Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Average Error: 7.5 → 1.8

Time: 18.1s

Precision: binary64

Cost: 4940

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

↓

\[\begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t_1}}{x + 1}\\ \mathbf{if}\;t_2 \leq -5000000000000:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{t_1}\\ \mathbf{elif}\;t_2 \leq 10^{-191}:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{x + 1}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+278}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{t \cdot \left(x + 1\right)} + \frac{x}{x + 1}\right) - \frac{x}{\left(z \cdot t\right) \cdot \left(x + 1\right)}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 -5000000000000.0)
     (* (/ y (+ x 1.0)) (/ z t_1))
     (if (<= t_2 1e-191)
       (/ (+ x (/ (- y (/ x z)) t)) (+ x 1.0))
       (if (<= t_2 5e+278)
         t_2
         (-
          (+ (/ y (* t (+ x 1.0))) (/ x (+ x 1.0)))
          (/ x (* (* z t) (+ x 1.0)))))))))

C

double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5000000000000.0) {
		tmp = (y / (x + 1.0)) * (z / t_1);
	} else if (t_2 <= 1e-191) {
		tmp = (x + ((y - (x / z)) / t)) / (x + 1.0);
	} else if (t_2 <= 5e+278) {
		tmp = t_2;
	} else {
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / ((z * t) * (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * t) - x
    t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0d0)
    if (t_2 <= (-5000000000000.0d0)) then
        tmp = (y / (x + 1.0d0)) * (z / t_1)
    else if (t_2 <= 1d-191) then
        tmp = (x + ((y - (x / z)) / t)) / (x + 1.0d0)
    else if (t_2 <= 5d+278) then
        tmp = t_2
    else
        tmp = ((y / (t * (x + 1.0d0))) + (x / (x + 1.0d0))) - (x / ((z * t) * (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5000000000000.0) {
		tmp = (y / (x + 1.0)) * (z / t_1);
	} else if (t_2 <= 1e-191) {
		tmp = (x + ((y - (x / z)) / t)) / (x + 1.0);
	} else if (t_2 <= 5e+278) {
		tmp = t_2;
	} else {
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / ((z * t) * (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)

↓

def code(x, y, z, t):
	t_1 = (z * t) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_2 <= -5000000000000.0:
		tmp = (y / (x + 1.0)) * (z / t_1)
	elif t_2 <= 1e-191:
		tmp = (x + ((y - (x / z)) / t)) / (x + 1.0)
	elif t_2 <= 5e+278:
		tmp = t_2
	else:
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / ((z * t) * (x + 1.0)))
	return tmp

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -5000000000000.0)
		tmp = Float64(Float64(y / Float64(x + 1.0)) * Float64(z / t_1));
	elseif (t_2 <= 1e-191)
		tmp = Float64(Float64(x + Float64(Float64(y - Float64(x / z)) / t)) / Float64(x + 1.0));
	elseif (t_2 <= 5e+278)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y / Float64(t * Float64(x + 1.0))) + Float64(x / Float64(x + 1.0))) - Float64(x / Float64(Float64(z * t) * Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = (z * t) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -5000000000000.0)
		tmp = (y / (x + 1.0)) * (z / t_1);
	elseif (t_2 <= 1e-191)
		tmp = (x + ((y - (x / z)) / t)) / (x + 1.0);
	elseif (t_2 <= 5e+278)
		tmp = t_2;
	else
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / ((z * t) * (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5000000000000.0], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-191], N[(N[(x + N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+278], t$95$2, N[(N[(N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(z * t), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	7.5
Target	0.3
Herbie	1.8

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < -5e12

   1. Initial program 18.7

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Simplified 18.7

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
      Proof
      (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 z t) x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 t z)) x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around inf 18.7

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

   4. Simplified 5.4

      \[\leadsto \color{blue}{\frac{y}{x + 1} \cdot \frac{z}{t \cdot z - x}} \]
      Proof
      (*.f64 (/.f64 y (+.f64 x 1)) (/.f64 z (-.f64 (*.f64 t z) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y (Rewrite<= +-commutative_binary64 (+.f64 1 x))) (/.f64 z (-.f64 (*.f64 t z) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y z) (*.f64 (+.f64 1 x) (-.f64 (*.f64 t z) x)))): 46 points increase in error, 24 points decrease in error
      (/.f64 (*.f64 y z) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 t z) x) (+.f64 1 x)))): 0 points increase in error, 0 points decrease in error

   if -5e12 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 1e-191

   1. Initial program 3.9

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Simplified 3.9

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
      Proof
      (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 z t) x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 t z)) x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around -inf 1.5

