Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I

Average Error: 3.8 → 2.4

Time: 1.2min

Precision: binary64

Cost: 22468

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]

↓

\[\begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(c - b\right) \cdot \left(\left(a + 0.8333333333333334\right) + \frac{-2}{t \cdot 3}\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y + y \cdot \left(\left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right) \cdot \left(\left(\left(c \cdot c\right) \cdot -2\right) \cdot \left(\left(-0.8333333333333334 - \frac{-0.6666666666666666}{t}\right) - a\right) + 2 \cdot c\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/
  x
  (+
   x
   (*
    y
    (exp
     (*
      2.0
      (-
       (/ (* z (sqrt (+ t a))) t)
       (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))

↓

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- c b) (+ (+ a 0.8333333333333334) (/ -2.0 (* t 3.0)))))))
   (if (<= t_1 INFINITY)
     (/ x (+ x (* y (exp (* 2.0 t_1)))))
     (/
      x
      (+
       x
       (+
        y
        (*
         y
         (*
          (+ a (+ 0.8333333333333334 (/ -0.6666666666666666 t)))
          (+
           (*
            (* (* c c) -2.0)
            (- (- -0.8333333333333334 (/ -0.6666666666666666 t)) a))
           (* 2.0 c))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) + ((c - b) * ((a + 0.8333333333333334) + (-2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = x / (x + (y * exp((2.0 * t_1))));
	} else {
		tmp = x / (x + (y + (y * ((a + (0.8333333333333334 + (-0.6666666666666666 / t))) * ((((c * c) * -2.0) * ((-0.8333333333333334 - (-0.6666666666666666 / t)) - a)) + (2.0 * c))))));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return x / (x + (y * Math.exp((2.0 * (((z * Math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
}

↓

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * Math.sqrt((t + a))) / t) + ((c - b) * ((a + 0.8333333333333334) + (-2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x / (x + (y * Math.exp((2.0 * t_1))));
	} else {
		tmp = x / (x + (y + (y * ((a + (0.8333333333333334 + (-0.6666666666666666 / t))) * ((((c * c) * -2.0) * ((-0.8333333333333334 - (-0.6666666666666666 / t)) - a)) + (2.0 * c))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	return x / (x + (y * math.exp((2.0 * (((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))

↓

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) + ((c - b) * ((a + 0.8333333333333334) + (-2.0 / (t * 3.0))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = x / (x + (y * math.exp((2.0 * t_1))))
	else:
		tmp = x / (x + (y + (y * ((a + (0.8333333333333334 + (-0.6666666666666666 / t))) * ((((c * c) * -2.0) * ((-0.8333333333333334 - (-0.6666666666666666 / t)) - a)) + (2.0 * c))))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	return Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))))))
end

↓

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(c - b) * Float64(Float64(a + 0.8333333333333334) + Float64(-2.0 / Float64(t * 3.0)))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * t_1)))));
	else
		tmp = Float64(x / Float64(x + Float64(y + Float64(y * Float64(Float64(a + Float64(0.8333333333333334 + Float64(-0.6666666666666666 / t))) * Float64(Float64(Float64(Float64(c * c) * -2.0) * Float64(Float64(-0.8333333333333334 - Float64(-0.6666666666666666 / t)) - a)) + Float64(2.0 * c)))))));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
end

↓

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((z * sqrt((t + a))) / t) + ((c - b) * ((a + 0.8333333333333334) + (-2.0 / (t * 3.0))));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = x / (x + (y * exp((2.0 * t_1))));
	else
		tmp = x / (x + (y + (y * ((a + (0.8333333333333334 + (-0.6666666666666666 / t))) * ((((c * c) * -2.0) * ((-0.8333333333333334 - (-0.6666666666666666 / t)) - a)) + (2.0 * c))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(c - b), $MachinePrecision] * N[(N[(a + 0.8333333333333334), $MachinePrecision] + N[(-2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y + N[(y * N[(N[(a + N[(0.8333333333333334 + N[(-0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(c * c), $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[(-0.8333333333333334 - N[(-0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision] + N[(2.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	3.8
Target	3.0
Herbie	2.4

\[\begin{array}{l} \mathbf{if}\;t < -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t < 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 5 6)) (/.f64 2 (*.f64 t 3))))) < +inf.0

