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

Average Error: 16.0 → 7.4

Time: 5.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.10120344687679351 \cdot 10^{108} \lor \neg \left(t \le 2.7030262370830962 \cdot 10^{76}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, x + y\right)\\ \end{array}\]

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 (((t <= -2.1012034468767935e+108) || !(t <= 2.7030262370830962e+76))) {
		temp = fma((z / t), y, x);
	} else {
		temp = fma(((cbrt((t - z)) * cbrt((t - z))) / ((cbrt((a - t)) * cbrt((a - t))) / (cbrt(y) * cbrt(y)))), (cbrt((t - z)) / (cbrt((a - t)) / cbrt(y))), (x + y));
	}
	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	16.0
Target	8.6
Herbie	7.4

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

Derivation

1. Split input into 2 regimes

2. if t < -2.1012034468767935e+108 or 2.7030262370830962e+76 < t

   1. Initial program 30.3

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

   2. Simplified 21.1

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

   4. Applied clear-num 21.1

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

   6. Applied fma-udef 21.2

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

   7. Simplified 21.2

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

   8. Taylor expanded around inf 17.6

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

   9. Simplified 12.1

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

   if -2.1012034468767935e+108 < t < 2.7030262370830962e+76

   1. Initial program 7.6

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

   2. Simplified 6.0

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

   4. Applied clear-num 6.1

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

   6. Applied fma-udef 6.1

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

   7. Simplified 5.9

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

   9. Applied add-cube-cbrt 6.1

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

   10. Applied add-cube-cbrt 6.2

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

   11. Applied times-frac 6.2

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

   12. Applied add-cube-cbrt 6.2

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

   13. Applied times-frac 4.6

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

   14. Applied fma-def 4.6

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, x + y\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.10120344687679351 \cdot 10^{108} \lor \neg \left(t \le 2.7030262370830962 \cdot 10^{76}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))))