Development.Shake.Progress:decay from shake-0.15.5

Average Error: 23.4 → 6.0

Time: 14.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{z}, \frac{y}{b - y}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ t_2 := z \cdot \left(t - a\right)\\ t_3 := y + z \cdot \left(b - y\right)\\ t_4 := \frac{x \cdot y + t_2}{t_3}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_4 \leq -1.9009386587944248 \cdot 10^{-274}:\\ \;\;\;\;\frac{z \cdot t}{t_3} + \frac{x \cdot y - z \cdot a}{t_3}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \frac{a}{{\left(y - b\right)}^{2}}, \frac{a}{y - b}\right) - \mathsf{fma}\left(\frac{y}{z}, \frac{x}{y - b}, \frac{t}{y - b}\right)\\ \mathbf{elif}\;t_4 \leq 7.970143228461605 \cdot 10^{+266}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t_2\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (fma (/ x z) (/ y (- b y)) (/ t (- b y)))
          (fma (/ y (pow (- b y) 2.0)) (/ t z) (/ a (- b y)))))
        (t_2 (* z (- t a)))
        (t_3 (+ y (* z (- b y))))
        (t_4 (/ (+ (* x y) t_2) t_3)))
   (if (<= t_4 (- INFINITY))
     t_1
     (if (<= t_4 -1.9009386587944248e-274)
       (+ (/ (* z t) t_3) (/ (- (* x y) (* z a)) t_3))
       (if (<= t_4 0.0)
         (-
          (fma (/ y z) (/ a (pow (- y b) 2.0)) (/ a (- y b)))
          (fma (/ y z) (/ x (- y b)) (/ t (- y b))))
         (if (<= t_4 7.970143228461605e+266)
           (/ (fma x y t_2) (fma z (- b y) y))
           t_1))))))

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((x / z), (y / (b - y)), (t / (b - y))) - fma((y / pow((b - y), 2.0)), (t / z), (a / (b - y)));
	double t_2 = z * (t - a);
	double t_3 = y + (z * (b - y));
	double t_4 = ((x * y) + t_2) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_4 <= -1.9009386587944248e-274) {
		tmp = ((z * t) / t_3) + (((x * y) - (z * a)) / t_3);
	} else if (t_4 <= 0.0) {
		tmp = fma((y / z), (a / pow((y - b), 2.0)), (a / (y - b))) - fma((y / z), (x / (y - b)), (t / (y - b)));
	} else if (t_4 <= 7.970143228461605e+266) {
		tmp = fma(x, y, t_2) / fma(z, (b - y), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	23.4
Target	18.1
Herbie	6.0

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 7.9701432284616051e266 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 61.0

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

   2. Simplified 61.0

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

   3. Taylor expanded in z around inf 41.8

      \[\leadsto \color{blue}{\left(\frac{y \cdot x}{\left(b - y\right) \cdot z} + \left(\frac{a \cdot y}{{\left(b - y\right)}^{2} \cdot z} + \frac{t}{b - y}\right)\right) - \left(\frac{y \cdot t}{{\left(b - y\right)}^{2} \cdot z} + \frac{a}{b - y}\right)} \]

   4. Simplified 17.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{{\left(b - y\right)}^{2}}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)} \]

   5. Taylor expanded in a around 0 30.5

      \[\leadsto \color{blue}{\left(\frac{y \cdot x}{\left(b - y\right) \cdot z} + \frac{t}{b - y}\right)} - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right) \]

   6. Simplified 16.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, \frac{y}{b - y}, \frac{t}{b - y}\right)} - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right) \]

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.9009386587944248e-274

   1. Initial program 0.3

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

   2. Simplified 0.3

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

   3. Taylor expanded in x around 0 0.3

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

   4. Simplified 0.3

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

   if -1.9009386587944248e-274 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 43.4

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

   2. Simplified 43.4

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

   3. Taylor expanded in z around -inf 20.4

      \[\leadsto \color{blue}{\left(\frac{y \cdot a}{z \cdot {\left(y - b\right)}^{2}} + \frac{a}{y - b}\right) - \left(\frac{y \cdot x}{z \cdot \left(y - b\right)} + \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\right)} \]

   4. Simplified 7.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, \frac{a}{{\left(y - b\right)}^{2}}, \frac{a}{y - b}\right) - \mathsf{fma}\left(\frac{y}{z}, \frac{x}{y - b}, \mathsf{fma}\left(\frac{y}{z}, \frac{t}{{\left(y - b\right)}^{2}}, \frac{t}{y - b}\right)\right)} \]

   5. Taylor expanded in z around inf 7.1

      \[\leadsto \mathsf{fma}\left(\frac{y}{z}, \frac{a}{{\left(y - b\right)}^{2}}, \frac{a}{y - b}\right) - \mathsf{fma}\left(\frac{y}{z}, \frac{x}{y - b}, \color{blue}{\frac{t}{y - b}}\right) \]

   if -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 7.9701432284616051e266

   1. Initial program 0.3

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{y}{b - y}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1.9009386587944248 \cdot 10^{-274}:\\ \;\;\;\;\frac{z \cdot t}{y + z \cdot \left(b - y\right)} + \frac{x \cdot y - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \frac{a}{{\left(y - b\right)}^{2}}, \frac{a}{y - b}\right) - \mathsf{fma}\left(\frac{y}{z}, \frac{x}{y - b}, \frac{t}{y - b}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 7.970143228461605 \cdot 10^{+266}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{y}{b - y}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

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

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