Development.Shake.Progress:decay from shake-0.15.5

Average Error: 23.3 → 4.0

Time: 1.2min

Precision: binary64

Cost: 28628

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot \left(b - y\right)\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{x}{1 - z}\\ t_4 := z \cdot \left(t - a\right)\\ t_5 := x \cdot \frac{y}{b - y}\\ t_6 := y + t_1\\ t_7 := \frac{t_4 + x \cdot y}{t_6}\\ t_8 := {\left(b - y\right)}^{2}\\ \mathbf{if}\;t_7 \leq -\infty:\\ \;\;\;\;t_3 + \frac{\frac{t}{y}}{-1 + \frac{1}{z}}\\ \mathbf{elif}\;t_7 \leq -2 \cdot 10^{-278}:\\ \;\;\;\;\frac{t_4}{t_6} + \frac{x \cdot y}{t_6}\\ \mathbf{elif}\;t_7 \leq 0:\\ \;\;\;\;\frac{t_5 + y \cdot \frac{a - t}{t_8}}{z} + t_2\\ \mathbf{elif}\;t_7 \leq 10^{+299}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;t_7 \leq \infty:\\ \;\;\;\;t_3 + \frac{a - t}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \frac{x}{b - y}, \frac{y}{z \cdot t_1} \cdot \left(y \cdot \frac{t - a}{t_8} - t_5\right)\right) + \left(t_2 + \frac{y}{t_8} \cdot \frac{a - t}{z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (- b y)))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (/ x (- 1.0 z)))
        (t_4 (* z (- t a)))
        (t_5 (* x (/ y (- b y))))
        (t_6 (+ y t_1))
        (t_7 (/ (+ t_4 (* x y)) t_6))
        (t_8 (pow (- b y) 2.0)))
   (if (<= t_7 (- INFINITY))
     (+ t_3 (/ (/ t y) (+ -1.0 (/ 1.0 z))))
     (if (<= t_7 -2e-278)
       (+ (/ t_4 t_6) (/ (* x y) t_6))
       (if (<= t_7 0.0)
         (+ (/ (+ t_5 (* y (/ (- a t) t_8))) z) t_2)
         (if (<= t_7 1e+299)
           t_7
           (if (<= t_7 INFINITY)
             (+ t_3 (/ (- a t) y))
             (+
              (fma
               (/ y z)
               (/ x (- b y))
               (* (/ y (* z t_1)) (- (* y (/ (- t a) t_8)) t_5)))
              (+ t_2 (* (/ y t_8) (/ (- a t) z)))))))))))

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (b - y);
	double t_2 = (t - a) / (b - y);
	double t_3 = x / (1.0 - z);
	double t_4 = z * (t - a);
	double t_5 = x * (y / (b - y));
	double t_6 = y + t_1;
	double t_7 = (t_4 + (x * y)) / t_6;
	double t_8 = pow((b - y), 2.0);
	double tmp;
	if (t_7 <= -((double) INFINITY)) {
		tmp = t_3 + ((t / y) / (-1.0 + (1.0 / z)));
	} else if (t_7 <= -2e-278) {
		tmp = (t_4 / t_6) + ((x * y) / t_6);
	} else if (t_7 <= 0.0) {
		tmp = ((t_5 + (y * ((a - t) / t_8))) / z) + t_2;
	} else if (t_7 <= 1e+299) {
		tmp = t_7;
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = t_3 + ((a - t) / y);
	} else {
		tmp = fma((y / z), (x / (b - y)), ((y / (z * t_1)) * ((y * ((t - a) / t_8)) - t_5))) + (t_2 + ((y / t_8) * ((a - t) / z)));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(b - y))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = Float64(x / Float64(1.0 - z))
	t_4 = Float64(z * Float64(t - a))
	t_5 = Float64(x * Float64(y / Float64(b - y)))
	t_6 = Float64(y + t_1)
	t_7 = Float64(Float64(t_4 + Float64(x * y)) / t_6)
	t_8 = Float64(b - y) ^ 2.0
	tmp = 0.0
	if (t_7 <= Float64(-Inf))
		tmp = Float64(t_3 + Float64(Float64(t / y) / Float64(-1.0 + Float64(1.0 / z))));
	elseif (t_7 <= -2e-278)
		tmp = Float64(Float64(t_4 / t_6) + Float64(Float64(x * y) / t_6));
	elseif (t_7 <= 0.0)
		tmp = Float64(Float64(Float64(t_5 + Float64(y * Float64(Float64(a - t) / t_8))) / z) + t_2);
	elseif (t_7 <= 1e+299)
		tmp = t_7;
	elseif (t_7 <= Inf)
		tmp = Float64(t_3 + Float64(Float64(a - t) / y));
	else
		tmp = Float64(fma(Float64(y / z), Float64(x / Float64(b - y)), Float64(Float64(y / Float64(z * t_1)) * Float64(Float64(y * Float64(Float64(t - a) / t_8)) - t_5))) + Float64(t_2 + Float64(Float64(y / t_8) * Float64(Float64(a - t) / z))));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(x * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(y + t$95$1), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$4 + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision]}, Block[{t$95$8 = N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$7, (-Infinity)], N[(t$95$3 + N[(N[(t / y), $MachinePrecision] / N[(-1.0 + N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, -2e-278], N[(N[(t$95$4 / t$95$6), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 0.0], N[(N[(N[(t$95$5 + N[(y * N[(N[(a - t), $MachinePrecision] / t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$7, 1e+299], t$95$7, If[LessEqual[t$95$7, Infinity], N[(t$95$3 + N[(N[(a - t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] * N[(x / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(y / N[(z * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(y * N[(N[(t - a), $MachinePrecision] / t$95$8), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[(y / t$95$8), $MachinePrecision] * N[(N[(a - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Target

