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

Average Error: 10.5 → 3.8

Time: 1.7min

Precision: binary64

Cost: 4112

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot a - t\\ t_2 := x - y \cdot z\\ t_3 := \frac{t_2}{t - a \cdot z}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;-\frac{z}{t_1} \cdot \left(-y\right)\\ \mathbf{elif}\;t_3 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;\frac{y \cdot \left(z + z\right) - \left(y \cdot z + x\right)}{t_1}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\frac{\left(\frac{x}{z} - y\right) \cdot \frac{-2}{a}}{2}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\frac{t_2}{\left(\left(t + t\right) - z \cdot a\right) + \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y - \left(\frac{x}{z} - \left(y - \frac{x}{z}\right)\right)}{a}}{2}\\ \end{array} \]

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_3 (/ t_2 (- t (* a z)))))
   (if (<= t_3 (- INFINITY))
     (- (* (/ z t_1) (- y)))
     (if (<= t_3 -4e-306)
       (/ (- (* y (+ z z)) (+ (* y z) x)) t_1)
       (if (<= t_3 0.0)
         (/ (* (- (/ x z) y) (/ -2.0 a)) 2.0)
         (if (<= t_3 5e+279)
           (/ t_2 (+ (- (+ t t) (* z a)) (- t)))
           (/ (/ (- y (- (/ x z) (- y (/ x z)))) a) 2.0)))))))

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);
	double t_3 = t_2 / (t - (a * z));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -((z / t_1) * -y);
	} else if (t_3 <= -4e-306) {
		tmp = ((y * (z + z)) - ((y * z) + x)) / t_1;
	} else if (t_3 <= 0.0) {
		tmp = (((x / z) - y) * (-2.0 / a)) / 2.0;
	} else if (t_3 <= 5e+279) {
		tmp = t_2 / (((t + t) - (z * a)) + -t);
	} else {
		tmp = ((y - ((x / z) - (y - (x / z)))) / a) / 2.0;
	}
	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);
	double t_3 = t_2 / (t - (a * z));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = -((z / t_1) * -y);
	} else if (t_3 <= -4e-306) {
		tmp = ((y * (z + z)) - ((y * z) + x)) / t_1;
	} else if (t_3 <= 0.0) {
		tmp = (((x / z) - y) * (-2.0 / a)) / 2.0;
	} else if (t_3 <= 5e+279) {
		tmp = t_2 / (((t + t) - (z * a)) + -t);
	} else {
		tmp = ((y - ((x / z) - (y - (x / z)))) / a) / 2.0;
	}
	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_3 = t_2 / (t - (a * z))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = -((z / t_1) * -y)
	elif t_3 <= -4e-306:
		tmp = ((y * (z + z)) - ((y * z) + x)) / t_1
	elif t_3 <= 0.0:
		tmp = (((x / z) - y) * (-2.0 / a)) / 2.0
	elif t_3 <= 5e+279:
		tmp = t_2 / (((t + t) - (z * a)) + -t)
	else:
		tmp = ((y - ((x / z) - (y - (x / z)))) / a) / 2.0
	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(x - Float64(y * z))
	t_3 = Float64(t_2 / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-Float64(Float64(z / t_1) * Float64(-y)));
	elseif (t_3 <= -4e-306)
		tmp = Float64(Float64(Float64(y * Float64(z + z)) - Float64(Float64(y * z) + x)) / t_1);
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(x / z) - y) * Float64(-2.0 / a)) / 2.0);
	elseif (t_3 <= 5e+279)
		tmp = Float64(t_2 / Float64(Float64(Float64(t + t) - Float64(z * a)) + Float64(-t)));
	else
		tmp = Float64(Float64(Float64(y - Float64(Float64(x / z) - Float64(y - Float64(x / z)))) / a) / 2.0);
	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_3 = t_2 / (t - (a * z));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = -((z / t_1) * -y);
	elseif (t_3 <= -4e-306)
		tmp = ((y * (z + z)) - ((y * z) + x)) / t_1;
	elseif (t_3 <= 0.0)
		tmp = (((x / z) - y) * (-2.0 / a)) / 2.0;
	elseif (t_3 <= 5e+279)
		tmp = t_2 / (((t + t) - (z * a)) + -t);
	else
		tmp = ((y - ((x / z) - (y - (x / z)))) / a) / 2.0;
	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[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], (-N[(N[(z / t$95$1), $MachinePrecision] * (-y)), $MachinePrecision]), If[LessEqual[t$95$3, -4e-306], N[(N[(N[(y * N[(z + z), $MachinePrecision]), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$3, 5e+279], N[(t$95$2 / N[(N[(N[(t + t), $MachinePrecision] - N[(z * a), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - N[(N[(x / z), $MachinePrecision] - N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]

