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

Average Error: 3.8 → 1.7

Time: 1.1min

Precision: binary64

Cost: 22340

Math

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

↓

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

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 (sqrt (+ t a))))
   (if (<=
        (-
         (/ (* z t_1) t)
         (* (- b c) (- (+ a 0.8333333333333334) (/ 2.0 (* t 3.0)))))
        INFINITY)
     (/
      x
      (+
       x
       (*
        y
        (exp
         (*
          2.0
          (-
           (* t_1 (/ z t))
           (*
            (- b c)
            (+ a (- 0.8333333333333334 (/ 0.6666666666666666 t))))))))))
     (/ 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 = sqrt((t + a));
	double tmp;
	if ((((z * t_1) / t) - ((b - c) * ((a + 0.8333333333333334) - (2.0 / (t * 3.0))))) <= ((double) INFINITY)) {
		tmp = x / (x + (y * exp((2.0 * ((t_1 * (z / t)) - ((b - c) * (a + (0.8333333333333334 - (0.6666666666666666 / t)))))))));
	} 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 = Math.sqrt((t + a));
	double tmp;
	if ((((z * t_1) / t) - ((b - c) * ((a + 0.8333333333333334) - (2.0 / (t * 3.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = x / (x + (y * Math.exp((2.0 * ((t_1 * (z / t)) - ((b - c) * (a + (0.8333333333333334 - (0.6666666666666666 / t)))))))));
	} 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 = math.sqrt((t + a))
	tmp = 0
	if (((z * t_1) / t) - ((b - c) * ((a + 0.8333333333333334) - (2.0 / (t * 3.0))))) <= math.inf:
		tmp = x / (x + (y * math.exp((2.0 * ((t_1 * (z / t)) - ((b - c) * (a + (0.8333333333333334 - (0.6666666666666666 / t)))))))))
	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 = sqrt(Float64(t + a))
	tmp = 0.0
	if (Float64(Float64(Float64(z * t_1) / t) - Float64(Float64(b - c) * Float64(Float64(a + 0.8333333333333334) - Float64(2.0 / Float64(t * 3.0))))) <= Inf)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(t_1 * Float64(z / t)) - Float64(Float64(b - c) * Float64(a + Float64(0.8333333333333334 - Float64(0.6666666666666666 / t))))))))));
	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 = sqrt((t + a));
	tmp = 0.0;
	if ((((z * t_1) / t) - ((b - c) * ((a + 0.8333333333333334) - (2.0 / (t * 3.0))))) <= Inf)
		tmp = x / (x + (y * exp((2.0 * ((t_1 * (z / t)) - ((b - c) * (a + (0.8333333333333334 - (0.6666666666666666 / t)))))))));
	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[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(z * t$95$1), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + 0.8333333333333334), $MachinePrecision] - 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[(N[(t$95$1 * N[(z / t), $MachinePrecision]), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(a + N[(0.8333333333333334 - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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	3.8
Target	2.9
Herbie	1.7

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

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

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

   2. Simplified 0.5

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

      [Start] 0.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      rational.json-simplify-49 [=>] 0.5	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\sqrt{t + a} \cdot \frac{z}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      rational.json-simplify-1 [=>] 0.5	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{t \cdot 3}\right)\right)}} \]
      rational.json-simplify-48 [=>] 0.5	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \color{blue}{\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)\right)}\right)}} \]
      metadata-eval [=>] 0.5	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \left(a + \left(\color{blue}{0.8333333333333334} - \frac{2}{t \cdot 3}\right)\right)\right)}} \]
      rational.json-simplify-46 [=>] 0.5	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \left(a + \left(0.8333333333333334 - \color{blue}{\frac{\frac{2}{t}}{3}}\right)\right)\right)}} \]
      rational.json-simplify-44 [=>] 0.5	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \left(a + \left(0.8333333333333334 - \color{blue}{\frac{\frac{2}{3}}{t}}\right)\right)\right)}} \]
      metadata-eval [=>] 0.5	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \left(a + \left(0.8333333333333334 - \frac{\color{blue}{0.6666666666666666}}{t}\right)\right)\right)}} \]

   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. Simplified 53.2

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

      [Start] 64.0	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      rational.json-simplify-49 [=>] 53.2	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\sqrt{t + a} \cdot \frac{z}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      rational.json-simplify-1 [=>] 53.2	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{t \cdot 3}\right)\right)}} \]
      rational.json-simplify-48 [=>] 53.2	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \color{blue}{\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)\right)}\right)}} \]
      metadata-eval [=>] 53.2	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \left(a + \left(\color{blue}{0.8333333333333334} - \frac{2}{t \cdot 3}\right)\right)\right)}} \]
      rational.json-simplify-46 [=>] 53.2	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \left(a + \left(0.8333333333333334 - \color{blue}{\frac{\frac{2}{t}}{3}}\right)\right)\right)}} \]
      rational.json-simplify-44 [=>] 53.2	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \left(a + \left(0.8333333333333334 - \color{blue}{\frac{\frac{2}{3}}{t}}\right)\right)\right)}} \]
      metadata-eval [=>] 53.2	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \left(a + \left(0.8333333333333334 - \frac{\color{blue}{0.6666666666666666}}{t}\right)\right)\right)}} \]

