Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Average Error: 24.6 → 7.1

Time: 20.7s

Precision: binary64

Cost: 8004

Math

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

↓

\[\begin{array}{l} t_1 := x - \frac{\left(y - z\right) \cdot \left(x - t\right)}{a - z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t + \frac{y - a}{z} \cdot \left(x - t\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t + \frac{a - y}{\frac{z}{t - x}}\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* (- y z) (- x t)) (- a z)))))
   (if (<= t_1 -2e-291)
     (fma (/ (- y z) (- a z)) (- t x) x)
     (if (<= t_1 0.0)
       (+ t (* (/ (- y a) z) (- x t)))
       (if (<= t_1 2e+300) t_1 (+ t (/ (- a y) (/ z (- t x)))))))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (((y - z) * (x - t)) / (a - z));
	double tmp;
	if (t_1 <= -2e-291) {
		tmp = fma(((y - z) / (a - z)), (t - x), x);
	} else if (t_1 <= 0.0) {
		tmp = t + (((y - a) / z) * (x - t));
	} else if (t_1 <= 2e+300) {
		tmp = t_1;
	} else {
		tmp = t + ((a - y) / (z / (t - x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(Float64(y - z) * Float64(x - t)) / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -2e-291)
		tmp = fma(Float64(Float64(y - z) / Float64(a - z)), Float64(t - x), x);
	elseif (t_1 <= 0.0)
		tmp = Float64(t + Float64(Float64(Float64(y - a) / z) * Float64(x - t)));
	elseif (t_1 <= 2e+300)
		tmp = t_1;
	else
		tmp = Float64(t + Float64(Float64(a - y) / Float64(z / Float64(t - x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(N[(y - z), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-291], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(t + N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+300], t$95$1, N[(t + N[(N[(a - y), $MachinePrecision] / N[(z / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	24.6
Target	12.1
Herbie	7.1

\[\begin{array}{l} \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -1.99999999999999992e-291

   1. Initial program 20.9

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

   2. Simplified 6.8

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

      [Start] 20.9	\[ x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
      +-commutative [=>] 20.9	\[ \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]
      associate-*l/ [<=] 6.8	\[ \color{blue}{\frac{y - z}{a - z} \cdot \left(t - x\right)} + x \]
      fma-def [=>] 6.8	\[ \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   if -1.99999999999999992e-291 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 60.1

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

   2. Simplified 59.9

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

      [Start] 60.1	\[ x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
      +-commutative [=>] 60.1	\[ \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]
      associate-*l/ [<=] 59.9	\[ \color{blue}{\frac{y - z}{a - z} \cdot \left(t - x\right)} + x \]
      fma-def [=>] 59.9	\[ \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   3. Taylor expanded in z around inf 1.6

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

   4. Simplified 1.7

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

      [Start] 1.6	\[ \frac{\left(-1 \cdot y - -1 \cdot a\right) \cdot \left(t - x\right)}{z} + t \]
      +-commutative [=>] 1.6	\[ \color{blue}{t + \frac{\left(-1 \cdot y - -1 \cdot a\right) \cdot \left(t - x\right)}{z}} \]
      distribute-lft-out-- [=>] 1.6	\[ t + \frac{\color{blue}{\left(-1 \cdot \left(y - a\right)\right)} \cdot \left(t - x\right)}{z} \]
      associate-*r* [<=] 1.6	\[ t + \frac{\color{blue}{-1 \cdot \left(\left(y - a\right) \cdot \left(t - x\right)\right)}}{z} \]
      associate-*r/ [<=] 1.6	\[ t + \color{blue}{-1 \cdot \frac{\left(y - a\right) \cdot \left(t - x\right)}{z}} \]
      mul-1-neg [=>] 1.6	\[ t + \color{blue}{\left(-\frac{\left(y - a\right) \cdot \left(t - x\right)}{z}\right)} \]
      unsub-neg [=>] 1.6	\[ \color{blue}{t - \frac{\left(y - a\right) \cdot \left(t - x\right)}{z}} \]
      associate-/l* [=>] 7.3	\[ t - \color{blue}{\frac{y - a}{\frac{z}{t - x}}} \]
      associate-/r/ [=>] 1.7	\[ t - \color{blue}{\frac{y - a}{z} \cdot \left(t - x\right)} \]

   if 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 2.0000000000000001e300

   1. Initial program 2.2

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

   if 2.0000000000000001e300 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 62.6

