Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A

Average Error: 10.8 → 2.9

Time: 12.2s

Precision: binary64

Cost: 8584

Math

\[\frac{x - y \cdot z}{t - a \cdot z} \]

↓

\[\begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ t_2 := t - z \cdot a\\ t_3 := \frac{x - z \cdot y}{t_2}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -z, x\right)}{t_2}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\frac{y}{a} + \frac{x}{a} \cdot \frac{-1}{z}\\ \mathbf{elif}\;t_3 \leq 10^{+287}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (/ (- x (* y z)) (- t (* a z))))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ y (- a (/ t z))))
        (t_2 (- t (* z a)))
        (t_3 (/ (- x (* z y)) t_2)))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 -1e-285)
       (/ (fma y (- z) x) t_2)
       (if (<= t_3 0.0)
         (+ (/ y a) (* (/ x a) (/ -1.0 z)))
         (if (<= t_3 1e+287) t_3 t_1))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = y / (a - (t / z));
	double t_2 = t - (z * a);
	double t_3 = (x - (z * y)) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= -1e-285) {
		tmp = fma(y, -z, x) / t_2;
	} else if (t_3 <= 0.0) {
		tmp = (y / a) + ((x / a) * (-1.0 / z));
	} else if (t_3 <= 1e+287) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	return Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)))
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(y / Float64(a - Float64(t / z)))
	t_2 = Float64(t - Float64(z * a))
	t_3 = Float64(Float64(x - Float64(z * y)) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= -1e-285)
		tmp = Float64(fma(y, Float64(-z), x) / t_2);
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(y / a) + Float64(Float64(x / a) * Float64(-1.0 / z)));
	elseif (t_3 <= 1e+287)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, -1e-285], N[(N[(y * (-z) + x), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(y / a), $MachinePrecision] + N[(N[(x / a), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+287], t$95$3, t$95$1]]]]]]]

Target

Original	10.8
Target	1.8
Herbie	2.9

\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or 1.0000000000000001e287 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 60.1

      \[\frac{x - y \cdot z}{t - a \cdot z} \]

   2. Simplified 60.1

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
      Proof
      (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 z a))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (Rewrite<= *-commutative_binary64 (*.f64 a z)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 62.6

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t - a \cdot z}} \]

   4. Simplified 2.7

      \[\leadsto \color{blue}{\frac{-y}{\frac{t}{z} - a}} \]
      Proof
      (/.f64 (neg.f64 y) (-.f64 (/.f64 t z) a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 y)) (-.f64 (/.f64 t z) a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1 y) (-.f64 (/.f64 t z) (Rewrite<= /-rgt-identity_binary64 (/.f64 a 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1 y) (-.f64 (/.f64 t z) (/.f64 a (Rewrite<= *-inverses_binary64 (/.f64 z z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1 y) (-.f64 (/.f64 t z) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a z) z)))): 26 points increase in error, 1 points decrease in error
      (/.f64 (*.f64 -1 y) (-.f64 (/.f64 t z) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 z a)) z))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1 y) (Rewrite<= div-sub_binary64 (/.f64 (-.f64 t (*.f64 z a)) z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 y (/.f64 (-.f64 t (*.f64 z a)) z)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1 (/.f64 y (/.f64 (-.f64 t (Rewrite=> *-commutative_binary64 (*.f64 a z))) z))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y z) (-.f64 t (*.f64 a z))))): 81 points increase in error, 33 points decrease in error

   5. Taylor expanded in y around 0 2.7

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{\frac{t}{z} - a}} \]

   6. Simplified 2.7

      \[\leadsto \color{blue}{\frac{y}{a - \frac{t}{z}}} \]
      Proof
      (/.f64 y (-.f64 a (/.f64 t z))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 y 1)) (-.f64 a (/.f64 t z))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 y 1) (Rewrite<= unsub-neg_binary64 (+.f64 a (neg.f64 (/.f64 t z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 y 1) (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 a))) (neg.f64 (/.f64 t z)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 y 1) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 a) (/.f64 t z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 y 1) (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 t z) (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 y 1) (neg.f64 (Rewrite=> unsub-neg_binary64 (-.f64 (/.f64 t z) a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 y (/.f64 1 (neg.f64 (-.f64 (/.f64 t z) a))))): 41 points increase in error, 11 points decrease in error
      (*.f64 y (/.f64 1 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 (/.f64 t z) a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 y (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 -1) (-.f64 (/.f64 t z) a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 y (/.f64 (Rewrite=> metadata-eval -1) (-.f64 (/.f64 t z) a))): 0 points increase in error, 0 points decrease in error
      (*.f64 y (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) (-.f64 (/.f64 t z) a))): 0 points increase in error, 0 points decrease in error
      (*.f64 y (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 (-.f64 (/.f64 t z) a))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y (/.f64 1 (-.f64 (/.f64 t z) a))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y 1) (-.f64 (/.f64 t z) a)))): 11 points increase in error, 41 points decrease in error
      (neg.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 y (-.f64 (/.f64 t z) a)) 1))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite=> *-rgt-identity_binary64 (/.f64 y (-.f64 (/.f64 t z) a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (/.f64 y (-.f64 (/.f64 t z) a)))): 0 points increase in error, 0 points decrease in error

