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

Average Error: 5.86% → 2.77%

Time: 36.2s

Precision: binary64

Cost: 28804

Math

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

↓

\[\begin{array}{l} t_1 := \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} + \left(-0.8333333333333334 - a\right)\right)\\ t_2 := \sqrt{t + a}\\ \mathbf{if}\;\frac{z \cdot t_2}{t} + t_1 \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z}{\frac{t}{t_2}} + t_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \end{array} \]

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 (* (- b c) (+ (/ 2.0 (* t 3.0)) (- -0.8333333333333334 a))))
        (t_2 (sqrt (+ t a))))
   (if (<= (+ (/ (* z t_2) t) t_1) INFINITY)
     (/ x (+ x (* y (pow (exp 2.0) (+ (/ z (/ t t_2)) t_1)))))
     (/ x (+ x (* y (exp (* (- b c) -1.6666666666666667))))))))

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 = (b - c) * ((2.0 / (t * 3.0)) + (-0.8333333333333334 - a));
	double t_2 = sqrt((t + a));
	double tmp;
	if ((((z * t_2) / t) + t_1) <= ((double) INFINITY)) {
		tmp = x / (x + (y * pow(exp(2.0), ((z / (t / t_2)) + t_1))));
	} else {
		tmp = x / (x + (y * exp(((b - c) * -1.6666666666666667))));
	}
	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 = (b - c) * ((2.0 / (t * 3.0)) + (-0.8333333333333334 - a));
	double t_2 = Math.sqrt((t + a));
	double tmp;
	if ((((z * t_2) / t) + t_1) <= Double.POSITIVE_INFINITY) {
		tmp = x / (x + (y * Math.pow(Math.exp(2.0), ((z / (t / t_2)) + t_1))));
	} else {
		tmp = x / (x + (y * Math.exp(((b - c) * -1.6666666666666667))));
	}
	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 = (b - c) * ((2.0 / (t * 3.0)) + (-0.8333333333333334 - a))
	t_2 = math.sqrt((t + a))
	tmp = 0
	if (((z * t_2) / t) + t_1) <= math.inf:
		tmp = x / (x + (y * math.pow(math.exp(2.0), ((z / (t / t_2)) + t_1))))
	else:
		tmp = x / (x + (y * math.exp(((b - c) * -1.6666666666666667))))
	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(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) + Float64(-0.8333333333333334 - a)))
	t_2 = sqrt(Float64(t + a))
	tmp = 0.0
	if (Float64(Float64(Float64(z * t_2) / t) + t_1) <= Inf)
		tmp = Float64(x / Float64(x + Float64(y * (exp(2.0) ^ Float64(Float64(z / Float64(t / t_2)) + t_1)))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(Float64(b - c) * -1.6666666666666667)))));
	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 = (b - c) * ((2.0 / (t * 3.0)) + (-0.8333333333333334 - a));
	t_2 = sqrt((t + a));
	tmp = 0.0;
	if ((((z * t_2) / t) + t_1) <= Inf)
		tmp = x / (x + (y * (exp(2.0) ^ ((z / (t / t_2)) + t_1))));
	else
		tmp = x / (x + (y * exp(((b - c) * -1.6666666666666667))));
	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[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] + N[(-0.8333333333333334 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(z * t$95$2), $MachinePrecision] / t), $MachinePrecision] + t$95$1), $MachinePrecision], Infinity], N[(x / N[(x + N[(y * N[Power[N[Exp[2.0], $MachinePrecision], N[(N[(z / N[(t / t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(N[(b - c), $MachinePrecision] * -1.6666666666666667), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	5.86%
Target	4.43%
Herbie	2.77%

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

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 1.08

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

   2. Simplified 0.9

      \[\leadsto \color{blue}{\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + 0.8333333333333334\right) - \frac{2}{t \cdot 3}\right)\right)}}} \]
      Proof

      [Start] 1.08	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      exp-prod [=>] 1.08	\[ \frac{x}{x + y \cdot \color{blue}{{\left(e^{2}\right)}^{\left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}} \]
      associate-/l* [=>] 0.9	\[ \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\color{blue}{\frac{z}{\frac{t}{\sqrt{t + a}}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      metadata-eval [=>] 0.9	\[ \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \color{blue}{0.8333333333333334}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]