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

   2. Simplified 17.4

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

      [Start] 62.6	\[ x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
      +-commutative [=>] 62.6	\[ \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]
      associate-*l/ [<=] 17.4	\[ \color{blue}{\frac{y - z}{a - z} \cdot \left(t - x\right)} + x \]
      fma-def [=>] 17.4	\[ \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   3. Taylor expanded in z around inf 40.7

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

   4. Simplified 21.2

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

      [Start] 40.7	\[ \frac{\left(-1 \cdot y - -1 \cdot a\right) \cdot \left(t - x\right)}{z} + t \]
      +-commutative [=>] 40.7	\[ \color{blue}{t + \frac{\left(-1 \cdot y - -1 \cdot a\right) \cdot \left(t - x\right)}{z}} \]
      associate-/l* [=>] 21.2	\[ t + \color{blue}{\frac{-1 \cdot y - -1 \cdot a}{\frac{z}{t - x}}} \]
      distribute-lft-out-- [=>] 21.2	\[ t + \frac{\color{blue}{-1 \cdot \left(y - a\right)}}{\frac{z}{t - x}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 7.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(y - z\right) \cdot \left(x - t\right)}{a - z} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{elif}\;x - \frac{\left(y - z\right) \cdot \left(x - t\right)}{a - z} \leq 0:\\ \;\;\;\;t + \frac{y - a}{z} \cdot \left(x - t\right)\\ \mathbf{elif}\;x - \frac{\left(y - z\right) \cdot \left(x - t\right)}{a - z} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;x - \frac{\left(y - z\right) \cdot \left(x - t\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{a - y}{\frac{z}{t - x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.1
Cost	3532

\[\begin{array}{l} t_1 := x - \frac{\left(y - z\right) \cdot \left(x - t\right)}{a - z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t + \frac{y - a}{z} \cdot \left(x - t\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t + \frac{a - y}{\frac{z}{t - x}}\\ \end{array} \]

Alternative 2
Error	27.4
Cost	1500

\[\begin{array}{l} t_1 := x + t \cdot \frac{y}{a}\\ t_2 := t \cdot \frac{y - z}{a - z}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+75}:\\ \;\;\;\;t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq -9 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+33}:\\ \;\;\;\;x \cdot \frac{y - a}{z}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	27.3
Cost	1368

\[\begin{array}{l} t_1 := t \cdot \frac{y - z}{a - z}\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{+75}:\\ \;\;\;\;t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{+30}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-270}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-139}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+34}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	35.6
Cost	1240

\[\begin{array}{l} t_1 := t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-226}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-188}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+32}:\\ \;\;\;\;x \cdot \frac{y - a}{z}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	35.7
Cost	1240

\[\begin{array}{l} t_1 := t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -1.08 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-226}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-183}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;\left(y - a\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	24.4
Cost	1104

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+74}:\\ \;\;\;\;t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-52}:\\ \;\;\;\;x - y \cdot \frac{x - t}{a}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+34}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 1.48 \cdot 10^{+35}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \end{array} \]

Alternative 7
Error	23.5
Cost	1104

\[\begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+75}:\\ \;\;\;\;t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-54}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+33}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \end{array} \]

Alternative 8
Error	8.7
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+211} \lor \neg \left(z \leq 1.26 \cdot 10^{+78}\right):\\ \;\;\;\;t + \frac{y - a}{z} \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \end{array} \]

Alternative 9
Error	34.1
Cost	976

\[\begin{array}{l} t_1 := t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-226}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-182}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+36}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	28.5
Cost	976

\[\begin{array}{l} t_1 := t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-23}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+32}:\\ \;\;\;\;x \cdot \frac{y - a}{z}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	28.8
Cost	976

\[\begin{array}{l} t_1 := t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+32}:\\ \;\;\;\;x \cdot \frac{y - a}{z}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+35}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	18.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+74} \lor \neg \left(z \leq 6.2 \cdot 10^{-56}\right):\\ \;\;\;\;t + \frac{y - a}{z} \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \end{array} \]

Alternative 13
Error	20.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+74} \lor \neg \left(z \leq 2.75 \cdot 10^{-55}\right):\\ \;\;\;\;t - \left(t - x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \end{array} \]

Alternative 14
Error	35.8
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+91}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-226}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-188}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 15
Error	35.8
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+92}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-226}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-188}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+42}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 16
Error	35.4
Cost	328

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+91}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+41}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 17
Error	62.1
Cost	64

\[0 \]

Alternative 18
Error	46.1
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023016 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

  (+ x (/ (* (- y z) (- t x)) (- a z))))