   3. Taylor expanded in a around inf 26.5

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + 0.8333333333333334\right) - \frac{2}{t \cdot 3}\right) \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot \left(a + \left(0.8333333333333334 - \frac{0.6666666666666666}{t}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.6
Cost	14672

\[\begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\sqrt{a} \cdot z - -0.6666666666666666 \cdot \left(b - c\right)}{t}}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(0.6666666666666666 \cdot \frac{1}{t} - \left(0.8333333333333334 + a\right)\right) \cdot b\right)}}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(a + 0.8333333333333334\right) - \frac{1}{t} \cdot 0.6666666666666666\right)\right)}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t} - \left(b - c\right) \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-\left(a + 0.8333333333333334\right)\right)\right)}}\\ \end{array} \]

Alternative 2
Error	9.6
Cost	14160

\[\begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-201}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\sqrt{a} \cdot z - -0.6666666666666666 \cdot \left(b - c\right)}{t}}}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(0.6666666666666666 \cdot \frac{1}{t} - \left(0.8333333333333334 + a\right)\right) \cdot b\right)}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-82}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(a + 0.8333333333333334\right) - \frac{1}{t} \cdot 0.6666666666666666\right)\right)}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{x + e^{2 \cdot \left(\sqrt{\frac{1}{t}} \cdot z\right)} \cdot y}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\frac{x}{x + e^{\left(b - c\right) \cdot -1.6666666666666667} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-\left(a + 0.8333333333333334\right)\right)\right)}}\\ \end{array} \]

Alternative 3
Error	12.7
Cost	8012

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-\left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-152}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(0.6666666666666666 \cdot \frac{b}{t}\right)}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(a + 0.8333333333333334\right) - \frac{1}{t} \cdot 0.6666666666666666\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	12.9
Cost	7880

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(0.6666666666666666 \cdot \frac{1}{t} - \left(0.8333333333333334 + a\right)\right) \cdot b\right)}}\\ \mathbf{if}\;b \leq -7.8 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 0.0024:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(a + 0.8333333333333334\right) - \frac{1}{t} \cdot 0.6666666666666666\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	13.5
Cost	7820

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-\left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-152}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(0.6666666666666666 \cdot \frac{b}{t}\right)}}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \frac{-0.6666666666666666}{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	17.1
Cost	7760

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(0.6666666666666666 \cdot \frac{b}{t}\right)}}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-260}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \frac{-0.6666666666666666}{t}\right)}}\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + e^{\left(b - c\right) \cdot -1.6666666666666667} \cdot y}\\ \end{array} \]

Alternative 7
Error	20.7
Cost	7632

\[\begin{array}{l} t_1 := \frac{x}{x + e^{\left(b - c\right) \cdot -1.6666666666666667} \cdot y}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-114}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{y \cdot e^{-1.3333333333333333 \cdot \frac{c}{t}}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{y \cdot e^{-2 \cdot \left(a \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	19.8
Cost	7632

\[\begin{array}{l} t_1 := \frac{x}{x + e^{\left(b - c\right) \cdot -1.6666666666666667} \cdot y}\\ \mathbf{if}\;t \leq -4.3 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-114}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{x + y}{x}}{\frac{\left(x + y\right) \cdot \left(2 \cdot \left(x + y\right)\right)}{x}} \cdot \frac{x}{0.5}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{y \cdot e^{-2 \cdot \left(a \cdot b\right)} + x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	18.2
Cost	7496

\[\begin{array}{l} t_1 := \frac{x}{x + e^{\left(b - c\right) \cdot -1.6666666666666667} \cdot y}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(0.6666666666666666 \cdot \frac{b}{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	17.1
Cost	7496

\[\begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-261}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(0.6666666666666666 \cdot \frac{b}{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + e^{\left(b - c\right) \cdot -1.6666666666666667} \cdot y}\\ \end{array} \]

Alternative 11
Error	28.1
Cost	7372

\[\begin{array}{l} t_1 := \frac{x}{e^{-1.6666666666666667 \cdot \left(b - c\right)} \cdot y}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3.4 \cdot 10^{+98}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -2.35 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-158}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-231}:\\ \;\;\;\;-1 + \left(1 - \frac{x}{-\left(x + \left(y + y \cdot \left(2 \cdot \left(c \cdot \left(0.8333333333333334 + \left(a - 0.6666666666666666 \cdot \frac{1}{t}\right)\right)\right)\right)\right)\right)}\right)\\ \mathbf{elif}\;b \leq 10^{-124}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 0.00175:\\ \;\;\;\;-1 + \left(1 - \frac{-x}{y + \left(x + y \cdot \left(\frac{c}{t} \cdot -1.3333333333333333\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	31.6
Cost	2456