   1. Initial program 0.7

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]

   if +inf.0 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 5 6)) (/.f64 2 (*.f64 t 3)))))

   1. Initial program 64.0

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]

   2. Taylor expanded in c around inf 26.5

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - 0.6666666666666666 \cdot \frac{1}{t}\right)\right)}}} \]

   3. Simplified 26.5

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\frac{-0.6666666666666666}{t} - \left(-0.8333333333333334 - a\right)\right)\right)}}} \]
      Proof
      (*.f64 c (-.f64 (/.f64 -2/3 t) (-.f64 -5/6 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (-.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 2/3)) t) (-.f64 -5/6 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (-.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2/3 t))) (-.f64 -5/6 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (-.f64 (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2/3 1)) t)) (-.f64 -5/6 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (-.f64 (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2/3 (/.f64 1 t)))) (-.f64 -5/6 a))): 6 points increase in error, 9 points decrease in error
      (*.f64 c (-.f64 (neg.f64 (*.f64 2/3 (/.f64 1 t))) (-.f64 (Rewrite<= metadata-eval (neg.f64 5/6)) a))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (-.f64 (neg.f64 (*.f64 2/3 (/.f64 1 t))) (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 5/6) (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (-.f64 (neg.f64 (*.f64 2/3 (/.f64 1 t))) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 5/6 a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (*.f64 2/3 (/.f64 1 t))) (neg.f64 (neg.f64 (+.f64 5/6 a)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (*.f64 2/3 (/.f64 1 t)) (neg.f64 (+.f64 5/6 a)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 2/3 (/.f64 1 t)) (+.f64 5/6 a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (Rewrite=> neg-sub0_binary64 (-.f64 0 (-.f64 (*.f64 2/3 (/.f64 1 t)) (+.f64 5/6 a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (*.f64 2/3 (/.f64 1 t))) (+.f64 5/6 a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (*.f64 2/3 (/.f64 1 t)))) (+.f64 5/6 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 5/6 a) (neg.f64 (*.f64 2/3 (/.f64 1 t)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (Rewrite<= sub-neg_binary64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in c around 0 44.3

      \[\leadsto \frac{x}{x + \color{blue}{\left(2 \cdot \left({c}^{2} \cdot \left(y \cdot {\left(\left(0.8333333333333334 + a\right) - 0.6666666666666666 \cdot \frac{1}{t}\right)}^{2}\right)\right) + \left(y + 2 \cdot \left(c \cdot \left(y \cdot \left(\left(0.8333333333333334 + a\right) - 0.6666666666666666 \cdot \frac{1}{t}\right)\right)\right)\right)\right)}} \]