Target

Original	10.5
Target	1.7
Herbie	3.8

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

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

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

      [Start] 64.0	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational_best-simplify-1 [=>] 64.0	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Taylor expanded in x around 0 64.0

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

   4. Simplified 64.0

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

      [Start] 64.0	\[ -1 \cdot \frac{y \cdot z}{t - a \cdot z} \]
      rational_best-simplify-55 [=>] 64.0	\[ \color{blue}{\left(y \cdot z\right) \cdot \frac{-1}{t - a \cdot z}} \]
      rational_best-simplify-1 [<=] 64.0	\[ \left(y \cdot z\right) \cdot \frac{-1}{t - \color{blue}{z \cdot a}} \]
      rational_best-simplify-55 [<=] 64.0	\[ \color{blue}{-1 \cdot \frac{y \cdot z}{t - z \cdot a}} \]
      rational_best-simplify-1 [=>] 64.0	\[ \color{blue}{\frac{y \cdot z}{t - z \cdot a} \cdot -1} \]
      rational_best-simplify-10 [=>] 64.0	\[ \color{blue}{-\frac{y \cdot z}{t - z \cdot a}} \]

   5. Applied egg-rr 0.3

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.00000000000000011e-306

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational_best-simplify-1 [=>] 0.2	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Applied egg-rr 0.2

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

   4. Simplified 0.2

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

      [Start] 0.2	\[ \frac{y \cdot \left(z + z\right)}{z \cdot a - t} - \frac{x + y \cdot z}{z \cdot a - t} \]
      rational_best-simplify-66 [=>] 0.2	\[ \color{blue}{\frac{y \cdot \left(z + z\right) - \left(x + y \cdot z\right)}{z \cdot a - t}} \]
      rational_best-simplify-3 [=>] 0.2	\[ \frac{y \cdot \left(z + z\right) - \color{blue}{\left(y \cdot z + x\right)}}{z \cdot a - t} \]

   if -4.00000000000000011e-306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 25.7

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

   2. Simplified 25.7

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

      [Start] 25.7	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational_best-simplify-1 [=>] 25.7	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Applied egg-rr 25.7

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

   4. Applied egg-rr 25.7

      \[\leadsto \color{blue}{\frac{1.5 \cdot \frac{x - y \cdot z}{t - z \cdot a}}{2} - \frac{\frac{x - y \cdot z}{2 \cdot \left(z \cdot a - t\right)}}{2}} \]

   5. Simplified 25.7

      \[\leadsto \color{blue}{\frac{\left(x - y \cdot z\right) \cdot \frac{1.5}{t - z \cdot a} - \frac{x - y \cdot z}{2 \cdot \left(z \cdot a - t\right)}}{2}} \]
      Proof

      [Start] 25.7	\[ \frac{1.5 \cdot \frac{x - y \cdot z}{t - z \cdot a}}{2} - \frac{\frac{x - y \cdot z}{2 \cdot \left(z \cdot a - t\right)}}{2} \]
      rational_best-simplify-66 [=>] 25.7	\[ \color{blue}{\frac{1.5 \cdot \frac{x - y \cdot z}{t - z \cdot a} - \frac{x - y \cdot z}{2 \cdot \left(z \cdot a - t\right)}}{2}} \]
      rational_best-simplify-55 [=>] 25.7	\[ \frac{\color{blue}{\left(x - y \cdot z\right) \cdot \frac{1.5}{t - z \cdot a}} - \frac{x - y \cdot z}{2 \cdot \left(z \cdot a - t\right)}}{2} \]

   6. Taylor expanded in a around inf 16.4

      \[\leadsto \frac{\color{blue}{\frac{-1.5 \cdot \frac{x - y \cdot z}{z} - 0.5 \cdot \frac{x - y \cdot z}{z}}{a}}}{2} \]

   7. Simplified 16.4

      \[\leadsto \frac{\color{blue}{\frac{\left(x - y \cdot z\right) \cdot \frac{-1.5}{z} - \left(x - y \cdot z\right) \cdot \frac{0.5}{z}}{a}}}{2} \]
      Proof