Original	23.3
Target	17.7
Herbie	4.0

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

Derivation

1. Split input into 6 regimes

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

   1. Initial program 64.0

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

   2. Taylor expanded in y around inf 38.2

      \[\leadsto \color{blue}{\left(\frac{\left(t - a\right) \cdot z}{\left(-1 \cdot z + 1\right) \cdot y} + \frac{x}{-1 \cdot z + 1}\right) - \frac{b \cdot \left(z \cdot x\right)}{{\left(-1 \cdot z + 1\right)}^{2} \cdot y}} \]

   3. Simplified 22.9

      \[\leadsto \color{blue}{\frac{x}{1 - z} + \frac{\frac{t - a}{\frac{1}{z} - 1} - \frac{x \cdot \left(z \cdot b\right)}{{\left(1 - z\right)}^{2}}}{y}} \]
      Proof
      (+.f64 (/.f64 x (-.f64 1 z)) (/.f64 (-.f64 (/.f64 (-.f64 t a) (-.f64 (/.f64 1 z) 1)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 z)))) (/.f64 (-.f64 (/.f64 (-.f64 t a) (-.f64 (/.f64 1 z) 1)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z)))) (/.f64 (-.f64 (/.f64 (-.f64 t a) (-.f64 (/.f64 1 z) 1)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 z) 1))) (/.f64 (-.f64 (/.f64 (-.f64 t a) (-.f64 (/.f64 1 z) 1)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (-.f64 t a) (-.f64 (/.f64 1 z) (Rewrite<= *-inverses_binary64 (/.f64 z z)))) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (-.f64 t a) (Rewrite<= div-sub_binary64 (/.f64 (-.f64 1 z) z))) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (-.f64 t a) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 z))) z)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (-.f64 t a) (/.f64 (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))) z)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (-.f64 t a) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 z) 1)) z)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1))) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 27 points increase in error, 7 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x z) b)) (pow.f64 (-.f64 1 z) 2))) y)): 7 points increase in error, 6 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 z x)) b) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 b (*.f64 z x))) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (*.f64 b (*.f64 z x)) (pow.f64 (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 z))) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (*.f64 b (*.f64 z x)) (pow.f64 (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (*.f64 b (*.f64 z x)) (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 z) 1)) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 b z) x)) (pow.f64 (+.f64 (*.f64 -1 z) 1) 2))) y)): 6 points increase in error, 7 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 z b)) x) (pow.f64 (+.f64 (*.f64 -1 z) 1) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 z (*.f64 b x))) (pow.f64 (+.f64 (*.f64 -1 z) 1) 2))) y)): 11 points increase in error, 3 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) y) (/.f64 (/.f64 (*.f64 z (*.f64 b x)) (pow.f64 (+.f64 (*.f64 -1 z) 1) 2)) y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (-.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y))) (/.f64 (/.f64 (*.f64 z (*.f64 b x)) (pow.f64 (+.f64 (*.f64 -1 z) 1) 2)) y))): 9 points increase in error, 5 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y)) (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 z (*.f64 b x)) (*.f64 (pow.f64 (+.f64 (*.f64 -1 z) 1) 2) y))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y))) (/.f64 (*.f64 z (*.f64 b x)) (*.f64 (pow.f64 (+.f64 (*.f64 -1 z) 1) 2) y)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y)) (/.f64 x (+.f64 (*.f64 -1 z) 1)))) (/.f64 (*.f64 z (*.f64 b x)) (*.f64 (pow.f64 (+.f64 (*.f64 -1 z) 1) 2) y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y)) (/.f64 x (+.f64 (*.f64 -1 z) 1))) (/.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 z b) x)) (*.f64 (pow.f64 (+.f64 (*.f64 -1 z) 1) 2) y))): 3 points increase in error, 11 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y)) (/.f64 x (+.f64 (*.f64 -1 z) 1))) (/.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 b z)) x) (*.f64 (pow.f64 (+.f64 (*.f64 -1 z) 1) 2) y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y)) (/.f64 x (+.f64 (*.f64 -1 z) 1))) (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 b (*.f64 z x))) (*.f64 (pow.f64 (+.f64 (*.f64 -1 z) 1) 2) y))): 7 points increase in error, 6 points decrease in error

   4. Taylor expanded in t around inf 26.2

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

   5. Simplified 26.2

      \[\leadsto \frac{x}{1 - z} + \color{blue}{\frac{\frac{t}{y}}{-1 + \frac{1}{z}}} \]
      Proof
      (/.f64 (/.f64 t y) (+.f64 -1 (/.f64 1 z))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 t y) (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 1 z) -1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 t y) (+.f64 (/.f64 1 z) (Rewrite<= metadata-eval (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 t y) (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 z) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 t (*.f64 y (-.f64 (/.f64 1 z) 1)))): 26 points increase in error, 23 points decrease in error

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.99999999999999988e-278

   1. Initial program 0.3

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

   2. Taylor expanded in x around inf 0.3

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

   if -1.99999999999999988e-278 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 44.8

