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

Average Error: 10.3 → 5.9

Time: 12.8s

Precision: binary64

Cost: 7812

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-302}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, y, -x\right)}{z \cdot a - t}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{y \cdot z - x}{a} \cdot \frac{1}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+248}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{a}}{z} + \frac{z}{z} \cdot \frac{y}{a}\\ \end{array} \]

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 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 -1e-302)
     (/ (fma z y (- x)) (- (* z a) t))
     (if (<= t_1 0.0)
       (* (/ (- (* y z) x) a) (/ 1.0 z))
       (if (<= t_1 1e+248)
         t_1
         (+ (* x (/ (/ -1.0 a) z)) (* (/ z 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 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_1 <= -1e-302) {
		tmp = fma(z, y, -x) / ((z * a) - t);
	} else if (t_1 <= 0.0) {
		tmp = (((y * z) - x) / a) * (1.0 / z);
	} else if (t_1 <= 1e+248) {
		tmp = t_1;
	} else {
		tmp = (x * ((-1.0 / a) / z)) + ((z / z) * (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(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_1 <= -1e-302)
		tmp = Float64(fma(z, y, Float64(-x)) / Float64(Float64(z * a) - t));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(y * z) - x) / a) * Float64(1.0 / z));
	elseif (t_1 <= 1e+248)
		tmp = t_1;
	else
		tmp = Float64(Float64(x * Float64(Float64(-1.0 / a) / z)) + Float64(Float64(z / z) * Float64(y / a)));
	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[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-302], N[(N[(z * y + (-x)), $MachinePrecision] / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / a), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+248], t$95$1, N[(N[(x * N[(N[(-1.0 / a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(N[(z / z), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	10.3
Target	1.6
Herbie	5.9

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -9.9999999999999996e-303

   1. Initial program 5.3

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

   2. Simplified 5.3

      \[\leadsto \color{blue}{\frac{y \cdot z - x}{z \cdot a - t}} \]
      Proof
      (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 z a) t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 a z)) t)): 0 points increase in error, 17 points decrease in error
      (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 a z) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 a z) t))): 17 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (*.f64 y z) x)) (*.f64 -1 (-.f64 (*.f64 a z) t)))): 0 points increase in error, 17 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (*.f64 y z) x))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 17 points decrease in error
      (/.f64 (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 y z) (neg.f64 x)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 y z)) (neg.f64 (neg.f64 x)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (neg.f64 x)) (neg.f64 (*.f64 y z)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> remove-double-neg_binary64 x) (neg.f64 (*.f64 y z))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 17 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (*.f64 y z))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 17 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (*.f64 a z) t)))): 0 points increase in error, 17 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 a z) (neg.f64 t))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 a z)) (neg.f64 (neg.f64 t))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (+.f64 (neg.f64 (*.f64 a z)) (Rewrite=> remove-double-neg_binary64 t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= +-commutative_binary64 (+.f64 t (neg.f64 (*.f64 a z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 a z)))): 17 points increase in error, 0 points decrease in error

   3. Applied egg-rr 5.3

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

   if -9.9999999999999996e-303 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 24.1

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

   2. Simplified 24.1

      \[\leadsto \color{blue}{\frac{y \cdot z - x}{z \cdot a - t}} \]
      Proof
      (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 z a) t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 a z)) t)): 0 points increase in error, 17 points decrease in error
      (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 a z) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 a z) t))): 17 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (*.f64 y z) x)) (*.f64 -1 (-.f64 (*.f64 a z) t)))): 0 points increase in error, 17 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (*.f64 y z) x))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 17 points decrease in error
      (/.f64 (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 y z) (neg.f64 x)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 y z)) (neg.f64 (neg.f64 x)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (neg.f64 x)) (neg.f64 (*.f64 y z)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> remove-double-neg_binary64 x) (neg.f64 (*.f64 y z))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 17 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (*.f64 y z))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 17 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (*.f64 a z) t)))): 0 points increase in error, 17 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 a z) (neg.f64 t))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 a z)) (neg.f64 (neg.f64 t))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (+.f64 (neg.f64 (*.f64 a z)) (Rewrite=> remove-double-neg_binary64 t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= +-commutative_binary64 (+.f64 t (neg.f64 (*.f64 a z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 a z)))): 17 points increase in error, 0 points decrease in error

