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

Average Error: 24.9 → 7.3

Time: 9.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -2.4909642387543203 \cdot 10^{-274} \lor \neg \left(x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0\right):\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(t + \left(\frac{x \cdot y}{z} + \frac{t \cdot a}{z}\right)\right) - \left(\frac{x \cdot a}{z} + \frac{y \cdot t}{z}\right)\\ \end{array}\]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))

↓

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (+ x (/ (* (- y z) (- t x)) (- a z))) -2.4909642387543203e-274)
         (not (<= (+ x (/ (* (- y z) (- t x)) (- a z))) 0.0)))
   (+
    x
    (*
     (*
      (- y z)
      (/ (* (cbrt (- t x)) (cbrt (- t x))) (* (cbrt (- a z)) (cbrt (- a z)))))
     (/ (cbrt (- t x)) (cbrt (- a z)))))
   (- (+ t (+ (/ (* x y) z) (/ (* t a) z))) (+ (/ (* x a) z) (/ (* y t) z)))))

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 tmp;
	if (((x + (((y - z) * (t - x)) / (a - z))) <= -2.4909642387543203e-274) || !((x + (((y - z) * (t - x)) / (a - z))) <= 0.0)) {
		tmp = x + (((y - z) * ((cbrt(t - x) * cbrt(t - x)) / (cbrt(a - z) * cbrt(a - z)))) * (cbrt(t - x) / cbrt(a - z)));
	} else {
		tmp = (t + (((x * y) / z) + ((t * a) / z))) - (((x * a) / z) + ((y * t) / z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.9
Target	12.0
Herbie	7.3

\[\begin{array}{l} \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -2.4909642387543203e-274 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 21.7

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

   3. Applied *-un-lft-identity_binary64 21.7

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

   4. Applied times-frac_binary64 10.3

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

   5. Simplified 10.3

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

   7. Applied add-cube-cbrt_binary64 10.9

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

   8. Applied add-cube-cbrt_binary64 11.0

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

   9. Applied times-frac_binary64 11.0

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

   10. Applied associate-*r*_binary64 7.7

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

   if -2.4909642387543203e-274 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 58.5

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

   2. Taylor expanded around inf 2.7

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

   3. Simplified 2.7

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

4. Final simplification 7.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -2.4909642387543203 \cdot 10^{-274} \lor \neg \left(x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0\right):\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(t + \left(\frac{x \cdot y}{z} + \frac{t \cdot a}{z}\right)\right) - \left(\frac{x \cdot a}{z} + \frac{y \cdot t}{z}\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))))