Development.Shake.Progress:decay from shake-0.15.5

Average Error: 23.7 → 5.9

Time: 16.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{y}{b - y}\\ t_3 := \frac{t}{b - y}\\ t_4 := {\left(b - y\right)}^{2}\\ t_5 := \mathsf{fma}\left(\frac{y}{t_4}, \frac{t}{z}, \frac{a}{b - y}\right)\\ t_6 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t_1}\\ t_7 := \mathsf{fma}\left(t_2, \frac{x}{z}, t_3\right) - t_5\\ \mathbf{if}\;t_6 \leq -\infty:\\ \;\;\;\;t_7\\ \mathbf{elif}\;t_6 \leq -2.5644078469744954 \cdot 10^{-215}:\\ \;\;\;\;\frac{z \cdot t}{t_1} + \frac{x \cdot y - z \cdot a}{t_1}\\ \mathbf{elif}\;t_6 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{t_4}, \frac{y}{z}, t_3\right)\right) - t_5\\ \mathbf{elif}\;t_6 \leq 1.8400111491673562 \cdot 10^{+267}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t - z \cdot a\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_7\\ \end{array} \]

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 (+ y (* z (- b y))))
        (t_2 (/ y (- b y)))
        (t_3 (/ t (- b y)))
        (t_4 (pow (- b y) 2.0))
        (t_5 (fma (/ y t_4) (/ t z) (/ a (- b y))))
        (t_6 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_7 (- (fma t_2 (/ x z) t_3) t_5)))
   (if (<= t_6 (- INFINITY))
     t_7
     (if (<= t_6 -2.5644078469744954e-215)
       (+ (/ (* z t) t_1) (/ (- (* x y) (* z a)) t_1))
       (if (<= t_6 0.0)
         (- (fma t_2 (/ x z) (fma (/ a t_4) (/ y z) t_3)) t_5)
         (if (<= t_6 1.8400111491673562e+267)
           (/ (fma x y (- (* z t) (* z a))) (fma z (- b y) y))
           t_7))))))

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 = y + (z * (b - y));
	double t_2 = y / (b - y);
	double t_3 = t / (b - y);
	double t_4 = pow((b - y), 2.0);
	double t_5 = fma((y / t_4), (t / z), (a / (b - y)));
	double t_6 = ((x * y) + (z * (t - a))) / t_1;
	double t_7 = fma(t_2, (x / z), t_3) - t_5;
	double tmp;
	if (t_6 <= -((double) INFINITY)) {
		tmp = t_7;
	} else if (t_6 <= -2.5644078469744954e-215) {
		tmp = ((z * t) / t_1) + (((x * y) - (z * a)) / t_1);
	} else if (t_6 <= 0.0) {
		tmp = fma(t_2, (x / z), fma((a / t_4), (y / z), t_3)) - t_5;
	} else if (t_6 <= 1.8400111491673562e+267) {
		tmp = fma(x, y, ((z * t) - (z * a))) / fma(z, (b - y), y);
	} else {
		tmp = t_7;
	}
	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	23.7
Target	18.5
Herbie	5.9

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

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 1.8400111491673562e267 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 62.0

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

   2. Simplified 62.0

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

   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 16.8

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

   5. Taylor expanded in a around 0 15.6

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -2.56440784697449545e-215

   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 -2.56440784697449545e-215 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 38.4

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

   2. Simplified 38.4

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

   3. Taylor expanded in z around inf 20.7

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

   4. Simplified 8.4

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.8400111491673562e267

   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. Applied sub-neg_binary64 0.3

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

   4. Applied distribute-rgt-in_binary64 0.3

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

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -2.5644078469744954 \cdot 10^{-215}:\\ \;\;\;\;\frac{z \cdot t}{y + z \cdot \left(b - y\right)} + \frac{x \cdot y - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{{\left(b - y\right)}^{2}}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 1.8400111491673562 \cdot 10^{+267}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t - z \cdot a\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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)))))