   5. Simplified 34.9

      \[\leadsto \frac{x}{x + \color{blue}{\left(y + y \cdot \left(\left(a + \left(0.8333333333333334 - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(\left(a + \left(0.8333333333333334 - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(2 \cdot \left(c \cdot c\right)\right) + c \cdot 2\right)\right)\right)}} \]
      Proof
      (+.f64 y (*.f64 y (*.f64 (+.f64 a (-.f64 5/6 (/.f64 2/3 t))) (+.f64 (*.f64 (+.f64 a (-.f64 5/6 (/.f64 2/3 t))) (*.f64 2 (*.f64 c c))) (*.f64 c 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (*.f64 (+.f64 a (-.f64 5/6 (/.f64 (Rewrite<= metadata-eval (*.f64 2/3 1)) t))) (+.f64 (*.f64 (+.f64 a (-.f64 5/6 (/.f64 2/3 t))) (*.f64 2 (*.f64 c c))) (*.f64 c 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (*.f64 (+.f64 a (-.f64 5/6 (Rewrite<= associate-*r/_binary64 (*.f64 2/3 (/.f64 1 t))))) (+.f64 (*.f64 (+.f64 a (-.f64 5/6 (/.f64 2/3 t))) (*.f64 2 (*.f64 c c))) (*.f64 c 2))))): 1 points increase in error, 2 points decrease in error
      (+.f64 y (*.f64 y (*.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 a 5/6) (*.f64 2/3 (/.f64 1 t)))) (+.f64 (*.f64 (+.f64 a (-.f64 5/6 (/.f64 2/3 t))) (*.f64 2 (*.f64 c c))) (*.f64 c 2))))): 1 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (*.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 5/6 a)) (*.f64 2/3 (/.f64 1 t))) (+.f64 (*.f64 (+.f64 a (-.f64 5/6 (/.f64 2/3 t))) (*.f64 2 (*.f64 c c))) (*.f64 c 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (*.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) (+.f64 (*.f64 (+.f64 a (-.f64 5/6 (/.f64 (Rewrite<= metadata-eval (*.f64 2/3 1)) t))) (*.f64 2 (*.f64 c c))) (*.f64 c 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (*.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) (+.f64 (*.f64 (+.f64 a (-.f64 5/6 (Rewrite<= associate-*r/_binary64 (*.f64 2/3 (/.f64 1 t))))) (*.f64 2 (*.f64 c c))) (*.f64 c 2))))): 0 points increase in error, 2 points decrease in error
      (+.f64 y (*.f64 y (*.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) (+.f64 (*.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 a 5/6) (*.f64 2/3 (/.f64 1 t)))) (*.f64 2 (*.f64 c c))) (*.f64 c 2))))): 1 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (*.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) (+.f64 (*.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 5/6 a)) (*.f64 2/3 (/.f64 1 t))) (*.f64 2 (*.f64 c c))) (*.f64 c 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (*.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) (+.f64 (*.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 c 2)))) (*.f64 c 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (*.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 (pow.f64 c 2)) (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))) (*.f64 c 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (*.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) (+.f64 (*.f64 (*.f64 2 (pow.f64 c 2)) (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t)))) (Rewrite<= *-commutative_binary64 (*.f64 2 c)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (*.f64 2 (pow.f64 c 2)) (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t)))) (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t)))) (*.f64 (*.f64 2 c) (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 2 (pow.f64 c 2)) (*.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t)))))) (*.f64 (*.f64 2 c) (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))))): 33 points increase in error, 1 points decrease in error
      (+.f64 y (*.f64 y (+.f64 (*.f64 (*.f64 2 (pow.f64 c 2)) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2))) (*.f64 (*.f64 2 c) (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 (pow.f64 c 2) (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2)))) (*.f64 (*.f64 2 c) (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (+.f64 (*.f64 2 (*.f64 (pow.f64 c 2) (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2))) (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 c (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 y (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 2 (*.f64 c (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))) (*.f64 2 (*.f64 (pow.f64 c 2) (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 2 (*.f64 c (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))) y) (*.f64 (*.f64 2 (*.f64 (pow.f64 c 2) (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2))) y)))): 1 points increase in error, 0 points decrease in error
      (+.f64 y (+.f64 (*.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 2 c) (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))) y) (*.f64 (*.f64 2 (*.f64 (pow.f64 c 2) (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 2 c) (*.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) y))) (*.f64 (*.f64 2 (*.f64 (pow.f64 c 2) (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2))) y))): 1 points increase in error, 0 points decrease in error
      (+.f64 y (+.f64 (*.f64 (*.f64 2 c) (Rewrite<= *-commutative_binary64 (*.f64 y (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t)))))) (*.f64 (*.f64 2 (*.f64 (pow.f64 c 2) (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 c (*.f64 y (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))))) (*.f64 (*.f64 2 (*.f64 (pow.f64 c 2) (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (+.f64 (*.f64 2 (*.f64 c (*.f64 y (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t)))))) (*.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 2 (pow.f64 c 2)) (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (+.f64 (*.f64 2 (*.f64 c (*.f64 y (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t)))))) (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 2 (pow.f64 c 2)) (*.f64 (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2) y))))): 5 points increase in error, 6 points decrease in error
      (+.f64 y (+.f64 (*.f64 2 (*.f64 c (*.f64 y (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t)))))) (*.f64 (*.f64 2 (pow.f64 c 2)) (Rewrite<= *-commutative_binary64 (*.f64 y (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (+.f64 (*.f64 2 (*.f64 c (*.f64 y (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t)))))) (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 (pow.f64 c 2) (*.f64 y (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 y (*.f64 2 (*.f64 c (*.f64 y (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))))) (*.f64 2 (*.f64 (pow.f64 c 2) (*.f64 y (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (*.f64 (pow.f64 c 2) (*.f64 y (pow.f64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))) 2)))) (+.f64 y (*.f64 2 (*.f64 c (*.f64 y (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))))))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(c - b\right) \cdot \left(\left(a + 0.8333333333333334\right) + \frac{-2}{t \cdot 3}\right) \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(c - b\right) \cdot \left(\left(a + 0.8333333333333334\right) + \frac{-2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y + y \cdot \left(\left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right) \cdot \left(\left(\left(c \cdot c\right) \cdot -2\right) \cdot \left(\left(-0.8333333333333334 - \frac{-0.6666666666666666}{t}\right) - a\right) + 2 \cdot c\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.1
Cost	33408