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

   2. Taylor expanded in z around inf 19.0

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

   3. Simplified 4.8

      \[\leadsto \color{blue}{\frac{\frac{y}{b - y} \cdot x - \frac{t - a}{{\left(b - y\right)}^{2}} \cdot y}{z} + \frac{t - a}{b - y}} \]
      Proof
      (+.f64 (/.f64 (-.f64 (*.f64 (/.f64 y (-.f64 b y)) x) (*.f64 (/.f64 (-.f64 t a) (pow.f64 (-.f64 b y) 2)) y)) z) (/.f64 (-.f64 t a) (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 y (/.f64 (-.f64 b y) x))) (*.f64 (/.f64 (-.f64 t a) (pow.f64 (-.f64 b y) 2)) y)) z) (/.f64 (-.f64 t a) (-.f64 b y))): 10 points increase in error, 16 points decrease in error
      (+.f64 (/.f64 (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y x) (-.f64 b y))) (*.f64 (/.f64 (-.f64 t a) (pow.f64 (-.f64 b y) 2)) y)) z) (/.f64 (-.f64 t a) (-.f64 b y))): 38 points increase in error, 9 points decrease in error
      (+.f64 (/.f64 (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 t a) (/.f64 (pow.f64 (-.f64 b y) 2) y)))) z) (/.f64 (-.f64 t a) (-.f64 b y))): 6 points increase in error, 4 points decrease in error
      (+.f64 (/.f64 (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) z) (/.f64 (-.f64 t a) (-.f64 b y))): 37 points increase in error, 4 points decrease in error
      (+.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2))))) z) (/.f64 (-.f64 t a) (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 (Rewrite<= metadata-eval (*.f64 -1 -1)) (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) z) (/.f64 (-.f64 t a) (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 -1 (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))))) z) (/.f64 (-.f64 t a) (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 -1 (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 -1 (/.f64 (*.f64 y x) (-.f64 b y))) (*.f64 -1 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))))) z) (/.f64 (-.f64 t a) (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (/.f64 (*.f64 y x) (-.f64 b y))) (*.f64 -1 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) z))) (/.f64 (-.f64 t a) (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (/.f64 (*.f64 y x) (-.f64 b y))) (*.f64 -1 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) z)) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 t (-.f64 b y)) (/.f64 a (-.f64 b y))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (/.f64 (*.f64 y x) (-.f64 b y))) (*.f64 -1 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) z)) (/.f64 t (-.f64 b y))) (/.f64 a (-.f64 b y)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 t (-.f64 b y)) (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (/.f64 (*.f64 y x) (-.f64 b y))) (*.f64 -1 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) z)))) (/.f64 a (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 t (-.f64 b y)) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1 (-.f64 (*.f64 -1 (/.f64 (*.f64 y x) (-.f64 b y))) (*.f64 -1 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2))))) z))) (/.f64 a (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 t (-.f64 b y)) (/.f64 (*.f64 -1 (Rewrite=> distribute-lft-out--_binary64 (*.f64 -1 (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))))) z)) (/.f64 a (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 t (-.f64 b y)) (/.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 -1 -1) (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2))))) z)) (/.f64 a (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 t (-.f64 b y)) (/.f64 (*.f64 (Rewrite=> metadata-eval 1) (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) z)) (/.f64 a (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 t (-.f64 b y)) (/.f64 (Rewrite=> *-lft-identity_binary64 (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) z)) (/.f64 a (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 t (-.f64 b y)) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (/.f64 (*.f64 y x) (-.f64 b y)) z) (/.f64 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)) z)))) (/.f64 a (-.f64 b y))): 0 points increase in error, 1 points decrease in error
      (-.f64 (+.f64 (/.f64 t (-.f64 b y)) (-.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 y x) (*.f64 (-.f64 b y) z))) (/.f64 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)) z))) (/.f64 a (-.f64 b y))): 13 points increase in error, 3 points decrease in error
      (-.f64 (+.f64 (/.f64 t (-.f64 b y)) (-.f64 (/.f64 (*.f64 y x) (*.f64 (-.f64 b y) z)) (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (-.f64 t a) y) (*.f64 (pow.f64 (-.f64 b y) 2) z))))) (/.f64 a (-.f64 b y))): 4 points increase in error, 3 points decrease in error
      (-.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 t (-.f64 b y)) (/.f64 (*.f64 y x) (*.f64 (-.f64 b y) z))) (/.f64 (*.f64 (-.f64 t a) y) (*.f64 (pow.f64 (-.f64 b y) 2) z)))) (/.f64 a (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (+.f64 (/.f64 t (-.f64 b y)) (/.f64 (*.f64 y x) (Rewrite=> *-commutative_binary64 (*.f64 z (-.f64 b y))))) (/.f64 (*.f64 (-.f64 t a) y) (*.f64 (pow.f64 (-.f64 b y) 2) z))) (/.f64 a (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 y x) (*.f64 z (-.f64 b y))) (/.f64 t (-.f64 b y)))) (/.f64 (*.f64 (-.f64 t a) y) (*.f64 (pow.f64 (-.f64 b y) 2) z))) (/.f64 a (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (+.f64 (/.f64 (*.f64 y x) (*.f64 z (-.f64 b y))) (/.f64 t (-.f64 b y))) (/.f64 (*.f64 (-.f64 t a) y) (Rewrite<= *-commutative_binary64 (*.f64 z (pow.f64 (-.f64 b y) 2))))) (/.f64 a (-.f64 b y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--r+_binary64 (-.f64 (+.f64 (/.f64 (*.f64 y x) (*.f64 z (-.f64 b y))) (/.f64 t (-.f64 b y))) (+.f64 (/.f64 (*.f64 (-.f64 t a) y) (*.f64 z (pow.f64 (-.f64 b y) 2))) (/.f64 a (-.f64 b y))))): 0 points increase in error, 0 points decrease in error

