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

Average Error: 4.2 → 2.5

Time: 38.0s

Precision: binary64

Cost: 22468

Math

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

↓

\[\begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} + \left(-0.8333333333333334 - a\right)\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{1.3333333333333333 \cdot \frac{b}{t}}}\\ \end{array} \]

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)
          (* (- b c) (+ (/ 2.0 (* t 3.0)) (- -0.8333333333333334 a))))))
   (if (<= t_1 INFINITY)
     (/ x (+ x (* y (exp (* 2.0 t_1)))))
     (/ x (+ x (* y (exp (* 1.3333333333333333 (/ b t)))))))))

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) + ((b - c) * ((2.0 / (t * 3.0)) + (-0.8333333333333334 - a)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = x / (x + (y * exp((2.0 * t_1))));
	} else {
		tmp = x / (x + (y * exp((1.3333333333333333 * (b / t)))));
	}
	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) + ((b - c) * ((2.0 / (t * 3.0)) + (-0.8333333333333334 - a)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x / (x + (y * Math.exp((2.0 * t_1))));
	} else {
		tmp = x / (x + (y * Math.exp((1.3333333333333333 * (b / t)))));
	}
	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) + ((b - c) * ((2.0 / (t * 3.0)) + (-0.8333333333333334 - a)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = x / (x + (y * math.exp((2.0 * t_1))))
	else:
		tmp = x / (x + (y * math.exp((1.3333333333333333 * (b / t)))))
	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(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) + Float64(-0.8333333333333334 - a))))
	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 * exp(Float64(1.3333333333333333 * Float64(b / t))))));
	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) + ((b - c) * ((2.0 / (t * 3.0)) + (-0.8333333333333334 - a)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = x / (x + (y * exp((2.0 * t_1))));
	else
		tmp = x / (x + (y * exp((1.3333333333333333 * (b / t)))));
	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[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] + N[(-0.8333333333333334 - a), $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[Exp[N[(1.3333333333333333 * N[(b / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	4.2
Target	3.2
Herbie	2.5

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

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 0.8

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

   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 b around inf 30.1

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

   3. Simplified 30.1

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(b \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right)\right)}}} \]
      Proof
      (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 2 (*.f64 b (-.f64 (/.f64 2/3 t) (+.f64 a 5/6)))))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 2 (*.f64 b (-.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2/3 1)) t) (+.f64 a 5/6)))))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 2 (*.f64 b (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2/3 (/.f64 1 t))) (+.f64 a 5/6)))))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 2 (*.f64 b (-.f64 (*.f64 2/3 (/.f64 1 t)) (Rewrite<= +-commutative_binary64 (+.f64 5/6 a))))))))): 5 points increase in error, 0 points decrease in error
      (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 2/3 (/.f64 1 t)) (+.f64 5/6 a)) b))))))): 0 points increase in error, 5 points decrease in error

   4. Taylor expanded in t around 0 33.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(0.6666666666666666 \cdot \frac{b}{t}\right)}}} \]

   5. Simplified 33.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{b \cdot 0.6666666666666666}{t}}}} \]
      Proof
      (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 2 (/.f64 (*.f64 b 2/3) t)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 2/3 b)) t)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 2/3 (/.f64 b t)))))))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in y around 0 33.6

      \[\leadsto \frac{x}{x + \color{blue}{e^{1.3333333333333333 \cdot \frac{b}{t}} \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} + \left(-0.8333333333333334 - a\right)\right) \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{1.3333333333333333 \cdot \frac{b}{t}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.9
Cost	14552

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{t} \cdot \sqrt{t + a}\right)}}\\ t_3 := \frac{x}{x + y \cdot e^{-2 \cdot \left(c \cdot \left(\left(\frac{0.6666666666666666}{t} + -0.8333333333333334\right) - a\right)\right)}}\\ \mathbf{if}\;z \leq -5 \cdot 10^{+128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-197}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10000000000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	30.1
Cost	8693

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{b \cdot -1.6666666666666667}}\\ t_2 := y \cdot y - x \cdot x\\ t_3 := \left(y - x\right) \cdot \left(\left(x + y\right) \cdot \frac{x}{t_2 \cdot \left(x + y\right)}\right)\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-151}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-289}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-281}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + -2 \cdot \left(b \cdot \left(\left(a + 0.8333333333333334\right) + 0.6666666666666666 \cdot \frac{-1}{t}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;0.5 \cdot \frac{x}{y \cdot \left(a \cdot \left(c - b\right)\right)}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-159}:\\ \;\;\;\;\frac{x}{t_2} \cdot \left(y - x\right)\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.4:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 210000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+160} \lor \neg \left(t \leq 2.25 \cdot 10^{+251}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	19.3
Cost	8412

