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

Average Error: 10.7 → 3.1

Time: 13.5s

Precision: binary64

Cost: 10128

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ t_2 := y \cdot z - x\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z \cdot a - t}{z}}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-314}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\frac{t_2}{a}}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+303}:\\ \;\;\;\;\frac{t_2}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

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 (/ (- x (* y z)) (- t (* z a)))) (t_2 (- (* y z) x)))
   (if (<= t_1 (- INFINITY))
     (/ y (/ (- (* z a) t) z))
     (if (<= t_1 -2e-314)
       t_1
       (if (<= t_1 0.0)
         (/ (/ t_2 a) z)
         (if (<= t_1 1e+303) (/ t_2 (fma z a (- t))) (/ 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 t_2 = (y * z) - x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y / (((z * a) - t) / z);
	} else if (t_1 <= -2e-314) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (t_2 / a) / z;
	} else if (t_1 <= 1e+303) {
		tmp = t_2 / fma(z, a, -t);
	} 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(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	t_2 = Float64(Float64(y * z) - x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y / Float64(Float64(Float64(z * a) - t) / z));
	elseif (t_1 <= -2e-314)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(t_2 / a) / z);
	elseif (t_1 <= 1e+303)
		tmp = Float64(t_2 / fma(z, a, Float64(-t)));
	else
		tmp = 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]}, Block[{t$95$2 = N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y / N[(N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-314], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(t$95$2 / a), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 1e+303], N[(t$95$2 / N[(z * a + (-t)), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]

Target

Original	10.7
Target	1.8
Herbie	3.1

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

Derivation

1. Split input into 5 regimes

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

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 64.0	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 64.0	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 64.0	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 64.0	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 64.0	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 64.0	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 64.0	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 64.0	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 64.0	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 64.0	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 64.0	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 64.0	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 64.0	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 64.0	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 64.0	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 64.0	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Applied egg-rr 64.0

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

   4. Taylor expanded in y around inf 64.0

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

   5. Simplified 0.2

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

      [Start] 64.0	\[ \frac{y \cdot z}{a \cdot z - t} \]
      associate-/l* [=>] 0.2	\[ \color{blue}{\frac{y}{\frac{a \cdot z - t}{z}}} \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.9999999999e-314

   1. Initial program 0.2

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

   if -1.9999999999e-314 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 26.0

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

   2. Simplified 26.0

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

      [Start] 26.0	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 26.0	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 26.0	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 26.0	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 26.0	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 26.0	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 26.0	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 26.0	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 26.0	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 26.0	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 26.0	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 26.0	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 26.0	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 26.0	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 26.0	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 26.0	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 26.0	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Applied egg-rr 26.0

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

   4. Taylor expanded in z around inf 42.7

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

   5. Applied egg-rr 14.1

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

   if -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1e303

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 0.2	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 0.2	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 0.2	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 0.2	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 0.2	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 0.2	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 0.2	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 0.2	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 0.2	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 0.2	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 0.2	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 0.2	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 0.2	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 0.2	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 0.2	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 0.2	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Applied egg-rr 0.2

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

   if 1e303 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 62.6

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

   2. Simplified 62.6

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

      [Start] 62.6	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 62.6	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 62.6	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 62.6	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 62.6	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 62.6	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 62.6	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 62.6	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 62.6	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 62.6	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 62.6	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 62.6	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 62.6	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 62.6	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 62.6	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 62.6	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 62.6	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Taylor expanded in z around inf 11.7

      \[\leadsto \color{blue}{\frac{y}{a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z \cdot a - t}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-314}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{\frac{y \cdot z - x}{a}}{z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{+303}:\\ \;\;\;\;\frac{y \cdot z - x}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.1
Cost	3792

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z \cdot a - t}{z}}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-314}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\frac{y \cdot z - x}{a}}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+303}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 2
Error	18.8
Cost	1369

\[\begin{array}{l} t_1 := \frac{y \cdot z - x}{-t}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ t_3 := z \cdot a - t\\ \mathbf{if}\;z \leq -2800000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-223}:\\ \;\;\;\;\frac{-x}{t_3}\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+130} \lor \neg \left(z \leq 7.8 \cdot 10^{+165}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{t_3}\\ \end{array} \]

Alternative 3
Error	18.7
Cost	1369

\[\begin{array}{l} t_1 := \frac{y \cdot z - x}{-t}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ t_3 := z \cdot a - t\\ \mathbf{if}\;z \leq -3900000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-223}:\\ \;\;\;\;\frac{-x}{t_3}\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+129} \lor \neg \left(z \leq 5.6 \cdot 10^{+166}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{t_3}{z}}\\ \end{array} \]

Alternative 4
Error	25.7
Cost	1108

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -2900000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.9 \cdot 10^{-55}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-180}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	18.6
Cost	1040

\[\begin{array}{l} t_1 := \frac{y \cdot z - x}{-t}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -2500000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-223}:\\ \;\;\;\;\frac{-x}{z \cdot a - t}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	31.6
Cost	912

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+98}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2500000000000:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+155}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7
Error	30.9
Cost	912

\[\begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+98}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3100000000000:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8
Error	31.9
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+109}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 9
Error	31.7
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+109}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+156}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 10
Error	18.7
Cost	777

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+108} \lor \neg \left(z \leq 3.25 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot a - t}\\ \end{array} \]

Alternative 11
Error	30.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+107}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 12
Error	41.7
Cost	192

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

Reproduce

herbie shell --seed 2023039 
(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))))