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.0000000000000001e299

   1. Initial program 0.3

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

   if 1.0000000000000001e299 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 62.2

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

   2. Taylor expanded in y around inf 37.0

      \[\leadsto \color{blue}{\left(\frac{\left(t - a\right) \cdot z}{\left(-1 \cdot z + 1\right) \cdot y} + \frac{x}{-1 \cdot z + 1}\right) - \frac{b \cdot \left(z \cdot x\right)}{{\left(-1 \cdot z + 1\right)}^{2} \cdot y}} \]

   3. Simplified 21.0

      \[\leadsto \color{blue}{\frac{x}{1 - z} + \frac{\frac{t - a}{\frac{1}{z} - 1} - \frac{x \cdot \left(z \cdot b\right)}{{\left(1 - z\right)}^{2}}}{y}} \]
      Proof
      (+.f64 (/.f64 x (-.f64 1 z)) (/.f64 (-.f64 (/.f64 (-.f64 t a) (-.f64 (/.f64 1 z) 1)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 z)))) (/.f64 (-.f64 (/.f64 (-.f64 t a) (-.f64 (/.f64 1 z) 1)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z)))) (/.f64 (-.f64 (/.f64 (-.f64 t a) (-.f64 (/.f64 1 z) 1)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 z) 1))) (/.f64 (-.f64 (/.f64 (-.f64 t a) (-.f64 (/.f64 1 z) 1)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (-.f64 t a) (-.f64 (/.f64 1 z) (Rewrite<= *-inverses_binary64 (/.f64 z z)))) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (-.f64 t a) (Rewrite<= div-sub_binary64 (/.f64 (-.f64 1 z) z))) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (-.f64 t a) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 z))) z)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (-.f64 t a) (/.f64 (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))) z)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (-.f64 t a) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 z) 1)) z)) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1))) (/.f64 (*.f64 x (*.f64 z b)) (pow.f64 (-.f64 1 z) 2))) y)): 27 points increase in error, 7 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x z) b)) (pow.f64 (-.f64 1 z) 2))) y)): 7 points increase in error, 6 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 z x)) b) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 b (*.f64 z x))) (pow.f64 (-.f64 1 z) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (*.f64 b (*.f64 z x)) (pow.f64 (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 z))) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (*.f64 b (*.f64 z x)) (pow.f64 (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (*.f64 b (*.f64 z x)) (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 z) 1)) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 b z) x)) (pow.f64 (+.f64 (*.f64 -1 z) 1) 2))) y)): 6 points increase in error, 7 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 z b)) x) (pow.f64 (+.f64 (*.f64 -1 z) 1) 2))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 z (*.f64 b x))) (pow.f64 (+.f64 (*.f64 -1 z) 1) 2))) y)): 11 points increase in error, 3 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (/.f64 (*.f64 (-.f64 t a) z) (+.f64 (*.f64 -1 z) 1)) y) (/.f64 (/.f64 (*.f64 z (*.f64 b x)) (pow.f64 (+.f64 (*.f64 -1 z) 1) 2)) y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (-.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y))) (/.f64 (/.f64 (*.f64 z (*.f64 b x)) (pow.f64 (+.f64 (*.f64 -1 z) 1) 2)) y))): 9 points increase in error, 5 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (-.f64 (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y)) (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 z (*.f64 b x)) (*.f64 (pow.f64 (+.f64 (*.f64 -1 z) 1) 2) y))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 x (+.f64 (*.f64 -1 z) 1)) (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y))) (/.f64 (*.f64 z (*.f64 b x)) (*.f64 (pow.f64 (+.f64 (*.f64 -1 z) 1) 2) y)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y)) (/.f64 x (+.f64 (*.f64 -1 z) 1)))) (/.f64 (*.f64 z (*.f64 b x)) (*.f64 (pow.f64 (+.f64 (*.f64 -1 z) 1) 2) y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y)) (/.f64 x (+.f64 (*.f64 -1 z) 1))) (/.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 z b) x)) (*.f64 (pow.f64 (+.f64 (*.f64 -1 z) 1) 2) y))): 3 points increase in error, 11 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y)) (/.f64 x (+.f64 (*.f64 -1 z) 1))) (/.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 b z)) x) (*.f64 (pow.f64 (+.f64 (*.f64 -1 z) 1) 2) y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (-.f64 t a) z) (*.f64 (+.f64 (*.f64 -1 z) 1) y)) (/.f64 x (+.f64 (*.f64 -1 z) 1))) (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 b (*.f64 z x))) (*.f64 (pow.f64 (+.f64 (*.f64 -1 z) 1) 2) y))): 7 points increase in error, 6 points decrease in error

   4. Taylor expanded in z around inf 17.1

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

   5. Simplified 17.1

      \[\leadsto \frac{x}{1 - z} + \color{blue}{\frac{a - t}{y}} \]
      Proof
      (/.f64 (-.f64 a t) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 a (neg.f64 t))) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 a (Rewrite<= mul-1-neg_binary64 (*.f64 -1 t))) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 t) a)) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 -1 t) (Rewrite<= *-lft-identity_binary64 (*.f64 1 a))) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 -1 t) (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) a)) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 -1 t) (*.f64 -1 a))) y): 0 points increase in error, 0 points decrease in error

