Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Average Error: 24.8 → 6.8

Time: 3.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -6.4366997153948992 \cdot 10^{-217} \lor \neg \left(a \le 1.41067993913888447 \cdot 10^{-238}\right):\\ \;\;\;\;\frac{t}{\frac{a - z}{y - z}} - \left(\frac{x}{\frac{a - z}{y - z}} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x - t, t\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((a <= -6.436699715394899e-217) || !(a <= 1.4106799391388845e-238))) {
		VAR = ((t / ((a - z) / (y - z))) - ((x / ((a - z) / (y - z))) - x));
	} else {
		VAR = fma((y / z), (x - t), t);
	}
	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	24.8
Target	11.8
Herbie	6.8

\[\begin{array}{l} \mathbf{if}\;z \lt -1.25361310560950359 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if a < -6.436699715394899e-217 or 1.4106799391388845e-238 < a

   1. Initial program 24.0

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

   2. Simplified 10.5

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

   4. Applied clear-num 10.6

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

   6. Applied fma-udef 10.6

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

   7. Simplified 10.5

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

   9. Applied div-sub 10.5

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

   10. Applied associate-+l- 7.0

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

   if -6.436699715394899e-217 < a < 1.4106799391388845e-238

   1. Initial program 30.8

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

   2. Simplified 21.9

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

   3. Taylor expanded around inf 9.2

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

   4. Simplified 5.7

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

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -6.4366997153948992 \cdot 10^{-217} \lor \neg \left(a \le 1.41067993913888447 \cdot 10^{-238}\right):\\ \;\;\;\;\frac{t}{\frac{a - z}{y - z}} - \left(\frac{x}{\frac{a - z}{y - z}} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x - t, t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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