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

Average Error: 4.0 → 2.2

Time: 49.8s

Precision: binary64

Cost: 22468

Math

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

↓

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

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) (+ (/ 2.0 (* t 3.0)) (- -0.8333333333333334 a))))))
   (if (<= t_1 INFINITY)
     (/ x (+ x (* y (exp (* 2.0 t_1)))))
     (/ x (+ x (* y (exp (* -2.0 (* a 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) * ((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((-2.0 * (a * 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) * ((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((-2.0 * (a * 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) * ((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((-2.0 * (a * 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(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(-2.0 * Float64(a * 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) * ((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((-2.0 * (a * 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[(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[(-2.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	4.0
Target	3.0
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 0.7

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

   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 a around inf 27.5

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

   3. Taylor expanded in c around 0 30.3

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

4. Final simplification 2.2

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

Alternatives

Alternative 1
Error	13.5
Cost	14028

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \frac{\left(b - c\right) \cdot \left(0.6944444444444444 - a \cdot a\right)}{a + -0.8333333333333334}}}\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{-288}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \frac{-0.6666666666666666}{t}\right)}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(z \cdot \sqrt{\frac{1}{t}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	17.3
Cost	8932

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{-2 \cdot \left(b \cdot \left(\left(a + 0.8333333333333334\right) + \frac{-0.6666666666666666}{t}\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right)\right)}}\\ \mathbf{if}\;a \leq -8 \cdot 10^{-142}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-306}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{0.6666666666666666}{\frac{t}{b}}}}\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(b - c\right) \cdot \left(0.6944444444444444 - a \cdot a\right)}{a + -0.8333333333333334}}}\\ \end{array} \]

Alternative 3
Error	17.2
Cost	8804

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{-2 \cdot \left(b \cdot \left(\left(a + 0.8333333333333334\right) + \frac{-0.6666666666666666}{t}\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right)\right)}}\\ \mathbf{if}\;a \leq -2.9 \cdot 10^{-135}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;a \leq -1.2 \cdot 10^{-308}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{-135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{0.6666666666666666}{\frac{t}{b}}}}\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(b - c\right) \cdot \left(0.6944444444444444 - a \cdot a\right)}{a}}}\\ \end{array} \]

Alternative 4
Error	17.3
Cost	8676

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{-2 \cdot \left(b \cdot \left(\left(a + 0.8333333333333334\right) + \frac{-0.6666666666666666}{t}\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right)\right)}}\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-306}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{0.6666666666666666}{\frac{t}{b}}}}\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \end{array} \]

Alternative 5
Error	32.6
Cost	8292

\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(c \cdot \left(a + c \cdot \left(a \cdot a\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-43}:\\ \;\;\;\;\left(y - x\right) \cdot \left(\left(x + y\right) \cdot \frac{\frac{x}{x + y}}{y \cdot y - x \cdot x}\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{x}{x + \left(c \cdot -2\right) \cdot \left(y \cdot \left(-0.8333333333333334 + \left(\frac{0.6666666666666666}{t} - a\right)\right)\right)}\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1500000:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot \left(c + a \cdot \left(c \cdot c\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+79}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+180}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{c \cdot \left(a + a\right)}}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+204} \lor \neg \left(x \leq 6 \cdot 10^{+283}\right):\\ \;\;\;\;\frac{x}{x \cdot \left(-x\right)} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 2 \cdot \left(c \cdot \left(a \cdot y\right)\right)}\\ \end{array} \]

Alternative 6
Error	32.0
Cost	8160

\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(c \cdot \left(a + c \cdot \left(a \cdot a\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{+37}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{x + \left(c \cdot -2\right) \cdot \left(y \cdot \left(-0.8333333333333334 + \left(\frac{0.6666666666666666}{t} - a\right)\right)\right)}\\ \mathbf{elif}\;x \leq -31000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(a \cdot b\right)}}\\ \mathbf{elif}\;x \leq 2050000000:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot \left(c + a \cdot \left(c \cdot c\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+79}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+179}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{c \cdot \left(a + a\right)}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+204} \lor \neg \left(x \leq 4.8 \cdot 10^{+264}\right):\\ \;\;\;\;\frac{x}{x \cdot \left(-x\right)} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	21.3
Cost	7632

