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

Average Error: 3.7 → 1.6

Time: 2.3min

Precision: binary64

Cost: 22724

Math

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

↓

\[\begin{array}{l} t_1 := a + \frac{5}{6}\\ t_2 := \frac{z \cdot \sqrt{t + a}}{t}\\ \mathbf{if}\;t_2 - \left(b - c\right) \cdot \left(t_1 - \frac{2}{t \cdot 3}\right) \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(t_2 - \left(b - c\right) \cdot \left(t_1 - \frac{\frac{2}{t}}{3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\sqrt{a} \cdot z - -0.6666666666666666 \cdot \left(b - c\right)}{t}}}\\ \end{array} \]

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 (+ a (/ 5.0 6.0))) (t_2 (/ (* z (sqrt (+ t a))) t)))
   (if (<= (- t_2 (* (- b c) (- t_1 (/ 2.0 (* t 3.0))))) INFINITY)
     (/
      x
      (+ x (* y (exp (* 2.0 (- t_2 (* (- b c) (- t_1 (/ (/ 2.0 t) 3.0)))))))))
     (/
      x
      (+
       x
       (*
        y
        (exp
         (*
          2.0
          (/ (- (* (sqrt a) z) (* -0.6666666666666666 (- b c))) 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 = a + (5.0 / 6.0);
	double t_2 = (z * sqrt((t + a))) / t;
	double tmp;
	if ((t_2 - ((b - c) * (t_1 - (2.0 / (t * 3.0))))) <= ((double) INFINITY)) {
		tmp = x / (x + (y * exp((2.0 * (t_2 - ((b - c) * (t_1 - ((2.0 / t) / 3.0))))))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (((sqrt(a) * z) - (-0.6666666666666666 * (b - c))) / 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 = a + (5.0 / 6.0);
	double t_2 = (z * Math.sqrt((t + a))) / t;
	double tmp;
	if ((t_2 - ((b - c) * (t_1 - (2.0 / (t * 3.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = x / (x + (y * Math.exp((2.0 * (t_2 - ((b - c) * (t_1 - ((2.0 / t) / 3.0))))))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * (((Math.sqrt(a) * z) - (-0.6666666666666666 * (b - c))) / 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 = a + (5.0 / 6.0)
	t_2 = (z * math.sqrt((t + a))) / t
	tmp = 0
	if (t_2 - ((b - c) * (t_1 - (2.0 / (t * 3.0))))) <= math.inf:
		tmp = x / (x + (y * math.exp((2.0 * (t_2 - ((b - c) * (t_1 - ((2.0 / t) / 3.0))))))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * (((math.sqrt(a) * z) - (-0.6666666666666666 * (b - c))) / 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(a + Float64(5.0 / 6.0))
	t_2 = Float64(Float64(z * sqrt(Float64(t + a))) / t)
	tmp = 0.0
	if (Float64(t_2 - Float64(Float64(b - c) * Float64(t_1 - Float64(2.0 / Float64(t * 3.0))))) <= Inf)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(t_2 - Float64(Float64(b - c) * Float64(t_1 - Float64(Float64(2.0 / t) / 3.0)))))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(sqrt(a) * z) - Float64(-0.6666666666666666 * Float64(b - c))) / 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 = a + (5.0 / 6.0);
	t_2 = (z * sqrt((t + a))) / t;
	tmp = 0.0;
	if ((t_2 - ((b - c) * (t_1 - (2.0 / (t * 3.0))))) <= Inf)
		tmp = x / (x + (y * exp((2.0 * (t_2 - ((b - c) * (t_1 - ((2.0 / t) / 3.0))))))));
	else
		tmp = x / (x + (y * exp((2.0 * (((sqrt(a) * z) - (-0.6666666666666666 * (b - c))) / 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[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[(N[(b - c), $MachinePrecision] * N[(t$95$1 - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(t$95$2 - N[(N[(b - c), $MachinePrecision] * N[(t$95$1 - N[(N[(2.0 / t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(N[Sqrt[a], $MachinePrecision] * z), $MachinePrecision] - N[(-0.6666666666666666 * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.7
Target	2.9
Herbie	1.6

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

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 0.6

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

   2. Applied egg-rr 0.6

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

   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 t around 0 21.2

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\sqrt{a} \cdot z - -0.6666666666666666 \cdot \left(b - c\right)}{t}}}} \]
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	1.6
Cost	22724

\[\begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\sqrt{a} \cdot z - -0.6666666666666666 \cdot \left(b - c\right)}{t}}}\\ \end{array} \]