\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, \left(b - c\right) \cdot \left(-0.8333333333333334 + \left(\frac{0.6666666666666666}{t} - a\right)\right)\right)\right)}, x\right)} \]

Alternative 2
Error	15.2
Cost	14156

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{if}\;c \leq -37.0774975637518:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -3.0540325163868543 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) - \frac{-0.6666666666666666}{t}\right)\right)}}\\ \mathbf{elif}\;c \leq 8.024201136326416 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{t} \cdot \sqrt{t + a}\right)}}\\ \mathbf{elif}\;c \leq 3.799117047267948 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	17.2
Cost	14028

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{if}\;c \leq -37.0774975637518:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -7.184548193424615 \cdot 10^{-268}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) - \frac{-0.6666666666666666}{t}\right)\right)}}\\ \mathbf{elif}\;c \leq 8.005710306279865 \cdot 10^{-191}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(z \cdot \sqrt{\frac{1}{t}}\right)}}\\ \mathbf{elif}\;c \leq 8.024201136326416 \cdot 10^{-57}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{elif}\;c \leq 3.799117047267948 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	18.9
Cost	8924

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{if}\;z \leq -6.13556300681305 \cdot 10^{+242}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -5.96742305000556 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot \frac{a \cdot a + -0.6944444444444444}{a + -0.8333333333333334}\right)}}\\ \mathbf{elif}\;z \leq -2.917180352735839 \cdot 10^{-128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.08186013158283 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4748981287489401 \cdot 10^{-143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.6555488286876983 \cdot 10^{+238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.156339107233164 \cdot 10^{+291}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(\left(z \cdot \frac{y}{t}\right) \cdot \left(\sqrt{t + a} + \frac{t + a}{\frac{t}{z}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \end{array} \]

Alternative 5
Error	22.5
Cost	8156

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{\left(c - b\right) \cdot 1.6666666666666667}}\\ \mathbf{if}\;a \leq -4.2 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-207}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-303}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{elif}\;a \leq 3.512602306270874 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.8855297165014061 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.386997871455298 \cdot 10^{+203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7.903813921686266 \cdot 10^{+208}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	23.9
Cost	7764

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{\left(c - b\right) \cdot 1.6666666666666667}}\\ \mathbf{if}\;a \leq -4.2 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-303}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{elif}\;a \leq 3.512602306270874 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.8855297165014061 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	12.8
Cost	7752

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \mathbf{if}\;t \leq -5.7 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) - \frac{-0.6666666666666666}{t}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	13.7
Cost	7752

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) - \frac{-0.6666666666666666}{t}\right)\right)}}\\ \mathbf{if}\;b \leq -7.534964359182189 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.3439860152124304 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	13.6
Cost	7624

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \mathbf{if}\;t \leq -5.7 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{0.6666666666666666}{\frac{t}{b}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	32.4
Cost	3188

