Development.Shake.Progress:decay from shake-0.15.5

Average Error: 23.1 → 5.7

Time: 16.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot \left(t - a\right)\\ t_2 := y + z \cdot \left(b - y\right)\\ t_3 := \frac{x \cdot y + t_1}{t_2}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;t_3 \leq -5.737180884435787 \cdot 10^{-281}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t_1\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_4 := \frac{y}{b - y}\\ t_5 := {\left(b - y\right)}^{2}\\ t_6 := \frac{t}{b - y}\\ t_7 := \mathsf{fma}\left(\frac{y}{t_5}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t_4, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{t_5}, \frac{y}{z}, t_6\right)\right) - t_7\\ \mathbf{elif}\;t_3 \leq 1.0968421907966705 \cdot 10^{+270}:\\ \;\;\;\;\frac{z \cdot t}{t_2} + \frac{x \cdot y - z \cdot a}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_4, \frac{x}{z}, t_6\right) - t_7\\ \end{array}\\ \end{array} \]

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 (* z (- t a)))
        (t_2 (+ y (* z (- b y))))
        (t_3 (/ (+ (* x y) t_1) t_2)))
   (if (<= t_3 (- INFINITY))
     (/ x (- 1.0 z))
     (if (<= t_3 -5.737180884435787e-281)
       (/ (fma x y t_1) (fma z (- b y) y))
       (let* ((t_4 (/ y (- b y)))
              (t_5 (pow (- b y) 2.0))
              (t_6 (/ t (- b y)))
              (t_7 (fma (/ y t_5) (/ t z) (/ a (- b y)))))
         (if (<= t_3 0.0)
           (- (fma t_4 (/ x z) (fma (/ a t_5) (/ y z) t_6)) t_7)
           (if (<= t_3 1.0968421907966705e+270)
             (+ (/ (* z t) t_2) (/ (- (* x y) (* z a)) t_2))
             (- (fma t_4 (/ x z) t_6) 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 = z * (t - a);
	double t_2 = y + (z * (b - y));
	double t_3 = ((x * y) + t_1) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = x / (1.0 - z);
	} else if (t_3 <= -5.737180884435787e-281) {
		tmp = fma(x, y, t_1) / fma(z, (b - y), y);
	} else {
		double t_4 = y / (b - y);
		double t_5 = pow((b - y), 2.0);
		double t_6 = t / (b - y);
		double t_7 = fma((y / t_5), (t / z), (a / (b - y)));
		double tmp_1;
		if (t_3 <= 0.0) {
			tmp_1 = fma(t_4, (x / z), fma((a / t_5), (y / z), t_6)) - t_7;
		} else if (t_3 <= 1.0968421907966705e+270) {
			tmp_1 = ((z * t) / t_2) + (((x * y) - (z * a)) / t_2);
		} else {
			tmp_1 = fma(t_4, (x / z), t_6) - t_7;
		}
		tmp = tmp_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	23.1
Target	17.8
Herbie	5.7

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

Derivation

1. Split input into 5 regimes

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

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded in y around inf 32.3

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -5.7371808844357872e-281

   1. Initial program 0.4

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

   2. Simplified 0.4

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

   if -5.7371808844357872e-281 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 42.7

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

   2. Simplified 42.7

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

   3. Taylor expanded in z around inf 19.0

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

   4. Simplified 3.5

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

   if -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.0968421907966705e270

   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 1.0968421907966705e270 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 60.8

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

   2. Simplified 60.8

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

   3. Taylor expanded in z around inf 40.5

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

   4. Simplified 13.1

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

   5. Taylor expanded in a around 0 12.2

      \[\leadsto \mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \color{blue}{\frac{t}{b - y}}\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -5.737180884435787 \cdot 10^{-281}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{{\left(b - y\right)}^{2}}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 1.0968421907966705 \cdot 10^{+270}:\\ \;\;\;\;\frac{z \cdot t}{y + z \cdot \left(b - y\right)} + \frac{x \cdot y - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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)))))