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

Average Error: 4.0 → 1.9

Time: 34.6s

Precision: binary64

Cost: 28804

Math

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

↓

\[\begin{array}{l} t_1 := \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} + \left(-0.8333333333333334 - a\right)\right)\\ t_2 := \sqrt{t + a}\\ \mathbf{if}\;\frac{z \cdot t_2}{t} + t_1 \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z}{\frac{t}{t_2}} + t_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ \end{array} \]

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 (* (- 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
         (*
          2.0
          (* b (+ (/ 0.6666666666666666 t) (- -0.8333333333333334 a)))))))))))

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((2.0 * (b * ((0.6666666666666666 / t) + (-0.8333333333333334 - a)))))));
	}
	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((2.0 * (b * ((0.6666666666666666 / t) + (-0.8333333333333334 - a)))))));
	}
	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((2.0 * (b * ((0.6666666666666666 / t) + (-0.8333333333333334 - a)))))))
	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(2.0 * Float64(b * Float64(Float64(0.6666666666666666 / t) + Float64(-0.8333333333333334 - a))))))));
	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((2.0 * (b * ((0.6666666666666666 / t) + (-0.8333333333333334 - a)))))));
	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[(2.0 * N[(b * N[(N[(0.6666666666666666 / t), $MachinePrecision] + N[(-0.8333333333333334 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	4.0
Target	3.2
Herbie	1.9

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

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 0.6

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

      [Start] 0.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      exp-prod [=>] 0.8	\[ \frac{x}{x + y \cdot \color{blue}{{\left(e^{2}\right)}^{\left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}} \]
      associate-/l* [=>] 0.6	\[ \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\color{blue}{\frac{z}{\frac{t}{\sqrt{t + a}}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      metadata-eval [=>] 0.6	\[ \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \color{blue}{0.8333333333333334}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]

   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 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 b around inf 26.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot \frac{1}{t} - \left(0.8333333333333334 + a\right)\right) \cdot b\right)}}} \]

   3. Simplified 26.8

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

      [Start] 26.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\left(0.6666666666666666 \cdot \frac{1}{t} - \left(0.8333333333333334 + a\right)\right) \cdot b\right)}} \]
      *-commutative [=>] 26.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(b \cdot \left(0.6666666666666666 \cdot \frac{1}{t} - \left(0.8333333333333334 + a\right)\right)\right)}}} \]
      associate-*r/ [=>] 26.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\color{blue}{\frac{0.6666666666666666 \cdot 1}{t}} - \left(0.8333333333333334 + a\right)\right)\right)}} \]
      metadata-eval [=>] 26.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{\color{blue}{0.6666666666666666}}{t} - \left(0.8333333333333334 + a\right)\right)\right)}} \]
      +-commutative [=>] 26.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} - \color{blue}{\left(a + 0.8333333333333334\right)}\right)\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.9

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

Alternatives

Alternative 1
Error	2.1
Cost	33408

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

Alternative 2
Error	2.1
Cost	22468

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

Alternative 3
Error	13.3
Cost	14156

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 - \frac{0.6666666666666666}{t}\right)\right)\right)}}\\ \mathbf{if}\;b \leq -1.4 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-237}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-204}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t}\right)}}\\ \mathbf{elif}\;b \leq 24000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	33.0
Cost	8560

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y + \left(y \cdot \frac{\left(b - c\right) \cdot \left(b + c\right)}{b + c}\right) \cdot \left(a \cdot -2\right)\right)}\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+86}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-141}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-300}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{x + \left(y + 1.3333333333333333 \cdot \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 3400000:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+297}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{1.3333333333333333 \cdot \frac{b}{t}}}\\ \end{array} \]

Alternative 5
Error	19.5
Cost	8156

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot -0.8333333333333334\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{if}\;a \leq -0.00044:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.16 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.95 \cdot 10^{-224}:\\ \;\;\;\;\frac{x}{y \cdot e^{1.3333333333333333 \cdot \frac{b}{t}}}\\ \mathbf{elif}\;a \leq -3.3 \cdot 10^{-242}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-181}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{x + 2 \cdot \left(\sqrt{a} \cdot \frac{y}{\frac{t}{z}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	21.5
Cost	8025

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{if}\;a \leq -24:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.16 \cdot 10^{-207}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq -1.95 \cdot 10^{-224}:\\ \;\;\;\;\frac{x}{y \cdot e^{1.3333333333333333 \cdot \frac{b}{t}}}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-181}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.86 \cdot 10^{-62} \lor \neg \left(a \leq 6 \cdot 10^{-35}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 2 \cdot \left(\sqrt{a} \cdot \frac{y}{\frac{t}{z}}\right)}\\ \end{array} \]