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot \left(1 + 2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + -0.6666666666666666 \cdot \frac{-1}{t}\right)\right)\right)}\\ t_2 := \frac{x}{x + y \cdot \left(1 + \frac{\left(a \cdot -2\right) \cdot \left(b \cdot b - c \cdot c\right)}{c + b}\right)}\\ \mathbf{if}\;z \leq -1.1088017331851014 \cdot 10^{+127}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -5.96742305000556 \cdot 10^{+54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.6173858079343105 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -4.158157480028268 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.63 \cdot 10^{-253}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 2.4285013130555147 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 8.106761548538017 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 4.18262877035384 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.222099550341434 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 3.6983572663462943 \cdot 10^{+176}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.4718231105540618 \cdot 10^{+186}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 3.3435009634678896 \cdot 10^{+186}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(2 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{elif}\;z \leq 1.307178790725156 \cdot 10^{+208}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	32.8
Cost	2928

\[\begin{array}{l} t_1 := \frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ t_2 := \left(a + 0.8333333333333334\right) + -0.6666666666666666 \cdot \frac{1}{t}\\ t_3 := \frac{x}{x + y \cdot \left(1 + 2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + -0.6666666666666666 \cdot \frac{-1}{t}\right)\right)\right)}\\ \mathbf{if}\;z \leq -7.234502649290478 \cdot 10^{+145}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -5.96742305000556 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(2 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{elif}\;z \leq -1.6173858079343105 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -4.158157480028268 \cdot 10^{-145}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.5230740374586051 \cdot 10^{-248}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 5.383940501490396 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.106761548538017 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 4.18262877035384 \cdot 10^{+22}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 4.222099550341434 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 3.359426801899943 \cdot 10^{+127}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(c \cdot \left(y \cdot t_2\right)\right)\right)}\\ \mathbf{elif}\;z \leq 3.6983572663462943 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4718231105540618 \cdot 10^{+186}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(c \cdot t_2\right)\right)}\\ \end{array} \]

Alternative 12
Error	33.9
Cost	2904

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y + y \cdot \left(\left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right) \cdot \left(\left(\left(c \cdot c\right) \cdot -2\right) \cdot \left(\left(-0.8333333333333334 - \frac{-0.6666666666666666}{t}\right) - a\right) + 2 \cdot c\right)\right)\right)}\\ \mathbf{if}\;a \leq -7.2 \cdot 10^{-151}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq -8.5 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(2 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.5308390563524665 \cdot 10^{-81}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 8.321363619954814 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.8855297165014061 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 7.066852181465908 \cdot 10^{+147}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + \frac{\left(a \cdot -2\right) \cdot \left(b \cdot b - c \cdot c\right)}{c + b}\right)}\\ \mathbf{elif}\;a \leq 1.4600600242799728 \cdot 10^{+205}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 5.037153313408607 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + -0.6666666666666666 \cdot \frac{-1}{t}\right)\right)\right)}\\ \mathbf{elif}\;a \leq 1.862835085644595 \cdot 10^{+291}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot \left(c - b\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	32.4
Cost	2676

\[\begin{array}{l} t_1 := \left(1 + \frac{x}{x + y}\right) + -1\\ t_2 := \frac{x}{x + y \cdot \left(1 + 2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + -0.6666666666666666 \cdot \frac{-1}{t}\right)\right)\right)}\\ \mathbf{if}\;z \leq -7.234502649290478 \cdot 10^{+145}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -5.96742305000556 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(2 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{elif}\;z \leq -1.6173858079343105 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -4.158157480028268 \cdot 10^{-145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.5230740374586051 \cdot 10^{-248}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 5.383940501490396 \cdot 10^{-216}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;z \leq 8.106761548538017 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 4.18262877035384 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.222099550341434 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 6.081964989481909 \cdot 10^{+164}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(c \cdot \left(\left(a + 0.8333333333333334\right) + -0.6666666666666666 \cdot \frac{1}{t}\right)\right)\right)}\\ \mathbf{elif}\;z \leq 3.6983572663462943 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4718231105540618 \cdot 10^{+186}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 1.1581984644981351 \cdot 10^{+249}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot \left(c - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	35.0
Cost	2400

