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

Average Error: 10.3 → 1.1

Time: 4.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le -1.61575697840551178 \cdot 10^{250} \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 7.9291238590814451 \cdot 10^{55}\right):\\ \;\;\;\;y \cdot \frac{t}{a - z} + \mathsf{fma}\left(-z, \frac{t}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\left(y - z\right) \cdot t}{a - z} + x\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double temp;
	if ((((((y - z) * t) / (a - z)) <= -1.6157569784055118e+250) || !((((y - z) * t) / (a - z)) <= 7.929123859081445e+55))) {
		temp = ((y * (t / (a - z))) + fma(-z, (t / (a - z)), x));
	} else {
		temp = ((1.0 * (((y - z) * t) / (a - z))) + x);
	}
	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	10.3
Target	0.5
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (* (- y z) t) (- a z)) < -1.6157569784055118e+250 or 7.929123859081445e+55 < (/ (* (- y z) t) (- a z))

   1. Initial program 35.5

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

   2. Simplified 2.6

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

   4. Applied clear-num 2.6

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

   6. Applied fma-udef 2.6

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

   7. Simplified 3.2

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

   9. Applied sub-neg 3.2

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

   10. Applied distribute-rgt-in 3.2

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

   11. Applied associate-+l+ 3.2

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

   12. Simplified 3.2

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

   if -1.6157569784055118e+250 < (/ (* (- y z) t) (- a z)) < 7.929123859081445e+55

   1. Initial program 0.3

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

   2. Simplified 1.0

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

   4. Applied clear-num 1.0

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

   6. Applied fma-udef 1.0

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

   7. Simplified 3.1

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

   9. Applied *-un-lft-identity 3.1

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

   10. Applied associate-*l* 3.1

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

   11. Simplified 0.3

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le -1.61575697840551178 \cdot 10^{250} \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 7.9291238590814451 \cdot 10^{55}\right):\\ \;\;\;\;y \cdot \frac{t}{a - z} + \mathsf{fma}\left(-z, \frac{t}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\left(y - z\right) \cdot t}{a - z} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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