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

Average Error: 16.01% → 2.65%

Time: 13.9s

Precision: binary64

Cost: 4177

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot a - t\\ t_2 := \frac{y}{a - \frac{t}{z}}\\ t_3 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-302}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq 0 \lor \neg \left(t_3 \leq 10^{+308}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{t_1} - \frac{x}{t_1}\\ \end{array} \]

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 (/ y (- a (/ t z))))
        (t_3 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_3 (- INFINITY))
     t_2
     (if (<= t_3 -2e-302)
       t_3
       (if (or (<= t_3 0.0) (not (<= t_3 1e+308)))
         t_2
         (- (/ (* y z) t_1) (/ x t_1)))))))

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 = y / (a - (t / z));
	double t_3 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_3 <= -2e-302) {
		tmp = t_3;
	} else if ((t_3 <= 0.0) || !(t_3 <= 1e+308)) {
		tmp = t_2;
	} else {
		tmp = ((y * z) / t_1) - (x / t_1);
	}
	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 = y / (a - (t / z));
	double t_3 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_3 <= -2e-302) {
		tmp = t_3;
	} else if ((t_3 <= 0.0) || !(t_3 <= 1e+308)) {
		tmp = t_2;
	} else {
		tmp = ((y * z) / t_1) - (x / t_1);
	}
	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 = y / (a - (t / z))
	t_3 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_2
	elif t_3 <= -2e-302:
		tmp = t_3
	elif (t_3 <= 0.0) or not (t_3 <= 1e+308):
		tmp = t_2
	else:
		tmp = ((y * z) / t_1) - (x / t_1)
	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(y / Float64(a - Float64(t / z)))
	t_3 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_3 <= -2e-302)
		tmp = t_3;
	elseif ((t_3 <= 0.0) || !(t_3 <= 1e+308))
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y * z) / t_1) - Float64(x / t_1));
	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 = y / (a - (t / z));
	t_3 = (x - (y * z)) / (t - (z * a));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_2;
	elseif (t_3 <= -2e-302)
		tmp = t_3;
	elseif ((t_3 <= 0.0) || ~((t_3 <= 1e+308)))
		tmp = t_2;
	else
		tmp = ((y * z) / t_1) - (x / t_1);
	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[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$2, If[LessEqual[t$95$3, -2e-302], t$95$3, If[Or[LessEqual[t$95$3, 0.0], N[Not[LessEqual[t$95$3, 1e+308]], $MachinePrecision]], t$95$2, N[(N[(N[(y * z), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	16.01%
Target	2.63%
Herbie	2.65%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or -1.9999999999999999e-302 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0 or 1e308 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 63.09

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

   2. Simplified 63.09

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

      [Start] 63.09	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 63.09	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 63.09	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 63.09	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 63.09	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 63.09	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 63.09	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 63.09	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 63.09	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 63.09	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 63.09	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 63.09	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 63.09	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 63.09	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 63.09	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 63.09	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 63.09	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Taylor expanded in y around inf 63.84

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

   4. Simplified 40.79

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

      [Start] 63.84	\[ \frac{y \cdot z}{a \cdot z - t} \]
      associate-/l* [=>] 40.79	\[ \color{blue}{\frac{y}{\frac{a \cdot z - t}{z}}} \]

   5. Taylor expanded in a around 0 9.67

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

   6. Simplified 9.67

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

      [Start] 9.67	\[ \frac{y}{a + -1 \cdot \frac{t}{z}} \]
      mul-1-neg [=>] 9.67	\[ \frac{y}{a + \color{blue}{\left(-\frac{t}{z}\right)}} \]
      distribute-neg-frac [=>] 9.67	\[ \frac{y}{a + \color{blue}{\frac{-t}{z}}} \]

   7. Taylor expanded in y around 0 9.67

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

   8. Simplified 9.67

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

      [Start] 9.67	\[ \frac{y}{a + -1 \cdot \frac{t}{z}} \]
      mul-1-neg [=>] 9.67	\[ \frac{y}{a + \color{blue}{\left(-\frac{t}{z}\right)}} \]
      sub-neg [<=] 9.67	\[ \frac{y}{\color{blue}{a - \frac{t}{z}}} \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.9999999999999999e-302

   1. Initial program 0.31

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1e308

   1. Initial program 0.31

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

   2. Simplified 0.31

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

      [Start] 0.31	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 0.31	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 0.31	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 0.31	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 0.31	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 0.31	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 0.31	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 0.31	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 0.31	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 0.31	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 0.31	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 0.31	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 0.31	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 0.31	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 0.31	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 0.31	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 0.31	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Applied egg-rr 0.31

      \[\leadsto \color{blue}{\frac{y \cdot z}{z \cdot a - t} - \frac{x}{z \cdot a - t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 2.65

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0 \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{+308}\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{z \cdot a - t} - \frac{x}{z \cdot a - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.65%
Cost	3794

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq -2 \cdot 10^{-302}\right) \land \left(t_1 \leq 0 \lor \neg \left(t_1 \leq 10^{+308}\right)\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	59.18%
Cost	1572

\[\begin{array}{l} t_1 := \frac{y \cdot \left(-z\right)}{t}\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-209}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;x \leq 1.38 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+214}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \end{array} \]

Alternative 3
Error	59.97%
Cost	1381

\[\begin{array}{l} t_1 := \frac{y \cdot \left(-z\right)}{t}\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-210}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-80} \lor \neg \left(x \leq 5.2 \cdot 10^{-37}\right) \land x \leq 7.2 \cdot 10^{+214}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 4
Error	42.73%
Cost	1372

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ t_2 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;x \leq -1.42 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+173}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+214}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	45.05%
Cost	1110

\[\begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+148} \lor \neg \left(x \leq -9.5 \cdot 10^{+90}\right) \land \left(x \leq -2.6 \cdot 10^{+15} \lor \neg \left(x \leq 8.5 \cdot 10^{-21}\right) \land x \leq 3.4 \cdot 10^{+159}\right):\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]

Alternative 6
Error	28.45%
Cost	972

\[\begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{t} - \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	30.09%
Cost	908

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot a - t}\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-87}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	28.36%
Cost	908

\[\begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{y \cdot z - x}{-t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	46.1%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -5.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 10
Error	65.34%
Cost	192

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

Reproduce

herbie shell --seed 2023102 
(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))))