. Simplified 5.8

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

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.331687176622029 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}}, x + y\right)\\ \mathbf{elif}\;a \le 1.92285616397135727 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a - t}, y, x\right) + y\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 +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))))