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

   1. Initial program 64.0

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

   2. Taylor expanded in z around inf 39.6

      \[\leadsto \color{blue}{\left(\frac{y \cdot x}{\left(b - y\right) \cdot z} + \left(-1 \cdot \frac{y \cdot \left(\frac{y \cdot x}{b - y} - \frac{\left(t - a\right) \cdot y}{{\left(b - y\right)}^{2}}\right)}{\left(b - y\right) \cdot {z}^{2}} + \frac{t}{b - y}\right)\right) - \left(\frac{a}{b - y} + \frac{\left(t - a\right) \cdot y}{{\left(b - y\right)}^{2} \cdot z}\right)} \]

   3. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, \frac{x}{b - y}, \frac{y}{z \cdot \left(z \cdot \left(b - y\right)\right)} \cdot \left(\frac{t - a}{{\left(b - y\right)}^{2}} \cdot y - \frac{y}{b - y} \cdot x\right)\right) + \left(\frac{t - a}{b - y} - \frac{t - a}{z} \cdot \frac{y}{{\left(b - y\right)}^{2}}\right)} \]
      Proof
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 z (*.f64 z (-.f64 b y)))) (-.f64 (*.f64 (/.f64 (-.f64 t a) (pow.f64 (-.f64 b y) 2)) y) (*.f64 (/.f64 y (-.f64 b y)) x)))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z z) (-.f64 b y)))) (-.f64 (*.f64 (/.f64 (-.f64 t a) (pow.f64 (-.f64 b y) 2)) y) (*.f64 (/.f64 y (-.f64 b y)) x)))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 5 points increase in error, 2 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) (-.f64 b y))) (-.f64 (*.f64 (/.f64 (-.f64 t a) (pow.f64 (-.f64 b y) 2)) y) (*.f64 (/.f64 y (-.f64 b y)) x)))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 b y) (pow.f64 z 2)))) (-.f64 (*.f64 (/.f64 (-.f64 t a) (pow.f64 (-.f64 b y) 2)) y) (*.f64 (/.f64 y (-.f64 b y)) x)))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 (-.f64 b y) (pow.f64 z 2))) (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 t a) (/.f64 (pow.f64 (-.f64 b y) 2) y))) (*.f64 (/.f64 y (-.f64 b y)) x)))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 1 points increase in error, 2 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 (-.f64 b y) (pow.f64 z 2))) (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2))) (*.f64 (/.f64 y (-.f64 b y)) x)))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 29 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 (-.f64 b y) (pow.f64 z 2))) (-.f64 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)) (Rewrite<= associate-/r/_binary64 (/.f64 y (/.f64 (-.f64 b y) x)))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 2 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 (-.f64 b y) (pow.f64 z 2))) (-.f64 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y x) (-.f64 b y)))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 15 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 (-.f64 b y) (pow.f64 z 2))) (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)) (neg.f64 (/.f64 (*.f64 y x) (-.f64 b y))))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 (-.f64 b y) (pow.f64 z 2))) (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2))))) (neg.f64 (/.f64 (*.f64 y x) (-.f64 b y)))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 (-.f64 b y) (pow.f64 z 2))) (+.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2))))) (neg.f64 (/.f64 (*.f64 y x) (-.f64 b y)))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 (-.f64 b y) (pow.f64 z 2))) (Rewrite=> distribute-neg-out_binary64 (neg.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2))) (/.f64 (*.f64 y x) (-.f64 b y))))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 (-.f64 b y) (pow.f64 z 2))) (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (*.f64 -1 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 (-.f64 b y) (pow.f64 z 2))) (neg.f64 (+.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (Rewrite=> mul-1-neg_binary64 (neg.f64 (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (*.f64 (/.f64 y (*.f64 (-.f64 b y) (pow.f64 z 2))) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2))))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 y (*.f64 (-.f64 b y) (pow.f64 z 2))) (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2))))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 y (/.f64 (*.f64 (-.f64 b y) (pow.f64 z 2)) (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 5 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (neg.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) (*.f64 (-.f64 b y) (pow.f64 z 2)))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 5 points increase in error, 1 points decrease in error
      (+.f64 (fma.f64 (/.f64 y z) (/.f64 x (-.f64 b y)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 y (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) (*.f64 (-.f64 b y) (pow.f64 z 2)))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 y z) (/.f64 x (-.f64 b y))) (*.f64 -1 (/.f64 (*.f64 y (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) (*.f64 (-.f64 b y) (pow.f64 z 2)))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 1 points decrease in error
      (+.f64 (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y x) (*.f64 z (-.f64 b y)))) (*.f64 -1 (/.f64 (*.f64 y (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) (*.f64 (-.f64 b y) (pow.f64 z 2))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 8 points increase in error, 9 points decrease in error
      (+.f64 (+.f64 (/.f64 (*.f64 y x) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 b y) z))) (*.f64 -1 (/.f64 (*.f64 y (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) (*.f64 (-.f64 b y) (pow.f64 z 2))))) (-.f64 (/.f64 (-.f64 t a) (-.f64 b y)) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 (*.f64 y x) (*.f64 (-.f64 b y) z)) (*.f64 -1 (/.f64 (*.f64 y (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) (*.f64 (-.f64 b y) (pow.f64 z 2))))) (-.f64 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 t (-.f64 b y)) (/.f64 a (-.f64 b y)))) (*.f64 (/.f64 (-.f64 t a) z) (/.f64 y (pow.f64 (-.f64 b y) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 (*.f64 y x) (*.f64 (-.f64 b y) z)) (*.f64 -1 (/.f64 (*.f64 y (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) (*.f64 (-.f64 b y) (pow.f64 z 2))))) (-.f64 (-.f64 (/.f64 t (-.f64 b y)) (/.f64 a (-.f64 b y))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 t a) y) (*.f64 z (pow.f64 (-.f64 b y) 2)))))): 0 points increase in error, 5 points decrease in error
      (+.f64 (+.f64 (/.f64 (*.f64 y x) (*.f64 (-.f64 b y) z)) (*.f64 -1 (/.f64 (*.f64 y (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) (*.f64 (-.f64 b y) (pow.f64 z 2))))) (-.f64 (-.f64 (/.f64 t (-.f64 b y)) (/.f64 a (-.f64 b y))) (/.f64 (*.f64 (-.f64 t a) y) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 (-.f64 b y) 2) z))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 (*.f64 y x) (*.f64 (-.f64 b y) z)) (*.f64 -1 (/.f64 (*.f64 y (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) (*.f64 (-.f64 b y) (pow.f64 z 2))))) (Rewrite<= associate--r+_binary64 (-.f64 (/.f64 t (-.f64 b y)) (+.f64 (/.f64 a (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (*.f64 (pow.f64 (-.f64 b y) 2) z)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (+.f64 (/.f64 (*.f64 y x) (*.f64 (-.f64 b y) z)) (*.f64 -1 (/.f64 (*.f64 y (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) (*.f64 (-.f64 b y) (pow.f64 z 2))))) (/.f64 t (-.f64 b y))) (+.f64 (/.f64 a (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (*.f64 (pow.f64 (-.f64 b y) 2) z))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (/.f64 (*.f64 y x) (*.f64 (-.f64 b y) z)) (+.f64 (*.f64 -1 (/.f64 (*.f64 y (-.f64 (/.f64 (*.f64 y x) (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (pow.f64 (-.f64 b y) 2)))) (*.f64 (-.f64 b y) (pow.f64 z 2)))) (/.f64 t (-.f64 b y))))) (+.f64 (/.f64 a (-.f64 b y)) (/.f64 (*.f64 (-.f64 t a) y) (*.f64 (pow.f64 (-.f64 b y) 2) z)))): 0 points increase in error, 0 points decrease in error
3. Recombined 6 regimes into one program.