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.00000000000000007e-285

   1. Initial program 0.2

      \[\frac{x - y \cdot z}{t - a \cdot z} \]

   2. Simplified 0.2

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, -z, x\right)}{t - z \cdot a}} \]
      Proof
      (/.f64 (fma.f64 y (neg.f64 z) x) (-.f64 t (*.f64 z a))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (neg.f64 z)) x)) (-.f64 t (*.f64 z a))): 2 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y z))) x) (-.f64 t (*.f64 z a))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 y) z)) x) (-.f64 t (*.f64 z a))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 (neg.f64 y) z))) (-.f64 t (*.f64 z a))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 x (*.f64 y z))) (-.f64 t (*.f64 z a))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (Rewrite<= *-commutative_binary64 (*.f64 a z)))): 0 points increase in error, 0 points decrease in error

   if -1.00000000000000007e-285 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 24.4

      \[\frac{x - y \cdot z}{t - a \cdot z} \]

   2. Simplified 24.4

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
      Proof
      (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 z a))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (Rewrite<= *-commutative_binary64 (*.f64 a z)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around 0 41.3

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

   4. Simplified 26.7

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{x}{z \cdot a}} \]
      Proof
      (-.f64 (/.f64 y a) (/.f64 x (*.f64 z a))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 y (Rewrite<= /-rgt-identity_binary64 (/.f64 a 1))) (/.f64 x (*.f64 z a))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 y (/.f64 a (Rewrite<= *-inverses_binary64 (/.f64 z z)))) (/.f64 x (*.f64 z a))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 y (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a z) z))) (/.f64 x (*.f64 z a))): 25 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 y (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 z a)) z)) (/.f64 x (*.f64 z a))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y z) (*.f64 z a))) (/.f64 x (*.f64 z a))): 48 points increase in error, 4 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 y z) x) (*.f64 z a))): 2 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 y z) (neg.f64 x))) (*.f64 z a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (*.f64 y z)))) (neg.f64 x)) (*.f64 z a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 y (neg.f64 z)))) (neg.f64 x)) (*.f64 z a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 y (neg.f64 z)))) (neg.f64 x)) (*.f64 z a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 -1 (*.f64 y (neg.f64 z))) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x))) (*.f64 z a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 -1 (+.f64 (*.f64 y (neg.f64 z)) x))) (*.f64 z a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1 (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 y (neg.f64 z))))) (*.f64 z a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1 (+.f64 x (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 y z))))) (*.f64 z a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1 (Rewrite<= sub-neg_binary64 (-.f64 x (*.f64 y z)))) (*.f64 z a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1 (-.f64 x (*.f64 y z))) (Rewrite=> *-commutative_binary64 (*.f64 a z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (-.f64 x (*.f64 y z)) (*.f64 a z)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 15.7

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

   if -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.0000000000000001e287

   1. Initial program 0.2

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - z \cdot a} \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -z, x\right)}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y}{a} + \frac{x}{a} \cdot \frac{-1}{z}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - z \cdot a} \leq 10^{+287}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.8
Cost	20872

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \sqrt[3]{t_1}\\ t_3 := \frac{\frac{x}{t_2}}{{t_2}^{2}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{if}\;z \leq -1200000000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -z, x\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	2.9
Cost	3792

\[\begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ t_2 := \frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{y}{a} + \frac{x}{a} \cdot \frac{-1}{z}\\ \mathbf{elif}\;t_2 \leq 10^{+287}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	18.4
Cost	1240

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ t_2 := \frac{y}{a - \frac{t}{z}}\\ t_3 := \frac{x - z \cdot y}{t}\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -0.09:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-290}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+19}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	18.4
Cost	1240

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ t_2 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{+164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -0.33:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-230}:\\ \;\;\;\;\frac{x}{t} - \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-290}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+18}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	18.6
Cost	1240

\[\begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -0.155:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-230}:\\ \;\;\;\;\frac{x}{t} - \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-290}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+132}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	30.3
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -480000000:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.9:\\ \;\;\;\;\left(-z\right) \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7
Error	30.3
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -200000000:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 8.5:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{t}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8
Error	22.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+38}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+70}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 9
Error	18.5
Cost	712

\[\begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	30.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -220000000:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 11
Error	42.8
Cost	192

\[\frac{x}{t} \]

Reproduce

herbie shell --seed 2022338 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))