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

Average Error: 10.8 → 4.1

Time: 18.7s

Precision: binary64

Cost: 2892

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot a - t\\ t_2 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t_1}{z}}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+287}:\\ \;\;\;\;\frac{y \cdot z}{t_1} - \frac{x}{t_1}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

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 (- (* z a) t)) (t_2 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_2 (- INFINITY))
     (/ y (/ t_1 z))
     (if (<= t_2 5e+287)
       (- (/ (* y z) t_1) (/ x t_1))
       (if (<= t_2 INFINITY) (* y (/ z t_1)) (/ 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 = (z * a) - t;
	double t_2 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y / (t_1 / z);
	} else if (t_2 <= 5e+287) {
		tmp = ((y * z) / t_1) - (x / t_1);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = y * (z / t_1);
	} 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 = (z * a) - t;
	double t_2 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = y / (t_1 / z);
	} else if (t_2 <= 5e+287) {
		tmp = ((y * z) / t_1) - (x / t_1);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = y * (z / t_1);
	} 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 = (z * a) - t
	t_2 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = y / (t_1 / z)
	elif t_2 <= 5e+287:
		tmp = ((y * z) / t_1) - (x / t_1)
	elif t_2 <= math.inf:
		tmp = y * (z / t_1)
	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(z * a) - t)
	t_2 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(y / Float64(t_1 / z));
	elseif (t_2 <= 5e+287)
		tmp = Float64(Float64(Float64(y * z) / t_1) - Float64(x / t_1));
	elseif (t_2 <= Inf)
		tmp = Float64(y * Float64(z / t_1));
	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 = (z * a) - t;
	t_2 = (x - (y * z)) / (t - (z * a));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = y / (t_1 / z);
	elseif (t_2 <= 5e+287)
		tmp = ((y * z) / t_1) - (x / t_1);
	elseif (t_2 <= Inf)
		tmp = y * (z / t_1);
	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[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(y / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+287], N[(N[(N[(y * z), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]

Target

Original	10.8
Target	1.7
Herbie	4.1

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

Derivation

1. Split input into 4 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

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

   4. Taylor expanded in y around inf 64.0

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

   5. Simplified 0.1

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

      [Start] 64.0	\[ \frac{y \cdot z}{a \cdot z - t} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{y}{\frac{a \cdot z - t}{z}}} \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5e287

   1. Initial program 4.3

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

   2. Simplified 4.3

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

      [Start] 4.3	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 4.3	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 4.3	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 4.3	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 4.3	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 4.3	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 4.3	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 4.3	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 4.3	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 4.3	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 4.3	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 4.3	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 4.3	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 4.3	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 4.3	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 4.3	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 4.3	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Applied egg-rr 4.3

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

   if 5e287 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 52.7

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

   2. Simplified 52.7

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

      [Start] 52.7	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 52.7	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 52.7	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 52.7	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 52.7	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 52.7	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 52.7	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 52.7	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 52.7	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 52.7	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 52.7	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 52.7	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 52.7	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 52.7	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 52.7	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 52.7	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 52.7	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Applied egg-rr 52.7

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

   4. Taylor expanded in y around inf 60.4

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

   5. Simplified 8.1

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

      [Start] 60.4	\[ \frac{y \cdot z}{a \cdot z - t} \]
      *-commutative [<=] 60.4	\[ \frac{\color{blue}{z \cdot y}}{a \cdot z - t} \]
      *-commutative [<=] 60.4	\[ \frac{z \cdot y}{\color{blue}{z \cdot a} - t} \]
      associate-*l/ [<=] 8.1	\[ \color{blue}{\frac{z}{z \cdot a - t} \cdot y} \]
      *-commutative [=>] 8.1	\[ \color{blue}{y \cdot \frac{z}{z \cdot a - t}} \]
      *-commutative [=>] 8.1	\[ y \cdot \frac{z}{\color{blue}{a \cdot z} - t} \]

   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

      \[\leadsto \color{blue}{\frac{y \cdot z - x}{z \cdot a - t}} \]
      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. Taylor expanded in z around inf 0.2

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

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z \cdot a - t}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 5 \cdot 10^{+287}:\\ \;\;\;\;\frac{y \cdot z}{z \cdot a - t} - \frac{x}{z \cdot a - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.1
Cost	2892

\[\begin{array}{l} t_1 := z \cdot a - t\\ t_2 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t_1}{z}}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+287}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;y \cdot \frac{z}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 2
Error	37.7
Cost	1968

\[\begin{array}{l} t_1 := \left(-y\right) \cdot \frac{z}{t}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{+229}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;t \leq -7.4 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{+51}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-289}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+257}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{+296}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 3
Error	37.7
Cost	1968

\[\begin{array}{l} t_1 := \left(-y\right) \cdot \frac{z}{t}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{+229}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;t \leq -2.05 \cdot 10^{+177}:\\ \;\;\;\;\frac{-y}{\frac{t}{z}}\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{+51}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-290}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;t \leq 10^{+101}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+257}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{+296}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 4
Error	30.4
Cost	1769

\[\begin{array}{l} t_1 := \left(-y\right) \cdot \frac{z}{t}\\ t_2 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{z \cdot \left(-y\right)}{t}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-117}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-211}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-275}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{-248}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+176} \lor \neg \left(x \leq 2.9 \cdot 10^{+189}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5
Error	21.5
Cost	1632

