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

Average Error: 11.1 → 5.0

Time: 36.9s

Precision: binary64

Cost: 3404

Math

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

↓

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t_1}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{x}{t_1} - \frac{y}{\frac{t_1}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

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 (- t (* z a))) (t_2 (/ (- x (* y z)) t_1)))
   (if (<= t_2 -2e-300)
     t_2
     (if (<= t_2 0.0)
       (- (/ y a) (/ (/ x a) z))
       (if (<= t_2 INFINITY) (- (/ x t_1) (/ y (/ t_1 z))) (/ y a))))))

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 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -2e-300) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (y / a) - ((x / a) / z);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (x / t_1) - (y / (t_1 / z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -2e-300) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (y / a) - ((x / a) / z);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (x / t_1) - (y / (t_1 / z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	return (x - (y * z)) / (t - (a * z))

↓

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	t_2 = (x - (y * z)) / t_1
	tmp = 0
	if t_2 <= -2e-300:
		tmp = t_2
	elif t_2 <= 0.0:
		tmp = (y / a) - ((x / a) / z)
	elif t_2 <= math.inf:
		tmp = (x / t_1) - (y / (t_1 / z))
	else:
		tmp = y / a
	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(t - Float64(z * a))
	t_2 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_2 <= -2e-300)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(y / a) - Float64(Float64(x / a) / z));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(x / t_1) - Float64(y / Float64(t_1 / z)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (x - (y * z)) / (t - (a * z));
end

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	t_2 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_2 <= -2e-300)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = (y / a) - ((x / a) / z);
	elseif (t_2 <= Inf)
		tmp = (x / t_1) - (y / (t_1 / z));
	else
		tmp = y / a;
	end
	tmp_2 = 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[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-300], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(y / a), $MachinePrecision] - N[(N[(x / a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]

Target

Original	11.1
Target	1.7
Herbie	5.0

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -2.00000000000000005e-300

   1. Initial program 4.7

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

   if -2.00000000000000005e-300 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 26.3

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

   2. Simplified 26.3

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
      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 z around inf 28.8

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{a \cdot z} + \frac{y}{a}\right) - -1 \cdot \frac{y \cdot t}{z \cdot {a}^{2}}} \]

   4. Simplified 16.5

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{y}{a} \cdot \frac{t}{a}}{z}} \]
      Proof
      (-.f64 (/.f64 y a) (/.f64 (-.f64 (/.f64 x a) (*.f64 (/.f64 y a) (/.f64 t a))) z)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 y a) (/.f64 (-.f64 (/.f64 x a) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y t) (*.f64 a a)))) z)): 31 points increase in error, 7 points decrease in error
      (-.f64 (/.f64 y a) (/.f64 (-.f64 (/.f64 x a) (/.f64 (*.f64 y t) (Rewrite<= unpow2_binary64 (pow.f64 a 2)))) z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 y a) (neg.f64 (/.f64 (-.f64 (/.f64 x a) (/.f64 (*.f64 y t) (pow.f64 a 2))) z)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 y a) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (-.f64 (/.f64 x a) (/.f64 (*.f64 y t) (pow.f64 a 2))) z)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 y a) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1 (-.f64 (/.f64 x a) (/.f64 (*.f64 y t) (pow.f64 a 2)))) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 y a) (/.f64 (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 -1 (/.f64 x a)) (*.f64 -1 (/.f64 (*.f64 y t) (pow.f64 a 2))))) z)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 y a) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 -1 (/.f64 x a)) z) (/.f64 (*.f64 -1 (/.f64 (*.f64 y t) (pow.f64 a 2))) z)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 y a) (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (/.f64 x a) z))) (/.f64 (*.f64 -1 (/.f64 (*.f64 y t) (pow.f64 a 2))) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 y a) (-.f64 (*.f64 -1 (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 a z)))) (/.f64 (*.f64 -1 (/.f64 (*.f64 y t) (pow.f64 a 2))) z))): 14 points increase in error, 13 points decrease in error
      (+.f64 (/.f64 y a) (-.f64 (*.f64 -1 (/.f64 x (*.f64 a z))) (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1 (*.f64 y t)) (pow.f64 a 2))) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 y a) (-.f64 (*.f64 -1 (/.f64 x (*.f64 a z))) (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 -1 (*.f64 y t)) (*.f64 (pow.f64 a 2) z))))): 5 points increase in error, 9 points decrease in error
      (+.f64 (/.f64 y a) (-.f64 (*.f64 -1 (/.f64 x (*.f64 a z))) (/.f64 (*.f64 -1 (*.f64 y t)) (Rewrite<= *-commutative_binary64 (*.f64 z (pow.f64 a 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 y a) (-.f64 (*.f64 -1 (/.f64 x (*.f64 a z))) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (*.f64 y t) (*.f64 z (pow.f64 a 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 y a) (*.f64 -1 (/.f64 x (*.f64 a z)))) (*.f64 -1 (/.f64 (*.f64 y t) (*.f64 z (pow.f64 a 2)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 x (*.f64 a z))) (/.f64 y a))) (*.f64 -1 (/.f64 (*.f64 y t) (*.f64 z (pow.f64 a 2))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in x around inf 16.0