\[\begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+187}:\\ \;\;\;\;\frac{x}{x + \left(y + y \cdot \left(2 \cdot \left(a \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{+37}:\\ \;\;\;\;-1 + \left(1 - \frac{-x}{y + \left(x + y \cdot \left(\frac{c}{t} \cdot -1.3333333333333333\right)\right)}\right)\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-69}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-126}:\\ \;\;\;\;-1 + \left(1 - \frac{x}{-\left(x + y\right)}\right)\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-172}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-250}:\\ \;\;\;\;-1 + \left(1 - \frac{x}{-\left(x + \left(y + y \cdot \left(2 \cdot \left(c \cdot \left(0.8333333333333334 + \left(a - 0.6666666666666666 \cdot \frac{1}{t}\right)\right)\right)\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	33.6
Cost	2400

\[\begin{array}{l} t_1 := \frac{x + y}{x}\\ t_2 := -1 + \left(1 - \frac{-x}{y + \left(x + y \cdot \left(\frac{c}{t} \cdot -1.3333333333333333\right)\right)}\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+168}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.55 \cdot 10^{-68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-273}:\\ \;\;\;\;-1 + \left(1 - \frac{x}{-\left(x + y\right)}\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-280}:\\ \;\;\;\;\left(2 \cdot \frac{y}{x}\right) \cdot \frac{x}{\frac{t_1}{\frac{0.5}{x + y}}}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{-1}{\frac{\left(x + y\right) \cdot \left(-\frac{x + y}{x \cdot x}\right)}{t_1}}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-130}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{t_1}{\frac{\left(x + y\right) \cdot \left(2 \cdot \left(x + y\right)\right)}{x}} \cdot \frac{x}{0.5}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+85}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{x + \left(y + \left(y \cdot \left(c - b\right)\right) \cdot \left(a + a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	30.9
Cost	2076

\[\begin{array}{l} \mathbf{if}\;z \leq -1.76 \cdot 10^{+171}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -9 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{x + y}{x}}{\frac{\left(x + y\right) \cdot \left(2 \cdot \left(x + y\right)\right)}{x}} \cdot \frac{x}{0.5}\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{+133}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(c \cdot \left(y \cdot \left(\left(0.8333333333333334 + a\right) - 0.6666666666666666 \cdot \frac{1}{t}\right)\right)\right)\right)}\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-289}:\\ \;\;\;\;\frac{x}{x + \left(y + 1.3333333333333333 \cdot \left(y \cdot \frac{b}{t}\right)\right)}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{-x}{y + \left(x + y \cdot \left(\frac{c}{t} \cdot -1.3333333333333333\right)\right)}\right)\\ \end{array} \]

Alternative 15
Error	30.6
Cost	2016

\[\begin{array}{l} t_1 := -1 + \left(1 - \frac{x}{-\left(x + y\right)}\right)\\ t_2 := \frac{x}{x + \left(y + \left(y \cdot \left(c - b\right)\right) \cdot \left(a + a\right)\right)}\\ \mathbf{if}\;c \leq -2.7 \cdot 10^{-130}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -8.6 \cdot 10^{-284}:\\ \;\;\;\;\frac{x}{x + \left(y + 1.3333333333333333 \cdot \left(y \cdot \frac{b}{t}\right)\right)}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{-142}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{+92}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+208}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{+250}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{+270}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	31.7
Cost	1944

\[\begin{array}{l} t_1 := -1 + \left(1 - \frac{-x}{y + \left(x + y \cdot \left(\frac{c}{t} \cdot -1.3333333333333333\right)\right)}\right)\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+187}:\\ \;\;\;\;\frac{x}{x + \left(y + y \cdot \left(2 \cdot \left(a \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-69}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-128}:\\ \;\;\;\;-1 + \left(1 - \frac{x}{-\left(x + y\right)}\right)\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-172}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	30.7
Cost	1360

\[\begin{array}{l} t_1 := -1 + \left(1 - \frac{x}{-\left(x + y\right)}\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+134}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-288}:\\ \;\;\;\;\frac{x}{x + \left(y + 1.3333333333333333 \cdot \left(y \cdot \frac{b}{t}\right)\right)}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+82}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	31.1
Cost	1360

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+133}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(y \cdot \left(c \cdot a\right)\right)\right)}\\ \mathbf{elif}\;z \leq -2.65 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-288}:\\ \;\;\;\;\frac{x}{x + \left(y + 1.3333333333333333 \cdot \left(y \cdot \frac{b}{t}\right)\right)}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{x}{-\left(x + y\right)}\right)\\ \end{array} \]

Alternative 19
Error	30.3
Cost	1300

\[\begin{array}{l} t_1 := -1 + \left(1 - \frac{x}{-\left(x + y\right)}\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{+133}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-289}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+23}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	32.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-179}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-275}:\\ \;\;\;\;-0.75 \cdot \left(x \cdot \frac{\frac{t}{c}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 21
Error	32.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-177}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-274}:\\ \;\;\;\;-0.75 \cdot \left(\frac{t}{c} \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 22
Error	32.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-179}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-275}:\\ \;\;\;\;-0.75 \cdot \frac{t \cdot x}{c \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 23
Error	32.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-294}:\\ \;\;\;\;\frac{0.5}{y \cdot \frac{c}{\frac{x}{a}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 24
Error	32.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-173}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 25
Error	31.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-226}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-289}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 26
Error	31.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023074 
(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)))))))))))