   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 100

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

   2. Taylor expanded in t around inf 36.05

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

   3. Simplified 36.05

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

      [Start] 36.05	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(-1 \cdot \left(\left(b - c\right) \cdot \left(0.8333333333333334 + a\right)\right)\right)}} \]
      mul-1-neg [=>] 36.05	\[ \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(-\left(b - c\right) \cdot \left(0.8333333333333334 + a\right)\right)}}} \]
      distribute-rgt-neg-in [=>] 36.05	\[ \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(b - c\right) \cdot \left(-\left(0.8333333333333334 + a\right)\right)\right)}}} \]
      neg-sub0 [=>] 36.05	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \color{blue}{\left(0 - \left(0.8333333333333334 + a\right)\right)}\right)}} \]
      associate--r+ [=>] 36.05	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \color{blue}{\left(\left(0 - 0.8333333333333334\right) - a\right)}\right)}} \]
      metadata-eval [=>] 36.05	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(\color{blue}{-0.8333333333333334} - a\right)\right)}} \]

   4. Taylor expanded in a around 0 39.55

      \[\leadsto \color{blue}{\frac{x}{e^{-1.6666666666666667 \cdot \left(b - c\right)} \cdot y + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.77

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} + \left(-0.8333333333333334 - a\right)\right) \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z}{\frac{t}{\sqrt{t + a}}} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.12%
Cost	33408

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

Alternative 2
Error	2.94%
Cost	22468

\[\begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} + \left(-0.8333333333333334 - a\right)\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \end{array} \]

Alternative 3
Error	20.17%
Cost	14288

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t}\right)}}\\ \mathbf{if}\;t \leq -4.15 \cdot 10^{-271}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{\frac{0.6666666666666666}{t}}\right)}^{\left(2 \cdot b\right)}}\\ \mathbf{elif}\;t \leq 1050000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \end{array} \]

Alternative 4
Error	20.28%
Cost	14160

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(z \cdot \sqrt{\frac{1}{t}}\right)}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-192}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{\frac{0.6666666666666666}{t}}\right)}^{\left(2 \cdot b\right)}}\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \end{array} \]

Alternative 5
Error	46.96%
Cost	8164

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot a\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{b \cdot -1.6666666666666667}}\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{+212}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.35 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 15000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+154}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+285}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	39.45%
Cost	8161

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+274}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.56 \cdot 10^{+97}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-152}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{0.6666666666666666}{\frac{t}{b}}}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+53} \lor \neg \left(y \leq 1.36 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	48%
Cost	8028

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{b \cdot -1.6666666666666667}}\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{+150}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-241}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-242}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 1.68 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+202}:\\ \;\;\;\;\frac{x}{y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	39.32%
Cost	7757

\[\begin{array}{l} \mathbf{if}\;b - c \leq -4 \cdot 10^{+136} \lor \neg \left(b - c \leq -5 \cdot 10^{+107}\right) \land b - c \leq -1 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	19.66%
Cost	7752

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-240}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 0.2:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \end{array} \]

Alternative 10
Error	29.93%
Cost	7628

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{if}\;a \leq -0.85:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-110}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	21.78%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-240}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{0.6666666666666666}{\frac{t}{b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \end{array} \]

Alternative 12
Error	31.46%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-271} \lor \neg \left(t \leq 6100000000\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	50.84%
Cost	1612

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(2 \cdot \left(a \cdot \left(c - b\right)\right) + 1\right)}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-243}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-290}:\\ \;\;\;\;\frac{x}{x + \left(y - \left(\left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right) \cdot \left(y \cdot b\right)\right) \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	47.21%
Cost	1608

\[\begin{array}{l} \mathbf{if}\;c \leq 1.3 \cdot 10^{-193}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{x + \left(y + -2 \cdot \left(c \cdot \left(y \cdot \left(-0.6666666666666666 \cdot \frac{-1}{t} - \left(a + 0.8333333333333334\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{+38}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(2 \cdot \left(a \cdot \left(c - b\right)\right) + 1\right)}\\ \end{array} \]

Alternative 15
Error	50.97%
Cost	1232

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+161}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{x + \left(y + \left(a \cdot \left(y \cdot b\right)\right) \cdot -2\right)}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-242}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-290}:\\ \;\;\;\;\frac{x}{y + 1.3333333333333333 \cdot \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	51.28%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(2 \cdot \left(a \cdot \left(c - b\right)\right) + 1\right)}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-243}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{-289}:\\ \;\;\;\;\frac{x}{y + 1.3333333333333333 \cdot \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	49.47%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	49.59%
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-242}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-290}:\\ \;\;\;\;\frac{x}{y + 1.3333333333333333 \cdot \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Error	49.57%
Cost	840

\[\begin{array}{l} \mathbf{if}\;a \leq 6.2 \cdot 10^{+173}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 2.25 \cdot 10^{+191}:\\ \;\;\;\;-0.5 \cdot \frac{x}{y \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Error	49.1%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023089 
(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)))))))))))