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 5.4

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

   2. Simplified 5.4

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
      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. Applied egg-rr 1.9

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

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

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
      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 z around inf 0

      \[\leadsto \color{blue}{\frac{y}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t - z \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.1
Cost	3020

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 2
Error	28.5
Cost	2296

\[\begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ t_2 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;t \leq -1.18 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-180}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-288}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.25 \cdot 10^{-164}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\frac{x}{z}}{a}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+161}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	36.9
Cost	1836

\[\begin{array}{l} t_1 := \frac{\frac{x}{-a}}{z}\\ \mathbf{if}\;a \leq -3.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq -3.8 \cdot 10^{+33}:\\ \;\;\;\;-\frac{x}{z \cdot a}\\ \mathbf{elif}\;a \leq -0.0008:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq -2.5 \cdot 10^{-155}:\\ \;\;\;\;y \cdot \frac{-z}{t}\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-178}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+93}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+171}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 4
Error	19.3
Cost	1636

\[\begin{array}{l} t_1 := y \cdot \frac{z}{z \cdot a - t}\\ t_2 := \frac{x}{t - z \cdot a}\\ t_3 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+168}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -8 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.4 \cdot 10^{+73}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{+61}:\\ \;\;\;\;\left(y \cdot z - x\right) \cdot \frac{-1}{t}\\ \mathbf{elif}\;z \leq -0.0148:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-141}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+112}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	19.3
Cost	1636

\[\begin{array}{l} t_1 := y \cdot \frac{z}{z \cdot a - t}\\ t_2 := \frac{x}{t - z \cdot a}\\ t_3 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -5 \cdot 10^{+168}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+80}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{+61}:\\ \;\;\;\;\left(y \cdot z - x\right) \cdot \frac{-1}{t}\\ \mathbf{elif}\;z \leq -0.00186:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{x}{t} - \frac{y \cdot z}{t}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Error	22.5
Cost	1504

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ t_2 := \frac{x - y \cdot z}{t}\\ t_3 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{-14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-238}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-115}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+110}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	20.0
Cost	1104

\[\begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+21}:\\ \;\;\;\;\frac{y}{a} - \frac{x}{z \cdot a}\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{t} - \frac{y \cdot z}{t}\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+95}:\\ \;\;\;\;\left(y \cdot z - x\right) \cdot \frac{-1}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \end{array} \]

Alternative 8
Error	36.6
Cost	984

\[\begin{array}{l} \mathbf{if}\;a \leq -3.25 \cdot 10^{+159}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{+34}:\\ \;\;\;\;-\frac{x}{z \cdot a}\\ \mathbf{elif}\;a \leq -0.00042:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-155}:\\ \;\;\;\;y \cdot \frac{-z}{t}\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-178}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 9
Error	23.2
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+131}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{+60}:\\ \;\;\;\;\frac{-y}{\frac{t}{z}}\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{+48}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 10
Error	30.9
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+131}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+61}:\\ \;\;\;\;\frac{-y}{\frac{t}{z}}\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+108}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 11
Error	30.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+108}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 12
Error	41.9
Cost	192

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

Reproduce

herbie shell --seed 2022325 
(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))))