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ \mathbf{if}\;b \leq -23:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -8.2 \cdot 10^{-39}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.2 \cdot 10^{-247}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{c \cdot 1.6666666666666667}}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-299}:\\ \;\;\;\;\frac{x}{x + \left(a \cdot \left(b \cdot y\right)\right) \cdot -2}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-158}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{c \cdot -0.6666666666666666}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	21.8
Cost	8292

\[\begin{array}{l} t_1 := y \cdot y - x \cdot x\\ t_2 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{if}\;t \leq -3 \cdot 10^{-290}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-279}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + -2 \cdot \left(b \cdot \left(\left(a + 0.8333333333333334\right) + 0.6666666666666666 \cdot \frac{-1}{t}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-173}:\\ \;\;\;\;0.5 \cdot \frac{x}{y \cdot \left(a \cdot \left(c - b\right)\right)}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{t_1} \cdot \left(y - x\right)\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6:\\ \;\;\;\;\left(y - x\right) \cdot \left(\left(x + y\right) \cdot \frac{x}{t_1 \cdot \left(x + y\right)}\right)\\ \mathbf{elif}\;t \leq 210000:\\ \;\;\;\;\frac{x}{x + y \cdot e^{b \cdot -1.6666666666666667}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	27.6
Cost	7900

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{c \cdot 1.6666666666666667}}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{b \cdot -1.6666666666666667}}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -2.2 \cdot 10^{-247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-301}:\\ \;\;\;\;\frac{x}{x + \left(a \cdot \left(b \cdot y\right)\right) \cdot -2}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 550000:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{+42}:\\ \;\;\;\;\left(y - x\right) \cdot \left(\left(x + y\right) \cdot \frac{x}{\left(y \cdot y - x \cdot x\right) \cdot \left(x + y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	14.8
Cost	7888

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{1.3333333333333333 \cdot \frac{b}{t}}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{elif}\;t \leq 1.46 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{c \cdot -0.6666666666666666}{t}}}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \end{array} \]

Alternative 7
Error	13.3
Cost	7753

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+29} \lor \neg \left(b \leq 68\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(c \cdot \left(\left(\frac{0.6666666666666666}{t} + -0.8333333333333334\right) - a\right)\right)}}\\ \end{array} \]

Alternative 8
Error	18.3
Cost	7632

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{1.3333333333333333 \cdot \frac{b}{t}}}\\ t_2 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-252}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{c \cdot -0.6666666666666666}{t}}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	18.2
Cost	7369

\[\begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-249} \lor \neg \left(t \leq 3.6 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{1.3333333333333333 \cdot \frac{b}{t}}}\\ \end{array} \]

Alternative 10
Error	32.9
Cost	2268

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-250}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + \left(c \cdot 2\right) \cdot \left(0.8333333333333334 + \left(a + \frac{-0.6666666666666666}{t}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-140}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{x + \left(y + -2 \cdot \left(a \cdot \left(\left(b - c\right) \cdot y\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-52}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + -2 \cdot \left(a \cdot \left(b - c\right)\right)\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+193}:\\ \;\;\;\;\left(y - x\right) \cdot \left(\left(x + y\right) \cdot \frac{x}{\left(y \cdot y - x \cdot x\right) \cdot \left(x + y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	32.6
Cost	2268

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-251}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + \left(c \cdot 2\right) \cdot \left(0.8333333333333334 + \left(a + \frac{-0.6666666666666666}{t}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-141}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \frac{\left(c \cdot c - b \cdot b\right) \cdot \left(a \cdot y\right)}{b + c}\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-28}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + -2 \cdot \left(a \cdot \left(b - c\right)\right)\right)}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+193}:\\ \;\;\;\;\left(y - x\right) \cdot \left(\left(x + y\right) \cdot \frac{x}{\left(y \cdot y - x \cdot x\right) \cdot \left(x + y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	30.8
Cost	1884

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y + 1.3333333333333333 \cdot \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{if}\;c \leq -5.1 \cdot 10^{-33}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -4.2 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -1 \cdot 10^{-139}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -8 \cdot 10^{-263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-301}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+123}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + -2 \cdot \left(a \cdot \left(b - c\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	33.1
Cost	1756

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-255}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-179}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + \left(c \cdot 2\right) \cdot \left(0.8333333333333334 + \left(a + \frac{-0.6666666666666666}{t}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-140}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{x + \left(y + -2 \cdot \left(a \cdot \left(\left(b - c\right) \cdot y\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-46}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + -2 \cdot \left(a \cdot \left(b - c\right)\right)\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+193}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	31.7
Cost	1632

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y + 1.3333333333333333 \cdot \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{if}\;c \leq -3.8 \cdot 10^{-31}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -4.7 \cdot 10^{-139}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-300}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{+189}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\frac{x}{a \cdot c} \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Error	31.4
Cost	1368

\[\begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-28}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;c \leq -2.6 \cdot 10^{-139}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -9 \cdot 10^{-263}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{+191}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{+287}:\\ \;\;\;\;\frac{x}{a \cdot c} \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	32.9
Cost	1168

\[\begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-248}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{c \cdot y} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+141}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+194}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{-x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	32.3
Cost	1168

\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-255}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-180}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{y}}{a \cdot \left(c - b\right)}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+141}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+193}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{-x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	32.5
Cost	1168

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-255}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{0.5 \cdot \frac{x}{c}}{y}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+141}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+194}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{-x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Error	31.8
Cost	976

\[\begin{array}{l} \mathbf{if}\;c \leq -2.35 \cdot 10^{-26}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -3.55 \cdot 10^{-99}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;c \leq -4.2 \cdot 10^{-140}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-262}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Error	32.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-255}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 21
Error	32.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-255}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 22
Error	30.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022343 
(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)))))))))))