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\frac{x}{1 - z} + \frac{\frac{t}{y}}{-1 + \frac{1}{z}}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -2 \cdot 10^{-278}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} + \frac{x \cdot y}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{x \cdot \frac{y}{b - y} + y \cdot \frac{a - t}{{\left(b - y\right)}^{2}}}{z} + \frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 10^{+299}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\frac{x}{1 - z} + \frac{a - t}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \frac{x}{b - y}, \frac{y}{z \cdot \left(z \cdot \left(b - y\right)\right)} \cdot \left(y \cdot \frac{t - a}{{\left(b - y\right)}^{2}} - x \cdot \frac{y}{b - y}\right)\right) + \left(\frac{t - a}{b - y} + \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{a - t}{z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	4.0
Cost	13972

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{x}{1 - z}\\ t_3 := z \cdot \left(t - a\right)\\ t_4 := \frac{t_3 + x \cdot y}{t_1}\\ t_5 := \frac{x \cdot \frac{y}{b - y} + y \cdot \frac{a - t}{{\left(b - y\right)}^{2}}}{z} + \frac{t - a}{b - y}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;t_2 + \frac{\frac{t}{y}}{-1 + \frac{1}{z}}\\ \mathbf{elif}\;t_4 \leq -2 \cdot 10^{-278}:\\ \;\;\;\;\frac{t_3}{t_1} + \frac{x \cdot y}{t_1}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_4 \leq 10^{+299}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_2 + \frac{a - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 2
Error	7.1
Cost	6484

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ t_3 := \frac{x}{1 - z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_3 + \frac{\frac{t}{y}}{-1 + \frac{1}{z}}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-278}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{+299}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_3 + \frac{a - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	7.1
Cost	6484

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{x}{1 - z}\\ t_3 := z \cdot \left(t - a\right)\\ t_4 := \frac{t_3 + x \cdot y}{t_1}\\ t_5 := \frac{t - a}{b - y}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;t_2 + \frac{\frac{t}{y}}{-1 + \frac{1}{z}}\\ \mathbf{elif}\;t_4 \leq -2 \cdot 10^{-278}:\\ \;\;\;\;\frac{t_3}{t_1} + \frac{x \cdot y}{t_1}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_4 \leq 10^{+299}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_2 + \frac{a - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 4
Error	23.0
Cost	2676