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y + 2 \cdot \left(y \cdot \left(a \cdot c\right)\right)\right)}\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{-291}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + -1.3333333333333333 \cdot \frac{c}{t}\right)}\\ \mathbf{elif}\;a \leq 2.5308390563524665 \cdot 10^{-81}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.512602306270874 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.8855297165014061 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.4412770845070837 \cdot 10^{+180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.4600600242799728 \cdot 10^{+205}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 5.037153313408607 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + -0.6666666666666666 \cdot \frac{-1}{t}\right)\right)\right)}\\ \mathbf{elif}\;a \leq 1.862835085644595 \cdot 10^{+291}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot \left(c - b\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	34.1
Cost	2284

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y + -1.3333333333333333 \cdot \frac{y \cdot c}{t}\right)}\\ t_2 := \frac{x}{x - y \cdot \left(\left(a \cdot c\right) \cdot -2\right)}\\ t_3 := \left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{-184}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-243}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-255}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-247}:\\ \;\;\;\;-0.75 \cdot \frac{x \cdot t}{y \cdot c}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-162}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6720254189456461:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.3181218041125334 \cdot 10^{+90}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2.72804881464451 \cdot 10^{+104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.316759687313724 \cdot 10^{+147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.1305197942424994 \cdot 10^{+211}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	35.0
Cost	2148

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y - \left(y \cdot \left(a \cdot c\right)\right) \cdot -2\right)}\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{-291}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - \frac{c}{t} \cdot 1.3333333333333333\right)}\\ \mathbf{elif}\;a \leq 2.5308390563524665 \cdot 10^{-81}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.512602306270874 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.8855297165014061 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.4412770845070837 \cdot 10^{+180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.4600600242799728 \cdot 10^{+205}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 5.037153313408607 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - 2 \cdot \left(a \cdot b\right)\right)}\\ \mathbf{elif}\;a \leq 1.862835085644595 \cdot 10^{+291}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + -2 \cdot \left(a \cdot \left(b - c\right)\right)\right)}\\ \end{array} \]

Alternative 17
Error	32.2
Cost	2024

\[\begin{array}{l} t_1 := \left(1 + \frac{x}{x + y}\right) + -1\\ t_2 := \frac{x}{x - y \cdot \left(\left(a \cdot c\right) \cdot -2\right)}\\ \mathbf{if}\;z \leq -7.234502649290478 \cdot 10^{+145}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -5.96742305000556 \cdot 10^{+54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.3347359027682898 \cdot 10^{-48}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -4.158157480028268 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - 2 \cdot \left(a \cdot b\right)\right)}\\ \mathbf{elif}\;z \leq -1.0548145485180503 \cdot 10^{-305}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 5.383940501490396 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.7669998744186706 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 3.1565169628162163 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.222099550341434 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 1.7096619642486624 \cdot 10^{+253}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	32.5
Cost	2024

\[\begin{array}{l} t_1 := \left(1 + \frac{x}{x + y}\right) + -1\\ t_2 := \frac{x}{x - y \cdot \left(\left(a \cdot c\right) \cdot -2\right)}\\ \mathbf{if}\;z \leq -7.234502649290478 \cdot 10^{+145}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -5.96742305000556 \cdot 10^{+54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.3347359027682898 \cdot 10^{-48}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -4.158157480028268 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - 2 \cdot \left(a \cdot b\right)\right)}\\ \mathbf{elif}\;z \leq 1.5230740374586051 \cdot 10^{-248}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 5.383940501490396 \cdot 10^{-216}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;z \leq 1.7669998744186706 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 3.1565169628162163 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.222099550341434 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 1.7096619642486624 \cdot 10^{+253}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	34.3
Cost	2020

