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

Average Error: 4.2 → 2.2

Time: 32.5s

Precision: binary64

Cost: 22468

Math

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

↓

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

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

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) * ((-0.8333333333333334 - a) - (-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 * exp((2.0 * (a * (c - b))))));
	}
	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) * ((-0.8333333333333334 - a) - (-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 * Math.exp((2.0 * (a * (c - b))))));
	}
	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) * ((-0.8333333333333334 - a) - (-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 * math.exp((2.0 * (a * (c - b))))))
	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(-0.8333333333333334 - a) - 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 * exp(Float64(2.0 * Float64(a * Float64(c - b)))))));
	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) * ((-0.8333333333333334 - a) - (-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 * exp((2.0 * (a * (c - b))))));
	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[(-0.8333333333333334 - a), $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[Exp[N[(2.0 * N[(a * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	4.2
Target	3.1
Herbie	2.2

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

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 a around inf 25.7

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(a \cdot \left(c - b\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.2

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

Alternatives

Alternative 1
Error	2.3
Cost	33408

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

Alternative 2
Error	13.6
Cost	14420

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t}\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a - \left(\frac{0.6666666666666666}{t} + -0.8333333333333334\right)\right)\right)}}\\ t_3 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ \mathbf{if}\;c \leq -2.8 \cdot 10^{+26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -5.6 \cdot 10^{-197}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq -8.2 \cdot 10^{-250}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-298}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-252}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+15}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	14.5
Cost	8280

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{-49}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a - \left(\frac{0.6666666666666666}{t} + -0.8333333333333334\right)\right)\right)}}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-129}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-34}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	14.1
Cost	8016

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ \mathbf{if}\;t \leq -4 \cdot 10^{-309}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-64}:\\ \;\;\;\;\left(x - y\right) \cdot \left(\left(x + y\right) \cdot \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}{x - y}\right)\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-33}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	21.6
Cost	7893

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{-260}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-157}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-34}:\\ \;\;\;\;\left(x - y\right) \cdot \left(\left(x + y\right) \cdot \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}{x - y}\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+81} \lor \neg \left(t \leq 10^{+104}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	19.5
Cost	7893

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \frac{0.6666666666666666}{t}\right)}}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-35}:\\ \;\;\;\;\left(x - y\right) \cdot \left(\left(x + y\right) \cdot \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}{x - y}\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+80} \lor \neg \left(t \leq 8 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	32.1
Cost	7768

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y + a \cdot \left(2 \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ t_2 := \left(x - y\right) \cdot \left(\left(x + y\right) \cdot \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}{x - y}\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{+54}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -4000000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-283}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{y \cdot e^{-2 \cdot \left(b \cdot a\right)}}\\ \mathbf{elif}\;z \leq 1.04 \cdot 10^{+68}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{+220}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+288}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	25.5
Cost	7764

\[\begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{+190}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x \cdot t}{z \cdot \sqrt{a}}}{y}\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -820000000000:\\ \;\;\;\;\left(x - y\right) \cdot \left(\left(x + y\right) \cdot \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}{x - y}\right)\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \end{array} \]

Alternative 9
Error	15.0
Cost	7756

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{if}\;t \leq -4.1 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \frac{0.6666666666666666}{t}\right)}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-39}:\\ \;\;\;\;\left(x - y\right) \cdot \left(\left(x + y\right) \cdot \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}{x - y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	23.9
Cost	7236

\[\begin{array}{l} \mathbf{if}\;b - c \leq -0.05:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{elif}\;b - c \leq -2 \cdot 10^{-287}:\\ \;\;\;\;\left(x - y\right) \cdot \left(\left(x + y\right) \cdot \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}{x - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	33.7
Cost	2016

\[\begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{x}{y \cdot y - x \cdot x}\\ \mathbf{if}\;x \leq -2.35 \cdot 10^{+206}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-56}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-240}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y - x\right)}{y - x}}{x + y}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-72}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{x + \left(y + a \cdot \left(2 \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	33.8
Cost	2016

\[\begin{array}{l} t_1 := \frac{x}{y \cdot y - x \cdot x}\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+206}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+57}:\\ \;\;\;\;\left(y - x\right) \cdot t_1\\ \mathbf{elif}\;x \leq -1.28 \cdot 10^{-56}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{t_1}{\frac{1}{y - x}}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-240}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y - x\right)}{y - x}}{x + y}\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{x + \left(y + a \cdot \left(2 \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	33.5
Cost	1884

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+57}:\\ \;\;\;\;\left(x - y\right) \cdot \left(\left(x + y\right) \cdot \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}{x - y}\right)\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-56}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{x}{y \cdot y - x \cdot x}}{\frac{1}{y - x}}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-240}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y - x\right)}{y - x}}{x + y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-72}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{x + \left(y + a \cdot \left(2 \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	33.8
Cost	1368

\[\begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{+239}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-24}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{x \cdot \left(-x\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+259}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Error	33.2
Cost	1368

\[\begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -2.95 \cdot 10^{+237}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-25}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{y \cdot y - x \cdot x}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+267}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	32.9
Cost	1236

\[\begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -1.86 \cdot 10^{+233}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{x - y \cdot \left(-1 - \frac{c}{t} \cdot -1.3333333333333333\right)}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+190}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{+256}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	31.8
Cost	1105

\[\begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+233}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{+111} \lor \neg \left(y \leq 2.3 \cdot 10^{+190}\right) \land y \leq 3 \cdot 10^{+259}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	32.3
Cost	1104

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+217}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{y + a \cdot \left(2 \cdot \left(y \cdot \left(c - b\right)\right)\right)}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{+256}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Error	32.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;c \leq 1.4 \cdot 10^{-275}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Error	32.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;c \leq 1.42 \cdot 10^{-275}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 3.6 \cdot 10^{-140}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 21
Error	33.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+201}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{+111}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 22
Error	31.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022364 
(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)))))))))))