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-222}:\\ \;\;\;\;\frac{x}{x + -1.3333333333333333 \cdot \frac{c \cdot y}{t}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(a \cdot b\right)}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	20.8
Cost	7632

\[\begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{-307}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-232}:\\ \;\;\;\;\frac{x}{x + -1.3333333333333333 \cdot \frac{c \cdot y}{t}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(a \cdot b\right)}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \end{array} \]

Alternative 9
Error	13.4
Cost	7625

\[\begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-290} \lor \neg \left(t \leq 4.4 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(c - b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \frac{-0.6666666666666666}{t}\right)}}\\ \end{array} \]

Alternative 10
Error	17.8
Cost	7496

\[\begin{array}{l} \mathbf{if}\;t \leq -6.1 \cdot 10^{-289}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \frac{-0.6666666666666666}{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \end{array} \]

Alternative 11
Error	31.2
Cost	3664

\[\begin{array}{l} t_1 := \left(c \cdot 2\right) \cdot \left(a + \frac{-0.6666666666666666}{t}\right)\\ t_2 := \frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot \left(c + a \cdot \left(c \cdot c\right)\right)\right)\right)}\\ \mathbf{if}\;c \leq -1.2 \cdot 10^{+232}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -2.15 \cdot 10^{+133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-21}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -5.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + \frac{\left(c \cdot 1.6666666666666667\right) \cdot \left(c \cdot 1.6666666666666667\right) - t_1 \cdot t_1}{c \cdot 1.6666666666666667 + \left(c \cdot 2\right) \cdot \left(\frac{0.6666666666666666}{t} - a\right)}\right)}\\ \mathbf{elif}\;c \leq -4.5 \cdot 10^{-142}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-227}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(c \cdot \left(a + c \cdot \left(a \cdot a\right)\right)\right)\right)}\\ \mathbf{elif}\;c \leq -7.7 \cdot 10^{-246}:\\ \;\;\;\;\frac{x}{x + 2 \cdot \left(c \cdot \left(a \cdot y\right)\right)}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+62}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 2.35 \cdot 10^{+234} \lor \neg \left(c \leq 3.7 \cdot 10^{+259}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	33.9
Cost	2280

\[\begin{array}{l} t_1 := \frac{x}{x + \left(c \cdot -2\right) \cdot \left(y \cdot \left(-0.8333333333333334 + \left(\frac{0.6666666666666666}{t} - a\right)\right)\right)}\\ \mathbf{if}\;x \leq -6 \cdot 10^{+198}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{y \cdot y - x \cdot x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{+38}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-212}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 18000000000:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot \left(c + a \cdot \left(c \cdot c\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+180}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+203}:\\ \;\;\;\;\frac{x}{x \cdot \left(-x\right)} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	33.5
Cost	2148

\[\begin{array}{l} t_1 := \frac{x}{x + \left(c \cdot -2\right) \cdot \left(y \cdot \left(-0.8333333333333334 + \left(\frac{0.6666666666666666}{t} - a\right)\right)\right)}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(c \cdot \left(a + c \cdot \left(a \cdot a\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-215}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2900000:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot \left(c + a \cdot \left(c \cdot c\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 10^{+180}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\frac{x}{x \cdot \left(-x\right)} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \end{array} \]