\[\begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-291}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - \frac{c}{t} \cdot 1.3333333333333333\right)}\\ \mathbf{elif}\;a \leq 2.5308390563524665 \cdot 10^{-81}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.512602306270874 \cdot 10^{-17}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{elif}\;a \leq 3.2829884978843216 \cdot 10^{+68}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.4412770845070837 \cdot 10^{+180}:\\ \;\;\;\;\frac{x}{x - y \cdot \left(\left(a \cdot c\right) \cdot -2\right)}\\ \mathbf{elif}\;a \leq 8.710907288652721 \cdot 10^{+198}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.1203529642154528 \cdot 10^{+223}:\\ \;\;\;\;0.5 \cdot \frac{x}{c \cdot \left(y \cdot a\right)}\\ \mathbf{elif}\;a \leq 5.037153313408607 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - 2 \cdot \left(a \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Error	35.0
Cost	2020

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y - \left(y \cdot \left(a \cdot c\right)\right) \cdot -2\right)}\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{-291}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - \frac{c}{t} \cdot 1.3333333333333333\right)}\\ \mathbf{elif}\;a \leq 2.5308390563524665 \cdot 10^{-81}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.512602306270874 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.8855297165014061 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.4412770845070837 \cdot 10^{+180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.4600600242799728 \cdot 10^{+205}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.1203529642154528 \cdot 10^{+223}:\\ \;\;\;\;0.5 \cdot \frac{x}{c \cdot \left(y \cdot a\right)}\\ \mathbf{elif}\;a \leq 5.037153313408607 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - 2 \cdot \left(a \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 21
Error	34.4
Cost	1896

\[\begin{array}{l} t_1 := \frac{x}{x + y}\\ t_2 := \left(1 + t_1\right) + -1\\ \mathbf{if}\;a \leq -7.2 \cdot 10^{-151}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-196}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-148}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.512602306270874 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 4.7433395258663346 \cdot 10^{+79}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 7.066852181465908 \cdot 10^{+147}:\\ \;\;\;\;\frac{0.5}{c} \cdot \frac{\frac{x}{y}}{a}\\ \mathbf{elif}\;a \leq 8.710907288652721 \cdot 10^{+198}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.1203529642154528 \cdot 10^{+223}:\\ \;\;\;\;0.5 \cdot \frac{x}{c \cdot \left(y \cdot a\right)}\\ \mathbf{elif}\;a \leq 5.037153313408607 \cdot 10^{+279}:\\ \;\;\;\;-0.5 \cdot \frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 22
Error	31.5
Cost	1760

\[\begin{array}{l} t_1 := \frac{x}{x - y \cdot \left(\left(a \cdot c\right) \cdot -2\right)}\\ t_2 := \left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{if}\;z \leq -7.234502649290478 \cdot 10^{+145}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -5.96742305000556 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.0548145485180503 \cdot 10^{-305}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 5.383940501490396 \cdot 10^{-216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.7669998744186706 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 3.1565169628162163 \cdot 10^{+38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.222099550341434 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 1.7096619642486624 \cdot 10^{+253}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 23
Error	31.9
Cost	972

\[\begin{array}{l} \mathbf{if}\;c \leq 4.9099548979838434 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 1.5244872759450812 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + -1\\ \mathbf{elif}\;c \leq 9.90076200569471 \cdot 10^{+97}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{c \cdot \left(y \cdot a\right)}\\ \end{array} \]

Alternative 24
Error	31.8
Cost	848

\[\begin{array}{l} \mathbf{if}\;c \leq 4.9099548979838434 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 1.5244872759450812 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + -1\\ \mathbf{elif}\;c \leq 3.826831177959319 \cdot 10^{+120}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 9.345051027900694 \cdot 10^{+221}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 25
Error	31.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;c \leq 3.826831177959319 \cdot 10^{+120}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 9.345051027900694 \cdot 10^{+221}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 26
Error	30.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2.0 (/ (- (* (* z (sqrt (+ t a))) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0) (* (- a (/ 5.0 6.0)) (* (- b c) t)))) (* (* (* t t) 3.0) (- a (/ 5.0 6.0))))))))) (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))

  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))