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ t_2 := z \cdot a - t\\ t_3 := \frac{x}{t - z \cdot a}\\ t_4 := \frac{x}{t} - \frac{z}{\frac{t}{y}}\\ \mathbf{if}\;a \leq -5.6 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.1 \cdot 10^{-17}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.5 \cdot 10^{-103}:\\ \;\;\;\;y \cdot \frac{z}{t_2}\\ \mathbf{elif}\;a \leq -3.2 \cdot 10^{-124}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -9.6 \cdot 10^{-307}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-223}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-178}:\\ \;\;\;\;z \cdot \frac{y}{t_2}\\ \mathbf{elif}\;a \leq 19000:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	21.5
Cost	1632

\[\begin{array}{l} t_1 := \frac{x}{t} - \frac{z}{\frac{t}{y}}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ t_3 := z \cdot a - t\\ \mathbf{if}\;a \leq -4.2 \cdot 10^{+76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;a \leq -3.8 \cdot 10^{-106}:\\ \;\;\;\;y \cdot \frac{z}{t_3}\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-123}:\\ \;\;\;\;\frac{-1}{\frac{t_3}{x}}\\ \mathbf{elif}\;a \leq -1.3 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-223}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-178}:\\ \;\;\;\;z \cdot \frac{y}{t_3}\\ \mathbf{elif}\;a \leq 18000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	21.6
Cost	1632

\[\begin{array}{l} t_1 := \frac{x}{t} - \frac{z}{\frac{t}{y}}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ t_3 := z \cdot a - t\\ \mathbf{if}\;a \leq -8.6 \cdot 10^{+74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{\frac{t_3}{z}}\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{-123}:\\ \;\;\;\;\frac{-1}{\frac{t_3}{x}}\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-223}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-178}:\\ \;\;\;\;z \cdot \frac{y}{t_3}\\ \mathbf{elif}\;a \leq 27000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	22.1
Cost	1632

\[\begin{array}{l} t_1 := \frac{x}{t} - \frac{z}{\frac{t}{y}}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ t_3 := z \cdot a - t\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.1 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;a \leq -2.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{\frac{t_3}{z}}\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{-127}:\\ \;\;\;\;\frac{-1}{\frac{t_3}{x}}\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-246}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;a \leq 2.25 \cdot 10^{-178}:\\ \;\;\;\;\frac{z}{\frac{t_3}{y}}\\ \mathbf{elif}\;a \leq 13500:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	21.6
Cost	1632

\[\begin{array}{l} t_1 := \frac{x}{t} - \frac{z}{\frac{t}{y}}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ t_3 := z \cdot a - t\\ \mathbf{if}\;a \leq -2.2 \cdot 10^{+76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{y}{\frac{t_3}{z}}\\ \mathbf{elif}\;a \leq -1.8 \cdot 10^{-123}:\\ \;\;\;\;\frac{-1}{\frac{t_3}{x}}\\ \mathbf{elif}\;a \leq -7 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-223}:\\ \;\;\;\;\frac{1}{\frac{-t}{y \cdot z - x}}\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-178}:\\ \;\;\;\;\frac{z}{\frac{t_3}{y}}\\ \mathbf{elif}\;a \leq 32000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	19.4
Cost	1368

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ t_2 := y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{if}\;z \leq -5.1 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 200000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+129}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	19.4
Cost	1368

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ t_2 := z \cdot a - t\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 2800000:\\ \;\;\;\;z \cdot \frac{y}{t_2}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+132}:\\ \;\;\;\;y \cdot \frac{z}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	27.5
Cost	1241

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ t_2 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.32 \cdot 10^{-210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-275}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;x \leq 680:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+176} \lor \neg \left(x \leq 2.9 \cdot 10^{+189}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 13
Error	20.0
Cost	978

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-24} \lor \neg \left(z \leq 9.8 \cdot 10^{-65}\right) \land \left(z \leq 3.95 \cdot 10^{+93} \lor \neg \left(z \leq 1.4 \cdot 10^{+124}\right)\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \]

Alternative 14
Error	30.7
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -0.014:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-112}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 15
Error	30.5
Cost	457

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \lor \neg \left(z \leq 1.36 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 16
Error	42.5
Cost	192

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

Reproduce

herbie shell --seed 2023060 
(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))))