Alternative 7
Error	13.5
Cost	7888

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \frac{b \cdot 0.6666666666666666}{t}}}\\ \mathbf{if}\;t \leq -4 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-236}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.32:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	11.7
Cost	7753

\[\begin{array}{l} \mathbf{if}\;t \leq -3.45 \cdot 10^{-279} \lor \neg \left(t \leq 0.48\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ \end{array} \]

Alternative 9
Error	12.4
Cost	7753

\[\begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+18} \lor \neg \left(b \leq 1800\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 - \frac{0.6666666666666666}{t}\right)\right)\right)}}\\ \end{array} \]

Alternative 10
Error	32.9
Cost	2924

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y + \left(y \cdot \frac{\left(b - c\right) \cdot \left(b + c\right)}{b + c}\right) \cdot \left(a \cdot -2\right)\right)}\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+86}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.45 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-304}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{x + \left(y + 1.3333333333333333 \cdot \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 20000:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+297}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y + 1.3333333333333333 \cdot \frac{y}{\frac{t}{b}}\right)}\\ \end{array} \]

Alternative 11
Error	31.7
Cost	2272

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot \left(1 + 2 \cdot \left(c \cdot \left(0.8333333333333334 + \left(a + \frac{-0.6666666666666666}{t}\right)\right)\right)\right)}\\ t_2 := \frac{x}{x - y \cdot \left(-1 - a \cdot \left(2 \cdot \left(c - b\right)\right)\right)}\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{+257}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{+211}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-253}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\frac{0.5}{c}}{y \cdot \frac{a}{x}}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	32.8
Cost	2140

\[\begin{array}{l} t_1 := \frac{x}{x - y \cdot \left(-1 - a \cdot \left(2 \cdot \left(c - b\right)\right)\right)}\\ \mathbf{if}\;b \leq -7.1 \cdot 10^{+252}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.7 \cdot 10^{+214}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -2.55 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.25 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.58 \cdot 10^{-254}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + 2 \cdot \left(c \cdot a + \left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	31.0
Cost	2016

\[\begin{array}{l} t_1 := \frac{x}{x - y \cdot \left(-1 - a \cdot \left(2 \cdot \left(c - b\right)\right)\right)}\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{+260}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -6 \cdot 10^{+210}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -7.8 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-93}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 3.45 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	32.4
Cost	1492

\[\begin{array}{l} t_1 := 0.5 \cdot \frac{x}{y \cdot \left(a \cdot \left(c - b\right)\right)}\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -4.85 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-274}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+249}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \end{array} \]

Alternative 15
Error	32.3
Cost	1492

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \frac{x}{y \cdot \left(a \cdot \left(c - b\right)\right)}\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{-273}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-302}:\\ \;\;\;\;\frac{x}{x + \left(y + 1.3333333333333333 \cdot \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+249}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \end{array} \]

Alternative 16
Error	30.8
Cost	1104

\[\begin{array}{l} \mathbf{if}\;c \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 1.55 \cdot 10^{+70}:\\ \;\;\;\;-0.5 \cdot \frac{x}{y \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{+239}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 2.55 \cdot 10^{+306}:\\ \;\;\;\;0.5 \cdot \frac{x}{c \cdot \left(y \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	30.6
Cost	1104

\[\begin{array}{l} \mathbf{if}\;c \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{+70}:\\ \;\;\;\;-0.5 \cdot \frac{x}{y \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{+243}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{+306}:\\ \;\;\;\;0.5 \cdot \frac{x}{a \cdot \left(y \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	30.5
Cost	1104

\[\begin{array}{l} \mathbf{if}\;c \leq 2 \cdot 10^{+51}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{+70}:\\ \;\;\;\;-0.5 \cdot \frac{x}{y \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{+221}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{x}{\frac{c \cdot a}{0.5}}}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Error	32.2
Cost	1101

\[\begin{array}{l} \mathbf{if}\;a \leq 1.08 \cdot 10^{+105}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+155} \lor \neg \left(a \leq 5 \cdot 10^{+267}\right):\\ \;\;\;\;0.5 \cdot \frac{x}{y \cdot \left(a \cdot \left(c - b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Error	31.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-274}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-302}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 21
Error	31.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-269}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-301}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 22
Error	30.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023031 
(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)))))))))))