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

Average Error: 1.5 → 1.0

Time: 6.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3.1778304487720257 \cdot 10^{37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)\\ \mathbf{elif}\;y \le 1.7883278228757425 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - t}, \frac{y}{\frac{1}{z - t}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((y <= -3.1778304487720257e+37)) {
		VAR = fma((y / (a - t)), (z - t), x);
	} else {
		double VAR_1;
		if ((y <= 1.7883278228757425e-99)) {
			VAR_1 = fma((1.0 / (a - t)), (y / (1.0 / (z - t))), x);
		} else {
			VAR_1 = fma(y, ((z - t) / (a - t)), x);
		}
		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	1.5
Target	0.4
Herbie	1.0

\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -3.1778304487720257e+37

   1. Initial program 0.6

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

   2. Simplified 0.6

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

   4. Applied clear-num 0.8

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

   6. Applied fma-udef 0.8

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

   7. Simplified 0.7

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

   9. Applied associate-/r/ 3.0

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

   10. Applied fma-def 3.0

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

   if -3.1778304487720257e+37 < y < 1.7883278228757425e-99

   1. Initial program 2.5

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

   2. Simplified 2.5

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

   4. Applied clear-num 2.5

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

   6. Applied fma-udef 2.5

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

   7. Simplified 2.1

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

   9. Applied div-inv 2.1

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

   10. Applied *-un-lft-identity 2.1

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

   11. Applied times-frac 0.3

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

   12. Applied fma-def 0.3

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

   if 1.7883278228757425e-99 < y

   1. Initial program 0.7

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

   2. Simplified 0.7

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.1778304487720257 \cdot 10^{37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)\\ \mathbf{elif}\;y \le 1.7883278228757425 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - t}, \frac{y}{\frac{1}{z - t}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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