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{t + \left(x \cdot \frac{y}{z} - a\right)}{b}\\ t_3 := \frac{t - a}{b - y}\\ t_4 := \frac{x}{1 - z} + \frac{\frac{t}{y}}{-1 + \frac{1}{z}}\\ t_5 := \frac{x \cdot y - z \cdot a}{t_1}\\ t_6 := z \cdot \left(t - a\right)\\ \mathbf{if}\;y \leq -1.6381747951157617 \cdot 10^{+35}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -22175.387832015713:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -2.182119900928099 \cdot 10^{-36}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -3.4784344918404805 \cdot 10^{-118}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-178}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;y \leq 10^{-162}:\\ \;\;\;\;\frac{\left(t + \frac{x \cdot y}{z}\right) - a}{b}\\ \mathbf{elif}\;y \leq 10^{-140}:\\ \;\;\;\;\frac{t_6}{t_1}\\ \mathbf{elif}\;y \leq 1.64039077451835 \cdot 10^{-109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.9210555687658528 \cdot 10^{-81}:\\ \;\;\;\;\frac{t - a}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.9195948794882283 \cdot 10^{-38}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 144399026.99696785:\\ \;\;\;\;\frac{t_6 + x \cdot y}{y}\\ \mathbf{elif}\;y \leq 3.400406445810975 \cdot 10^{+54}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;y \leq 5.924051927507992 \cdot 10^{+68}:\\ \;\;\;\;\frac{a - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 5
Error	22.5
Cost	2676

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := -1 + \frac{1}{z}\\ t_3 := \frac{t + \left(x \cdot \frac{y}{z} - a\right)}{b}\\ t_4 := \frac{t - a}{b - y}\\ t_5 := \frac{x}{1 - z}\\ t_6 := t_5 + \frac{\frac{t}{y}}{t_2}\\ t_7 := \frac{x \cdot y - z \cdot a}{t_1}\\ \mathbf{if}\;y \leq -1.6381747951157617 \cdot 10^{+35}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y \leq -22175.387832015713:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -2.182119900928099 \cdot 10^{-36}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y \leq -3.4784344918404805 \cdot 10^{-118}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-178}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;y \leq 10^{-162}:\\ \;\;\;\;\frac{\left(t + \frac{x \cdot y}{z}\right) - a}{b}\\ \mathbf{elif}\;y \leq 10^{-140}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{t_1}\\ \mathbf{elif}\;y \leq 1.64039077451835 \cdot 10^{-109}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 1.9210555687658528 \cdot 10^{-81}:\\ \;\;\;\;\frac{t - a}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.9195948794882283 \cdot 10^{-38}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 4.532956432875503 \cdot 10^{+50}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;y \leq 1.0402046530572708 \cdot 10^{+83}:\\ \;\;\;\;t_5 + \frac{a - t}{y}\\ \mathbf{elif}\;y \leq 7.330868067330012 \cdot 10^{+155}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;t_5 - \frac{a}{y \cdot t_2}\\ \end{array} \]

Alternative 6
Error	22.9
Cost	2420

\[\begin{array}{l} t_1 := \frac{z \cdot \left(t - a\right) + x \cdot y}{y}\\ t_2 := y + z \cdot \left(b - y\right)\\ t_3 := \frac{t - a}{b - y}\\ t_4 := \frac{x}{1 - z}\\ \mathbf{if}\;z \leq -9.164738037909063 \cdot 10^{+188}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -6.1248293140821585 \cdot 10^{+158}:\\ \;\;\;\;\frac{y}{\frac{b - y}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq -2.673799948175928 \cdot 10^{+25}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -4.439444253216878 \cdot 10^{-44}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq -7.691225292828483 \cdot 10^{-61}:\\ \;\;\;\;\frac{z \cdot t}{t_2}\\ \mathbf{elif}\;z \leq -1.7605152714842728 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y}{t_2}\\ \mathbf{elif}\;z \leq -1.2716360930707328 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.729839584454672 \cdot 10^{-115}:\\ \;\;\;\;x - z \cdot \frac{a}{y}\\ \mathbf{elif}\;z \leq 1.2941242290313986 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.569618547490513 \cdot 10^{-47}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;z \leq 8.825541203631617 \cdot 10^{-44}:\\ \;\;\;\;a \cdot \left(-\frac{z}{y}\right)\\ \mathbf{elif}\;z \leq 7978868.683290316:\\ \;\;\;\;\frac{t + \left(x \cdot \frac{y}{z} - a\right)}{b}\\ \mathbf{elif}\;z \leq 2.202763629591919 \cdot 10^{+48}:\\ \;\;\;\;t_4 + \frac{a - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	23.7
Cost	2412

\[\begin{array}{l} t_1 := \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_2 := \frac{t + \left(x \cdot \frac{y}{z} - a\right)}{b}\\ t_3 := \frac{t - a}{b - y}\\ t_4 := \frac{x}{1 - z} + \frac{\frac{t}{y}}{-1 + \frac{1}{z}}\\ \mathbf{if}\;y \leq -1.6381747951157617 \cdot 10^{+35}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -22175.387832015713:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -2.182119900928099 \cdot 10^{-36}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -3.4784344918404805 \cdot 10^{-118}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.55 \cdot 10^{-163}:\\ \;\;\;\;x - z \cdot \frac{a}{y}\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-162}:\\ \;\;\;\;\frac{\left(t + \frac{x \cdot y}{z}\right) - a}{b}\\ \mathbf{elif}\;y \leq 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.64039077451835 \cdot 10^{-109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.9210555687658528 \cdot 10^{-81}:\\ \;\;\;\;\frac{t - a}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 5.924051927507992 \cdot 10^{+68}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 8
Error	22.1
Cost	2020

