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

Average Error: 15.7 → 9.9

Time: 7.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)\\ t_2 := \frac{x}{t_1}\\ \mathbf{if}\;t \leq -2.1055779123004298 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{t_1}, \frac{z}{t}, t_2\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{if}\;t \leq 6.2810140716487435 \cdot 10^{-263}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.046109116160516 \cdot 10^{-31}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)} + \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;t \leq 31525899.08026222:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t_1}, \frac{z}{t}, t_2\right)\\ \end{array}\\ \end{array} \]

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 (+ 1.0 (fma b (/ y t) a))) (t_2 (/ x t_1)))
   (if (<= t -2.1055779123004298e-85)
     (fma (* y (/ 1.0 t_1)) (/ z t) t_2)
     (let* ((t_3 (/ (+ z (/ (* t x) y)) b)))
       (if (<= t 6.2810140716487435e-263)
         t_3
         (if (<= t 1.046109116160516e-31)
           (+
            (/ (* y z) (fma y b (fma a t t)))
            (/ x (+ 1.0 (+ a (/ (* y b) t)))))
           (if (<= t 31525899.08026222) t_3 (fma (/ y t_1) (/ z t) t_2))))))))

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 = 1.0 + fma(b, (y / t), a);
	double t_2 = x / t_1;
	double tmp;
	if (t <= -2.1055779123004298e-85) {
		tmp = fma((y * (1.0 / t_1)), (z / t), t_2);
	} else {
		double t_3 = (z + ((t * x) / y)) / b;
		double tmp_1;
		if (t <= 6.2810140716487435e-263) {
			tmp_1 = t_3;
		} else if (t <= 1.046109116160516e-31) {
			tmp_1 = ((y * z) / fma(y, b, fma(a, t, t))) + (x / (1.0 + (a + ((y * b) / t))));
		} else if (t <= 31525899.08026222) {
			tmp_1 = t_3;
		} else {
			tmp_1 = fma((y / t_1), (z / t), t_2);
		}
		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	15.7
Target	13.2
Herbie	9.9

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

Derivation

1. Split input into 4 regimes

2. if t < -2.1055779123004298e-85

   1. Initial program 11.6

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

   2. Simplified 6.5

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

   3. Taylor expanded in z around 0 12.7

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

   4. Simplified 5.4

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

   5. Applied div-inv_binary64 5.4

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

   if -2.1055779123004298e-85 < t < 6.28101407164874353e-263 or 1.046109116160516e-31 < t < 31525899.080262221

   1. Initial program 23.8

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

   2. Simplified 32.2

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

   3. Taylor expanded in z around 0 18.9

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

   4. Simplified 32.5

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

   5. Taylor expanded in b around inf 22.5

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

   if 6.28101407164874353e-263 < t < 1.046109116160516e-31

   1. Initial program 19.1

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

   2. Simplified 28.1

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

   3. Taylor expanded in z around 0 15.6

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

   4. Taylor expanded in z around inf 10.7

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

   5. Simplified 10.7

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

   if 31525899.080262221 < t

   1. Initial program 10.5

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

   2. Simplified 3.7

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

   3. Taylor expanded in z around 0 11.9

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

   4. Simplified 2.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{z}{t}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1055779123004298 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{z}{t}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)\\ \mathbf{elif}\;t \leq 6.2810140716487435 \cdot 10^{-263}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;t \leq 1.046109116160516 \cdot 10^{-31}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)} + \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;t \leq 31525899.08026222:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{z}{t}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))))