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

Average Error: 16.3 → 8.2

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.69603370545272536 \cdot 10^{73} \lor \neg \left(t \le 1.9681526611044615 \cdot 10^{149}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{\sqrt[3]{\frac{t - z}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{t - z}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{\frac{t - z}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{a - t}}} + \left(x + y\right)\\ \end{array}\]

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 VAR;
	if (((t <= -5.696033705452725e+73) || !(t <= 1.9681526611044615e+149))) {
		VAR = fma((z / t), y, x);
	} else {
		VAR = (((y * ((cbrt(((t - z) / (cbrt((a - t)) * cbrt((a - t))))) * cbrt(((t - z) / (cbrt((a - t)) * cbrt((a - t)))))) / cbrt((cbrt((a - t)) * cbrt((a - t)))))) * (cbrt(((t - z) / (cbrt((a - t)) * cbrt((a - t))))) / cbrt(cbrt((a - t))))) + (x + y));
	}
	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	16.3
Target	8.1
Herbie	8.2

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

Derivation

1. Split input into 2 regimes

2. if t < -5.696033705452725e+73 or 1.9681526611044615e+149 < t

   1. Initial program 30.2

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

   2. Simplified 21.3

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

   4. Applied fma-udef 21.4

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

   6. Applied div-inv 21.4

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

   7. Applied associate-*l* 20.1

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

   8. Simplified 20.1

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

   9. Taylor expanded around inf 16.5

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

   10. Simplified 11.5

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

   if -5.696033705452725e+73 < t < 1.9681526611044615e+149

   1. Initial program 9.2

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

   2. Simplified 6.9

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

   4. Applied fma-udef 6.9

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

   6. Applied div-inv 7.0

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

   7. Applied associate-*l* 6.8

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

   8. Simplified 6.8

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

   10. Applied add-cube-cbrt 7.0

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

   11. Applied associate-/r* 7.0

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

   13. Applied add-cube-cbrt 7.0

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

   14. Applied cbrt-prod 7.1

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

   15. Applied add-cube-cbrt 7.1

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

   16. Applied times-frac 7.1

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

   17. Applied associate-*r* 6.5

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

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.69603370545272536 \cdot 10^{73} \lor \neg \left(t \le 1.9681526611044615 \cdot 10^{149}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{\sqrt[3]{\frac{t - z}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{t - z}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{\frac{t - z}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{a - t}}} + \left(x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 +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))))