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

Average Error: 10.5 → 0.5

Time: 4.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4.16821903091809114 \cdot 10^{-47} \lor \neg \left(y \le 1.5894593416801174 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{z - t}{a - t} \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a - t} \cdot \left(\left(z - t\right) \cdot y\right) + x\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((y <= -4.168219030918091e-47) || !(y <= 1.5894593416801174e-57))) {
		VAR = ((double) (((double) (((double) (((double) (z - t)) / ((double) (a - t)))) * y)) + x));
	} else {
		VAR = ((double) (((double) (((double) (1.0 / ((double) (a - t)))) * ((double) (((double) (z - t)) * y)))) + x));
	}
	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	10.5
Target	1.3
Herbie	0.5

\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

1. Split input into 2 regimes

2. if y < -4.168219030918091e-47 or 1.5894593416801174e-57 < y

   1. Initial program 18.9

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

   2. Simplified 2.6

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

   4. Applied *-un-lft-identity 2.6

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

   5. Applied add-cube-cbrt 3.3

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

   6. Applied times-frac 3.3

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

   7. Simplified 3.3

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

   9. Applied fma-udef 3.3

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

   10. Simplified 2.5

       \[\leadsto \color{blue}{\frac{z - t}{\frac{a - t}{y}}} + x\]
   11. Using strategy rm

   12. Applied associate-/r/ 0.6

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

   if -4.168219030918091e-47 < y < 1.5894593416801174e-57

   1. Initial program 0.4

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

   2. Simplified 3.6

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

   4. Applied *-un-lft-identity 3.6

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

   5. Applied add-cube-cbrt 3.9

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

   6. Applied times-frac 3.9

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

   7. Simplified 3.9

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

   9. Applied fma-udef 3.9

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

   10. Simplified 4.0

       \[\leadsto \color{blue}{\frac{z - t}{\frac{a - t}{y}}} + x\]
   11. Using strategy rm

   12. Applied div-inv 4.0

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

   13. Applied *-un-lft-identity 4.0

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

   14. Applied times-frac 0.4

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

   15. Simplified 0.4

       \[\leadsto \frac{1}{a - t} \cdot \color{blue}{\left(\left(z - t\right) \cdot y\right)} + x\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.16821903091809114 \cdot 10^{-47} \lor \neg \left(y \le 1.5894593416801174 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{z - t}{a - t} \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a - t} \cdot \left(\left(z - t\right) \cdot y\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- a t) (- z t))))

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