\[\begin{array}{l} t_1 := z \cdot \left(t - a\right)\\ t_2 := y + z \cdot \left(b - y\right)\\ t_3 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9.164738037909063 \cdot 10^{+188}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -6.1248293140821585 \cdot 10^{+158}:\\ \;\;\;\;\frac{y}{\frac{b - y}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq -2.673799948175928 \cdot 10^{+25}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -4.439444253216878 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;z \leq -7.691225292828483 \cdot 10^{-61}:\\ \;\;\;\;\frac{z \cdot t}{t_2}\\ \mathbf{elif}\;z \leq -1.7605152714842728 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y}{t_2}\\ \mathbf{elif}\;z \leq -1.2716360930707328 \cdot 10^{-176}:\\ \;\;\;\;\frac{t_1 + x \cdot y}{y}\\ \mathbf{elif}\;z \leq 5.729839584454672 \cdot 10^{-115}:\\ \;\;\;\;x - z \cdot \frac{a}{y}\\ \mathbf{elif}\;z \leq 1.5776368865925943 \cdot 10^{-19}:\\ \;\;\;\;\frac{t_1}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	23.1
Cost	1892

\[\begin{array}{l} t_1 := \frac{t + \left(x \cdot \frac{y}{z} - a\right)}{b}\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{x}{1 - z}\\ \mathbf{if}\;z \leq -9.164738037909063 \cdot 10^{+188}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -6.1248293140821585 \cdot 10^{+158}:\\ \;\;\;\;\frac{y}{\frac{b - y}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq -2.673799948175928 \cdot 10^{+25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.439444253216878 \cdot 10^{-44}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.7605152714842728 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.974208734562379 \cdot 10^{-93}:\\ \;\;\;\;\frac{t - a}{\frac{y}{z}}\\ \mathbf{elif}\;z \leq 3.1643621269339064 \cdot 10^{-85}:\\ \;\;\;\;x - z \cdot \frac{a}{y}\\ \mathbf{elif}\;z \leq 7978868.683290316:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.202763629591919 \cdot 10^{+48}:\\ \;\;\;\;t_3 + \frac{a - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	22.7
Cost	1760

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{x}{1 - z}\\ \mathbf{if}\;z \leq -9.164738037909063 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.1248293140821585 \cdot 10^{+158}:\\ \;\;\;\;\frac{y}{\frac{b - y}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq -2.673799948175928 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.439444253216878 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -9.359232360229099 \cdot 10^{-49}:\\ \;\;\;\;\frac{z \cdot t}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;z \leq 3.1643621269339064 \cdot 10^{-85}:\\ \;\;\;\;x - z \cdot \frac{a}{y}\\ \mathbf{elif}\;z \leq 7978868.683290316:\\ \;\;\;\;\frac{t + \left(x \cdot \frac{y}{z} - a\right)}{b}\\ \mathbf{elif}\;z \leq 2.202763629591919 \cdot 10^{+48}:\\ \;\;\;\;t_2 + \frac{a - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	23.3
Cost	1372

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9.164738037909063 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.1248293140821585 \cdot 10^{+158}:\\ \;\;\;\;\frac{y}{\frac{b - y}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq -2.673799948175928 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.439444253216878 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;z \leq -2.3621952886931014 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.974208734562379 \cdot 10^{-93}:\\ \;\;\;\;\frac{t - a}{\frac{y}{z}}\\ \mathbf{elif}\;z \leq 3.1643621269339064 \cdot 10^{-85}:\\ \;\;\;\;x - z \cdot \frac{a}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	23.2
Cost	1372

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9.164738037909063 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.1248293140821585 \cdot 10^{+158}:\\ \;\;\;\;\frac{y}{\frac{b - y}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq -2.673799948175928 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.439444253216878 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;z \leq -1.7605152714842728 \cdot 10^{-71}:\\ \;\;\;\;\frac{t + \left(x \cdot \frac{y}{z} - a\right)}{b}\\ \mathbf{elif}\;z \leq -9.974208734562379 \cdot 10^{-93}:\\ \;\;\;\;\frac{t - a}{\frac{y}{z}}\\ \mathbf{elif}\;z \leq 3.1643621269339064 \cdot 10^{-85}:\\ \;\;\;\;x - z \cdot \frac{a}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	22.3
Cost	1108

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.673799948175928 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.439444253216878 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;z \leq -2.3621952886931014 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.974208734562379 \cdot 10^{-93}:\\ \;\;\;\;\frac{t - a}{\frac{y}{z}}\\ \mathbf{elif}\;z \leq 3.1643621269339064 \cdot 10^{-85}:\\ \;\;\;\;x - z \cdot \frac{a}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	35.8
Cost	848

\[\begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -9.164738037909063 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.1248293140821585 \cdot 10^{+158}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq -24.138764329864582:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2812846117610095 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	35.0
Cost	848

\[\begin{array}{l} t_1 := \frac{t - a}{b}\\ \mathbf{if}\;z \leq -1.947742986826229 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2812846117610095 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7978868.683290316:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.713046935232841 \cdot 10^{+244}:\\ \;\;\;\;\frac{a - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	30.6
Cost	716

\[\begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -1.836092380198332 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.046339022322917 \cdot 10^{+24}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;y \leq 7.197886278675535 \cdot 10^{+82}:\\ \;\;\;\;\frac{a - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	40.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -24.138764329864582:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 2.2812846117610095 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]

Alternative 18
Error	47.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

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

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