Alternative 2
Error	3.1
Cost	14920

\[\begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot a\right)}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\sqrt{a} \cdot z - -0.6666666666666666 \cdot \left(b - c\right)}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{\frac{2}{t}}{3}\right)\right)}}\\ \end{array} \]

Alternative 3
Error	6.1
Cost	14800

\[\begin{array}{l} t_1 := \frac{-0.6666666666666666}{t} + \left(0.8333333333333334 + a\right)\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot a\right)}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\sqrt{a} \cdot z - -0.6666666666666666 \cdot \left(b - c\right)}{t}}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\begin{array}{l} \mathbf{if}\;0.8333333333333334 + a \ne 0:\\ \;\;\;\;\left(0.8333333333333334 + a\right) \cdot \left(1 + \frac{-0.6666666666666666}{\left(0.8333333333333334 + a\right) \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t}}{t} + \left(c - b\right) \cdot \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(t_1 \cdot \left(c - b\right)\right)}}\\ \end{array} \]

Alternative 4
Error	7.8
Cost	14280

\[\begin{array}{l} t_1 := \frac{-0.6666666666666666}{t} + \left(0.8333333333333334 + a\right)\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot a\right)}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\sqrt{a} \cdot z - -0.6666666666666666 \cdot \left(b - c\right)}{t}}}\\ \mathbf{elif}\;t \leq 180000000000:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\begin{array}{l} \mathbf{if}\;0.8333333333333334 + a \ne 0:\\ \;\;\;\;\left(0.8333333333333334 + a\right) \cdot \left(1 + \frac{-0.6666666666666666}{\left(0.8333333333333334 + a\right) \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+62}:\\ \;\;\;\;3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(t_1 \cdot \left(c - b\right)\right)}}\\ \end{array} \]

Alternative 5
Error	6.9
Cost	13892

\[\begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\begin{array}{l} \mathbf{if}\;0.8333333333333334 + a \ne 0:\\ \;\;\;\;\left(0.8333333333333334 + a\right) \cdot \left(1 + \frac{-0.6666666666666666}{\left(0.8333333333333334 + a\right) \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.6666666666666666}{t} + \left(0.8333333333333334 + a\right)\\ \end{array} \cdot \left(c - b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\sqrt{a + t} \cdot z}{t}}}\\ \end{array} \]

Alternative 6
Error	9.3
Cost	8392

\[\begin{array}{l} t_1 := \frac{-0.6666666666666666}{t} + \left(0.8333333333333334 + a\right)\\ \mathbf{if}\;t \leq 180000000000:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\begin{array}{l} \mathbf{if}\;0.8333333333333334 + a \ne 0:\\ \;\;\;\;\left(0.8333333333333334 + a\right) \cdot \left(1 + \frac{-0.6666666666666666}{\left(0.8333333333333334 + a\right) \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+62}:\\ \;\;\;\;3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(t_1 \cdot \left(c - b\right)\right)}}\\ \end{array} \]

Alternative 7
Error	31.0
Cost	8296

\[\begin{array}{l} t_1 := 3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ t_2 := \frac{x}{e^{1.6666666666666667 \cdot c} \cdot y + x}\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + \left(c - b\right) \cdot \left(a + a\right)\right)}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-41}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{y + \left(1.3333333333333333 \cdot \frac{\left(b - c\right) \cdot y}{t} + x\right)}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-245}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-220}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+194}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	27.0
Cost	8160

\[\begin{array}{l} t_1 := \frac{x}{e^{1.6666666666666667 \cdot c} \cdot y + x}\\ t_2 := \frac{x}{x + e^{-2 \cdot \left(a \cdot b\right)} \cdot y}\\ \mathbf{if}\;a \leq -2.2 \cdot 10^{+16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.55 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-257}:\\ \;\;\;\;3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+45}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+181}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	24.2
Cost	8144

