Development.Shake.Progress:decay from shake-0.15.5

Average Error: 22.5 → 14.7

Time: 8.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right) - \frac{a}{\left(\frac{y}{z} + b\right) - y}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 1.3775399847857508 \cdot 10^{279}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(b - y, z, y\right)} - \frac{a}{\left(\frac{y}{z} + b\right) - y}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{a}{\left(\frac{y}{z} + b\right) - y}\\ \end{array}\]

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 VAR;
	if (((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= -inf.0)) {
		VAR = (fma(x, z, x) - (a / (((y / z) + b) - y)));
	} else {
		double VAR_1;
		if (((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= 1.3775399847857508e+279)) {
			VAR_1 = ((fma(t, z, (x * y)) * (1.0 / fma((b - y), z, y))) - (a / (((y / z) + b) - y)));
		} else {
			VAR_1 = (0.0 - (a / (((y / z) + b) - y)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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	22.5
Target	17.4
Herbie	14.7

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

Derivation

1. Split input into 3 regimes

2. if (/ (+ (* x y) (* z (- t a))) (+ y (* z (- 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. Using strategy rm

   3. Applied sub-neg 64.0

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

   4. Applied distribute-lft-in 64.0

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

   5. Applied associate-+r+ 64.0

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

   6. Simplified 64.0

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

   8. Applied distribute-rgt-neg-out 64.0

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

   9. Applied unsub-neg 64.0

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

   10. Applied div-sub 64.0

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

   11. Simplified 64.0

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

   12. Simplified 53.0

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

   13. Taylor expanded around 0 53.0

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

   14. Taylor expanded around 0 28.9

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

   15. Simplified 28.9

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

   if -inf.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 1.3775399847857508e+279

   1. Initial program 6.1

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

   3. Applied sub-neg 6.1

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

   4. Applied distribute-lft-in 6.1

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

   5. Applied associate-+r+ 6.1

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

   6. Simplified 6.1

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

   8. Applied distribute-rgt-neg-out 6.1

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

   9. Applied unsub-neg 6.1

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

   10. Applied div-sub 6.1

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

   11. Simplified 6.1

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

   12. Simplified 6.5

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

   13. Taylor expanded around 0 4.5

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

   15. Applied div-inv 4.5

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

   if 1.3775399847857508e+279 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))

   1. Initial program 61.6

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

   3. Applied sub-neg 61.6

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

   4. Applied distribute-lft-in 61.6

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

   5. Applied associate-+r+ 61.6

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

   6. Simplified 61.6

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

   8. Applied distribute-rgt-neg-out 61.6

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

   9. Applied unsub-neg 61.6

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

   10. Applied div-sub 61.6

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

   11. Simplified 61.6

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

   12. Simplified 57.5

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

   13. Taylor expanded around 0 47.5

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

   14. Taylor expanded around inf 42.8

       \[\leadsto \color{blue}{0} - \frac{a}{\left(\frac{y}{z} + b\right) - y}\]
3. Recombined 3 regimes into one program.

4. Final simplification 14.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right) - \frac{a}{\left(\frac{y}{z} + b\right) - y}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 1.3775399847857508 \cdot 10^{279}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(b - y, z, y\right)} - \frac{a}{\left(\frac{y}{z} + b\right) - y}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{a}{\left(\frac{y}{z} + b\right) - y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +o rules:numerics
(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)))))