   3. Taylor expanded in a around inf 41.9

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

   4. Simplified 41.9

      \[\leadsto \color{blue}{\frac{z \cdot y - x}{a \cdot z}} \]
      Proof
      (/.f64 (-.f64 (*.f64 z y) x) (*.f64 a z)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 y z)) x) (*.f64 a z)): 0 points increase in error, 2 points decrease in error

   5. Applied egg-rr 15.0

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

   if -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.00000000000000005e248

   1. Initial program 0.2

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

   if 1.00000000000000005e248 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 49.1

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

   2. Simplified 49.1

      \[\leadsto \color{blue}{\frac{y \cdot z - x}{z \cdot a - t}} \]
      Proof
      (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 z a) t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 a z)) t)): 0 points increase in error, 17 points decrease in error
      (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 a z) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 a z) t))): 17 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (*.f64 y z) x)) (*.f64 -1 (-.f64 (*.f64 a z) t)))): 0 points increase in error, 17 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (*.f64 y z) x))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 17 points decrease in error
      (/.f64 (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 y z) (neg.f64 x)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 y z)) (neg.f64 (neg.f64 x)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (neg.f64 x)) (neg.f64 (*.f64 y z)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> remove-double-neg_binary64 x) (neg.f64 (*.f64 y z))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 17 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (*.f64 y z))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 17 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (*.f64 a z) t)))): 0 points increase in error, 17 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 a z) (neg.f64 t))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 a z)) (neg.f64 (neg.f64 t))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (+.f64 (neg.f64 (*.f64 a z)) (Rewrite=> remove-double-neg_binary64 t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= +-commutative_binary64 (+.f64 t (neg.f64 (*.f64 a z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 a z)))): 17 points increase in error, 0 points decrease in error

   3. Taylor expanded in a around inf 57.9

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

   4. Simplified 57.9

      \[\leadsto \color{blue}{\frac{z \cdot y - x}{a \cdot z}} \]
      Proof
      (/.f64 (-.f64 (*.f64 z y) x) (*.f64 a z)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 y z)) x) (*.f64 a z)): 0 points increase in error, 2 points decrease in error

   5. Applied egg-rr 16.3

      \[\leadsto \color{blue}{\frac{\frac{1}{a}}{z} \cdot \left(-x\right) + \frac{z}{z} \cdot \frac{y}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1 \cdot 10^{-302}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, y, -x\right)}{z \cdot a - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y \cdot z - x}{a} \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{+248}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{a}}{z} + \frac{z}{z} \cdot \frac{y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.9
Cost	3276

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{y \cdot z - x}{a} \cdot \frac{1}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+248}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{a}}{z} + \frac{z}{z} \cdot \frac{y}{a}\\ \end{array} \]

Alternative 2
Error	5.9
Cost	3276

\[\begin{array}{l} t_1 := z \cdot a - t\\ t_2 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-302}:\\ \;\;\;\;\frac{y \cdot z}{t_1} - \frac{x}{t_1}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{y \cdot z - x}{a} \cdot \frac{1}{z}\\ \mathbf{elif}\;t_2 \leq 10^{+248}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{a}}{z} + \frac{z}{z} \cdot \frac{y}{a}\\ \end{array} \]

Alternative 3
Error	5.6
Cost	3020

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{y \cdot z - x}{a} \cdot \frac{1}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+277}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 4
Error	23.8
Cost	976

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5
Error	18.1
Cost	976

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	30.6
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{t} \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7
Error	23.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-20}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8
Error	30.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-22}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 9
Error	42.2
Cost	192

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

Reproduce

herbie shell --seed 2022340 
(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))))