      [Start] 16.4	\[ \frac{\frac{-1.5 \cdot \frac{x - y \cdot z}{z} - 0.5 \cdot \frac{x - y \cdot z}{z}}{a}}{2} \]
      rational_best-simplify-55 [=>] 16.4	\[ \frac{\frac{\color{blue}{\left(x - y \cdot z\right) \cdot \frac{-1.5}{z}} - 0.5 \cdot \frac{x - y \cdot z}{z}}{a}}{2} \]
      rational_best-simplify-55 [=>] 16.4	\[ \frac{\frac{\left(x - y \cdot z\right) \cdot \frac{-1.5}{z} - \color{blue}{\left(x - y \cdot z\right) \cdot \frac{0.5}{z}}}{a}}{2} \]

   8. Applied egg-rr 16.4

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{\frac{x}{z} - y}{a} + 0}}{2} \]

   9. Simplified 16.5

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

      [Start] 16.4	\[ \frac{-2 \cdot \frac{\frac{x}{z} - y}{a} + 0}{2} \]
      rational_best-simplify-3 [=>] 16.4	\[ \frac{\color{blue}{0 + -2 \cdot \frac{\frac{x}{z} - y}{a}}}{2} \]
      rational_best-simplify-6 [=>] 16.4	\[ \frac{\color{blue}{-2 \cdot \frac{\frac{x}{z} - y}{a}}}{2} \]
      rational_best-simplify-55 [=>] 16.5	\[ \frac{\color{blue}{\left(\frac{x}{z} - y\right) \cdot \frac{-2}{a}}}{2} \]

   if -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5.0000000000000002e279

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational_best-simplify-1 [=>] 0.2	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Applied egg-rr 0.2

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

   if 5.0000000000000002e279 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 57.1

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

   2. Simplified 57.1

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

      [Start] 57.1	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational_best-simplify-1 [=>] 57.1	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Applied egg-rr 57.1

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

   4. Applied egg-rr 57.1

      \[\leadsto \color{blue}{\frac{1.5 \cdot \frac{x - y \cdot z}{t - z \cdot a}}{2} - \frac{\frac{x - y \cdot z}{2 \cdot \left(z \cdot a - t\right)}}{2}} \]

   5. Simplified 57.2

      \[\leadsto \color{blue}{\frac{\left(x - y \cdot z\right) \cdot \frac{1.5}{t - z \cdot a} - \frac{x - y \cdot z}{2 \cdot \left(z \cdot a - t\right)}}{2}} \]
      Proof

      [Start] 57.1	\[ \frac{1.5 \cdot \frac{x - y \cdot z}{t - z \cdot a}}{2} - \frac{\frac{x - y \cdot z}{2 \cdot \left(z \cdot a - t\right)}}{2} \]
      rational_best-simplify-66 [=>] 57.1	\[ \color{blue}{\frac{1.5 \cdot \frac{x - y \cdot z}{t - z \cdot a} - \frac{x - y \cdot z}{2 \cdot \left(z \cdot a - t\right)}}{2}} \]
      rational_best-simplify-55 [=>] 57.2	\[ \frac{\color{blue}{\left(x - y \cdot z\right) \cdot \frac{1.5}{t - z \cdot a}} - \frac{x - y \cdot z}{2 \cdot \left(z \cdot a - t\right)}}{2} \]

   6. Taylor expanded in a around inf 61.4

      \[\leadsto \frac{\color{blue}{\frac{-1.5 \cdot \frac{x - y \cdot z}{z} - 0.5 \cdot \frac{x - y \cdot z}{z}}{a}}}{2} \]

   7. Simplified 61.4

      \[\leadsto \frac{\color{blue}{\frac{\left(x - y \cdot z\right) \cdot \frac{-1.5}{z} - \left(x - y \cdot z\right) \cdot \frac{0.5}{z}}{a}}}{2} \]
      Proof

      [Start] 61.4	\[ \frac{\frac{-1.5 \cdot \frac{x - y \cdot z}{z} - 0.5 \cdot \frac{x - y \cdot z}{z}}{a}}{2} \]
      rational_best-simplify-55 [=>] 61.4	\[ \frac{\frac{\color{blue}{\left(x - y \cdot z\right) \cdot \frac{-1.5}{z}} - 0.5 \cdot \frac{x - y \cdot z}{z}}{a}}{2} \]
      rational_best-simplify-55 [=>] 61.4	\[ \frac{\frac{\left(x - y \cdot z\right) \cdot \frac{-1.5}{z} - \color{blue}{\left(x - y \cdot z\right) \cdot \frac{0.5}{z}}}{a}}{2} \]

   8. Applied egg-rr 12.9

      \[\leadsto \frac{\color{blue}{\frac{y - \frac{x}{z}}{a} - \frac{\frac{x}{z} - y}{a}}}{2} \]