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

   if 1e-191 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 5.00000000000000029e278

   1. Initial program 0.1

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
      Proof
      (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 z t) x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 t z)) x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error

   if 5.00000000000000029e278 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 60.6

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Simplified 60.6

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
      Proof
      (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 z t) x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 t z)) x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around inf 10.9

      \[\leadsto \color{blue}{\left(\frac{x}{1 + x} + \frac{y}{\left(1 + x\right) \cdot t}\right) - \frac{x}{\left(1 + x\right) \cdot \left(t \cdot z\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -5000000000000:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{z \cdot t - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{-191}:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+278}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{t \cdot \left(x + 1\right)} + \frac{x}{x + 1}\right) - \frac{x}{\left(z \cdot t\right) \cdot \left(x + 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.8
Cost	4556

\[\begin{array}{l} t_1 := x + \frac{y}{t}\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t_2}}{x + 1}\\ \mathbf{if}\;t_3 \leq -5000000000000:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{t_2}\\ \mathbf{elif}\;t_3 \leq 10^{-191}:\\ \;\;\;\;\frac{t_1 - \frac{x}{z \cdot t}}{x + 1}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+278}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{x + 1}\\ \end{array} \]

Alternative 2
Error	1.8
Cost	4556

\[\begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t_1}}{x + 1}\\ \mathbf{if}\;t_2 \leq -5000000000000:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{t_1}\\ \mathbf{elif}\;t_2 \leq 10^{-191}:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{x + 1}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+278}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]

Alternative 3
Error	14.3
Cost	1232

\[\begin{array}{l} t_1 := \frac{x - \frac{y}{\frac{x}{z}}}{x + 1}\\ t_2 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t \leq -2.0096535261717547 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.65750662961582 \cdot 10^{-140}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \mathbf{elif}\;t \leq 2.6274856072389656 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	11.3
Cost	1228

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t \leq -2.1067046578638508 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.3432509834992011 \cdot 10^{-150}:\\ \;\;\;\;\frac{x - \frac{x}{z \cdot t - x}}{x + 1}\\ \mathbf{elif}\;t \leq 2.6274856072389656 \cdot 10^{-10}:\\ \;\;\;\;\frac{x + \left(1 - \frac{y}{\frac{x}{z}}\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	22.8
Cost	1112

\[\begin{array}{l} t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;t \leq -2.494843498104904 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-279}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-228}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \mathbf{elif}\;t \leq 2.6274856072389656 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 30110.42145131298:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;t \leq 1.5556262245297528 \cdot 10^{+43}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	22.9
Cost	1108

\[\begin{array}{l} t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;t \leq -2.494843498104904 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-279}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 10^{-214}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \mathbf{elif}\;t \leq 2.6274856072389656 \cdot 10^{-10}:\\ \;\;\;\;1 - \frac{z}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;t \leq 30110.42145131298:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	11.8
Cost	1096

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t \leq -2.0096535261717547 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.6274856072389656 \cdot 10^{-10}:\\ \;\;\;\;\frac{x + \left(1 - \frac{y \cdot z}{x}\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	10.7
Cost	1096

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t \leq -2.0096535261717547 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.6274856072389656 \cdot 10^{-10}:\\ \;\;\;\;\frac{x + \left(1 - \frac{y}{\frac{x}{z}}\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	14.0
Cost	972

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t \leq -2.0096535261717547 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-302}:\\ \;\;\;\;1 - \frac{z}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;t \leq 4.899553875339569 \cdot 10^{-101}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	20.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3306986714421017 \cdot 10^{-68}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.7703782424888698 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 11
Error	20.5
Cost	460

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3306986714421017 \cdot 10^{-68}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.7703782424888698 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 1.765357389481764 \cdot 10^{-60}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	26.4
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3306986714421017 \cdot 10^{-68}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.765357389481764 \cdot 10^{-60}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	28.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022294 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))