Alternative 14
Error	32.7
Cost	2148

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(c \cdot \left(a + c \cdot \left(a \cdot a\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+36}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-43}:\\ \;\;\;\;\left(y - x\right) \cdot \left(\left(x + y\right) \cdot \frac{\frac{x}{x + y}}{y \cdot y - x \cdot x}\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-215}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-153}:\\ \;\;\;\;\frac{x}{x + \left(c \cdot -2\right) \cdot \left(y \cdot \left(-0.8333333333333334 + \left(\frac{0.6666666666666666}{t} - a\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot \left(c + a \cdot \left(c \cdot c\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+180}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+203}:\\ \;\;\;\;\frac{x}{x \cdot \left(-x\right)} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	33.2
Cost	2016

\[\begin{array}{l} t_1 := \frac{x}{x + \left(c \cdot -2\right) \cdot \left(y \cdot \left(-0.8333333333333334 + \left(\frac{0.6666666666666666}{t} - a\right)\right)\right)}\\ \mathbf{if}\;x \leq -8 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{+122}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{y \cdot y - x \cdot x}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{+36}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-211}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+179}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+203}:\\ \;\;\;\;\frac{x}{x \cdot \left(-x\right)} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \end{array} \]

Alternative 16
Error	32.1
Cost	1752

\[\begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{x}{y \cdot y - x \cdot x}\\ \mathbf{if}\;x \leq -6 \cdot 10^{+196}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+178}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+203}:\\ \;\;\;\;\frac{x}{x \cdot \left(-x\right)} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y + -2 \cdot \left(a \cdot \left(\left(b - c\right) \cdot y\right)\right)\right)}\\ \end{array} \]

Alternative 17
Error	32.2
Cost	1624

\[\begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{x}{y \cdot y - x \cdot x}\\ t_2 := \frac{x}{x + y \cdot \left(1 + 2 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+196}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{+38}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{+179}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+204}:\\ \;\;\;\;\frac{x}{x \cdot \left(-x\right)} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	31.2
Cost	1360

\[\begin{array}{l} \mathbf{if}\;c \leq -1.3 \cdot 10^{-150}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -2.55 \cdot 10^{-244}:\\ \;\;\;\;\frac{x}{x + 2 \cdot \left(c \cdot \left(a \cdot y\right)\right)}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot \frac{x}{\left(c \cdot c\right) \cdot \left(y \cdot \left(a \cdot a\right)\right)}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{+258}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{x}{c}}{a}}{y}\\ \end{array} \]

Alternative 19
Error	31.5
Cost	1360

\[\begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{x}{y \cdot y - x \cdot x}\\ t_2 := \frac{x}{x + 2 \cdot \left(c \cdot \left(a \cdot y\right)\right)}\\ \mathbf{if}\;b \leq -1.02 \cdot 10^{+179}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -8.5 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{+40}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Error	31.1
Cost	1236

\[\begin{array}{l} t_1 := \frac{x}{x + 2 \cdot \left(c \cdot \left(a \cdot y\right)\right)}\\ \mathbf{if}\;c \leq -1.35 \cdot 10^{-150}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+96}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.45 \cdot 10^{+258}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{x}{c}}{a}}{y}\\ \end{array} \]

Alternative 21
Error	32.8
Cost	1104

\[\begin{array}{l} \mathbf{if}\;a \leq -1.18 \cdot 10^{-157}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.12 \cdot 10^{-255}:\\ \;\;\;\;\frac{x}{x + -1.3333333333333333 \cdot \frac{c \cdot y}{t}}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+219}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.75 \cdot 10^{+242}:\\ \;\;\;\;\frac{x \cdot 0.5}{c \cdot \left(a \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 22
Error	31.3
Cost	973

\[\begin{array}{l} \mathbf{if}\;c \leq 1.9 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{+228} \lor \neg \left(c \leq 3.7 \cdot 10^{+258}\right):\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{x}{c}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 23
Error	32.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-265}:\\ \;\;\;\;-0.75 \cdot \left(\frac{t}{c} \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 24
Error	31.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{+219}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.75 \cdot 10^{+242}:\\ \;\;\;\;-0.75 \cdot \frac{t \cdot x}{c \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 25
Error	31.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;a \leq 6.8 \cdot 10^{+219}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+242}:\\ \;\;\;\;0.5 \cdot \frac{x}{c \cdot \left(a \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 26
Error	31.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+219}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+242}:\\ \;\;\;\;\frac{x \cdot 0.5}{c \cdot \left(a \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 27
Error	31.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023018 
(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)))))))))))