   9. Simplified 12.9

      \[\leadsto \frac{\color{blue}{\frac{y - \left(\frac{x}{z} - \left(y - \frac{x}{z}\right)\right)}{a}}}{2} \]
      Proof

      [Start] 12.9	\[ \frac{\frac{y - \frac{x}{z}}{a} - \frac{\frac{x}{z} - y}{a}}{2} \]
      rational_best-simplify-66 [=>] 12.9	\[ \frac{\color{blue}{\frac{\left(y - \frac{x}{z}\right) - \left(\frac{x}{z} - y\right)}{a}}}{2} \]
      rational_best-simplify-51 [=>] 12.9	\[ \frac{\frac{\color{blue}{y - \left(\frac{x}{z} - \left(y - \frac{x}{z}\right)\right)}}{a}}{2} \]

3. Recombined 5 regimes into one program.

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -\infty:\\ \;\;\;\;-\frac{z}{z \cdot a - t} \cdot \left(-y\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -4 \cdot 10^{-306}:\\ \;\;\;\;\frac{y \cdot \left(z + z\right) - \left(y \cdot z + x\right)}{z \cdot a - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{\left(\frac{x}{z} - y\right) \cdot \frac{-2}{a}}{2}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\frac{x - y \cdot z}{\left(\left(t + t\right) - z \cdot a\right) + \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y - \left(\frac{x}{z} - \left(y - \frac{x}{z}\right)\right)}{a}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.8
Cost	4112

\[\begin{array}{l} t_1 := x - y \cdot z\\ t_2 := \frac{t_1}{t - a \cdot z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;-\frac{z}{z \cdot a - t} \cdot \left(-y\right)\\ \mathbf{elif}\;t_2 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{\left(\frac{x}{z} - y\right) \cdot \frac{-2}{a}}{2}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\frac{t_1}{\left(\left(t + t\right) - z \cdot a\right) + \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y - \left(\frac{x}{z} - \left(y - \frac{x}{z}\right)\right)}{a}}{2}\\ \end{array} \]

Alternative 2
Error	3.7
Cost	4048

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-\frac{z}{z \cdot a - t} \cdot \left(-y\right)\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\left(\frac{x}{z} - y\right) \cdot \frac{-2}{a}}{2}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+279}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y - \left(\frac{x}{z} - \left(y - \frac{x}{z}\right)\right)}{a}}{2}\\ \end{array} \]

Alternative 3
Error	3.4
Cost	3792

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-\frac{z}{z \cdot a - t} \cdot \left(-y\right)\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\left(\frac{x}{z} - y\right) \cdot \frac{-2}{a}}{2}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+305}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 4
Error	21.8
Cost	1564

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := -z \cdot \frac{y}{t_1}\\ t_3 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+183}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{+83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{t_1} \cdot x\\ \mathbf{elif}\;z \leq 10^{-219}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+60}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+224}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5
Error	22.7
Cost	1496

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := -\frac{z}{z \cdot a - t} \cdot \left(-y\right)\\ t_3 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.36 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{t_1} \cdot x\\ \mathbf{elif}\;z \leq 10^{-222}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+60}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+267}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 6
Error	22.7
Cost	1496

\[\begin{array}{l} t_1 := -\frac{z}{z \cdot a - t} \cdot \left(-y\right)\\ t_2 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{2 \cdot t - \left(t + z \cdot a\right)}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-223}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+267}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7
Error	29.9
Cost	1108

\[\begin{array}{l} t_1 := -1 \cdot \frac{\frac{x}{a}}{z}\\ \mathbf{if}\;z \leq -5.9 \cdot 10^{+113}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-11}:\\ \;\;\;\;-\frac{y \cdot z}{t}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8
Error	22.5
Cost	1108

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ t_2 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-180}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-95}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 9
Error	22.5
Cost	1108

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;z \leq -1.36 \cdot 10^{+100}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{t_1} \cdot x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-214}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+127}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 10
Error	19.6
Cost	1100

\[\begin{array}{l} t_1 := \frac{\left(\frac{x}{z} - y\right) \cdot \frac{-2}{a}}{2}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+60}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+207}:\\ \;\;\;\;-\frac{z}{z \cdot a - t} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	29.7
Cost	912

\[\begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+89}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{z \cdot \left(-a\right)}\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+33}:\\ \;\;\;\;-\frac{y \cdot z}{t}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 12
Error	29.7
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -1.42 \cdot 10^{-72}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+32}:\\ \;\;\;\;-\frac{y \cdot z}{t}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 13
Error	22.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+127}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 14
Error	29.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-72}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 15
Error	41.8
Cost	192

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

Reproduce

herbie shell --seed 2023099 
(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))))