\[\begin{array}{l} t_1 := 3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot a\right)}}\\ t_3 := \left(a + a\right) \cdot y\\ \mathbf{if}\;b - c \leq -4000000:\\ \;\;\;\;\frac{x}{y \cdot e^{1.6666666666666667 \cdot \left(c - b\right)}}\\ \mathbf{elif}\;b - c \leq -5 \cdot 10^{-113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{-301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq 2 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq 10^{+131}:\\ \;\;\;\;1\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq 4 \cdot 10^{+173}:\\ \;\;\;\;\frac{x}{x + \left(y + \begin{array}{l} \mathbf{if}\;t_3 \cdot c - t_3 \cdot \left(-b\right) \ne 0:\\ \;\;\;\;\frac{a \cdot \left(4 \cdot \left(\left(y \cdot y\right) \cdot \left(c \cdot c - b \cdot b\right)\right)\right)}{2 \cdot \left(c \cdot y + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(y \cdot \left(c - b\right)\right) \cdot 2\right)\\ \end{array}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	23.8
Cost	7896

\[\begin{array}{l} t_1 := \frac{x}{y \cdot e^{1.6666666666666667 \cdot \left(c - b\right)} + x}\\ t_2 := \frac{x}{x + e^{-2 \cdot \left(a \cdot b\right)} \cdot y}\\ \mathbf{if}\;a \leq -3600000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.12 \cdot 10^{-234}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-248}:\\ \;\;\;\;3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+230}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	16.9
Cost	7892

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot a\right)}}\\ \mathbf{if}\;t \leq -0.0002:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.14 \cdot 10^{-125}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{1.3333333333333333 \cdot \frac{b - c}{t}}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+62}:\\ \;\;\;\;3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+242}:\\ \;\;\;\;\frac{x}{y \cdot e^{1.6666666666666667 \cdot \left(c - b\right)} + x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	13.4
Cost	7888

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot \left(0.8333333333333334 + a\right)\right)}}\\ \mathbf{if}\;t \leq -4 \cdot 10^{-225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-125}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{1.3333333333333333 \cdot \frac{b - c}{t}}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot a\right)}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+62}:\\ \;\;\;\;3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	9.3
Cost	7880

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\frac{-0.6666666666666666}{t} + \left(0.8333333333333334 + a\right)\right) \cdot \left(c - b\right)\right)}}\\ \mathbf{if}\;t \leq 180000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+62}:\\ \;\;\;\;3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	17.7
Cost	7632

\[\begin{array}{l} t_1 := \frac{x}{y \cdot e^{1.6666666666666667 \cdot \left(c - b\right)} + x}\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{1.3333333333333333 \cdot \frac{b - c}{t}}}\\ \mathbf{elif}\;t \leq 61000000000:\\ \;\;\;\;\frac{x}{e^{1.6666666666666667 \cdot c} \cdot y + x}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+62}:\\ \;\;\;\;3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	28.8
Cost	7504

\[\begin{array}{l} t_1 := \left(a + a\right) \cdot y\\ t_2 := t_1 \cdot c\\ t_3 := t_1 \cdot \left(-b\right)\\ t_4 := t_2 - t_3\\ \mathbf{if}\;c \leq -5 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -2.72 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{x + \left(y + \begin{array}{l} \mathbf{if}\;t_4 \ne 0:\\ \;\;\;\;\frac{t_2 \cdot t_2 - t_3 \cdot t_3}{t_4}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(y \cdot \left(c - b\right)\right) \cdot 2\right)\\ \end{array}\right)}\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-173}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{+103}:\\ \;\;\;\;3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{1.6666666666666667 \cdot \left(c - b\right)}}\\ \end{array} \]

Alternative 16
Error	30.8
Cost	2016

\[\begin{array}{l} t_1 := 3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{-116}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-229}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-285}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{\left(y + x\right) - \frac{-1.3333333333333333 \cdot \left(y \cdot b\right)}{t}}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + \left(c - b\right) \cdot \left(a + a\right)\right)}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+148}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	30.5
Cost	1696

\[\begin{array}{l} t_1 := 3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{-115}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-285}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-63}:\\ \;\;\;\;\frac{x}{\left(y + x\right) - \frac{-1.3333333333333333 \cdot \left(y \cdot b\right)}{t}}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{+147}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	31.4
Cost	1236

\[\begin{array}{l} t_1 := 3 - \left(3 + \frac{x}{\left(-x\right) - y}\right)\\ \mathbf{if}\;c \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.6 \cdot 10^{-173}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{+179}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-0.75 \cdot \frac{t \cdot x}{c \cdot y}\\ \end{array} \]

Alternative 19
Error	31.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;c \leq 7.5 \cdot 10^{+237}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-0.75 \cdot \frac{t \cdot x}{c \cdot y}\\ \end{array} \]

Alternative 20
Error	32.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-220}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-129}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 21
Error	31.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023033 
(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)))))))))))