Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B

Average Error: 16.6 → 5.3

Time: 9.7s

Precision: binary64

Math

\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

↓

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := \frac{x}{1 + \left(a + t_1\right)}\\ t_3 := \left(a + 1\right) + t_1\\ t_4 := \frac{x + \frac{y \cdot z}{t}}{t_3}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}, \frac{z}{t}, t_2\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_5 := x + \left(y \cdot z\right) \cdot \frac{1}{t}\\ \mathbf{if}\;t_4 \leq -5.7172199925063 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_5}{t_3}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;t_4 \leq 9.35937687403725 \cdot 10^{+270}:\\ \;\;\;\;\frac{t_5}{\left(a + 1\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + \left(a + y \cdot \frac{b}{t}\right)}, \frac{z}{t}, t_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* y b) t))
        (t_2 (/ x (+ 1.0 (+ a t_1))))
        (t_3 (+ (+ a 1.0) t_1))
        (t_4 (/ (+ x (/ (* y z) t)) t_3)))
   (if (<= t_4 (- INFINITY))
     (fma (/ y (+ 1.0 (+ a (/ y (/ t b))))) (/ z t) t_2)
     (let* ((t_5 (+ x (* (* y z) (/ 1.0 t)))))
       (if (<= t_4 -5.7172199925063e-310)
         (/ t_5 t_3)
         (if (<= t_4 0.0)
           (/ (+ z (/ (* x t) y)) b)
           (if (<= t_4 9.35937687403725e+270)
             (/ t_5 (+ (+ a 1.0) (/ 1.0 (/ t (* y b)))))
             (if (<= t_4 INFINITY)
               (fma (/ y (+ 1.0 (+ a (* y (/ b t))))) (/ z t) t_2)
               (/ z b)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * b) / t;
	double t_2 = x / (1.0 + (a + t_1));
	double t_3 = (a + 1.0) + t_1;
	double t_4 = (x + ((y * z) / t)) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = fma((y / (1.0 + (a + (y / (t / b))))), (z / t), t_2);
	} else {
		double t_5 = x + ((y * z) * (1.0 / t));
		double tmp_1;
		if (t_4 <= -5.7172199925063e-310) {
			tmp_1 = t_5 / t_3;
		} else if (t_4 <= 0.0) {
			tmp_1 = (z + ((x * t) / y)) / b;
		} else if (t_4 <= 9.35937687403725e+270) {
			tmp_1 = t_5 / ((a + 1.0) + (1.0 / (t / (y * b))));
		} else if (t_4 <= ((double) INFINITY)) {
			tmp_1 = fma((y / (1.0 + (a + (y * (b / t))))), (z / t), t_2);
		} else {
			tmp_1 = z / b;
		}
		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	16.6
Target	13.5
Herbie	5.3

\[\begin{array}{l} \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array} \]

Derivation

1. Split input into 6 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -inf.0

   1. Initial program 64.0

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. Taylor expanded in x around 0 39.5

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

   3. Simplified 16.8

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

   4. Applied associate-/l*_binary64 16.8

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -5.7172199925063e-310

   1. Initial program 0.4

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. Applied div-inv_binary64 0.4

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

   if -5.7172199925063e-310 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 31.0

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. Taylor expanded in x around 0 31.0

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

   3. Simplified 31.1

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

   4. Taylor expanded in b around inf 20.1

      \[\leadsto \color{blue}{\frac{\frac{t \cdot x}{y} + z}{b}} \]

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 9.35937687403724969e270

   1. Initial program 0.6

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. Applied div-inv_binary64 0.6

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

   3. Applied clear-num_binary64 0.6

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

   if 9.35937687403724969e270 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 50.4

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. Taylor expanded in x around 0 30.0

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

   3. Simplified 14.8

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

   4. Applied *-un-lft-identity_binary64 14.8

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

   5. Applied times-frac_binary64 14.8

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

   6. Simplified 14.8

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 64.0

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. Taylor expanded in y around inf 3.0

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}, \frac{z}{t}, \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -5.7172199925063 \cdot 10^{-310}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 9.35937687403725 \cdot 10^{+270}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + \left(a + y \cdot \frac{b}{t}\right)}, \frac{z}{t}, \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2021329 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))