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

Average Error: 3.8 → 2.1

Time: 39.4s

Precision: binary64

Cost: 41220

Math

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

↓

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

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 (+ (/ 0.6666666666666666 t) (- -0.8333333333333334 a)))
        (t_2 (sqrt (+ t a))))
   (if (<=
        (+
         (/ (* z t_2) t)
         (* (+ (+ a 0.8333333333333334) (/ -2.0 (* t 3.0))) (- c b)))
        INFINITY)
     (/ x (fma y (pow (exp 2.0) (fma (- b c) t_1 (* z (/ t_2 t)))) x))
     (/ x (+ x (* y (exp (* 2.0 (* b t_1)))))))))

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 = (0.6666666666666666 / t) + (-0.8333333333333334 - a);
	double t_2 = sqrt((t + a));
	double tmp;
	if ((((z * t_2) / t) + (((a + 0.8333333333333334) + (-2.0 / (t * 3.0))) * (c - b))) <= ((double) INFINITY)) {
		tmp = x / fma(y, pow(exp(2.0), fma((b - c), t_1, (z * (t_2 / t)))), x);
	} else {
		tmp = x / (x + (y * exp((2.0 * (b * t_1)))));
	}
	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(0.6666666666666666 / t) + Float64(-0.8333333333333334 - a))
	t_2 = sqrt(Float64(t + a))
	tmp = 0.0
	if (Float64(Float64(Float64(z * t_2) / t) + Float64(Float64(Float64(a + 0.8333333333333334) + Float64(-2.0 / Float64(t * 3.0))) * Float64(c - b))) <= Inf)
		tmp = Float64(x / fma(y, (exp(2.0) ^ fma(Float64(b - c), t_1, Float64(z * Float64(t_2 / t)))), x));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(b * t_1))))));
	end
	return 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[(0.6666666666666666 / t), $MachinePrecision] + N[(-0.8333333333333334 - a), $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] + N[(N[(N[(a + 0.8333333333333334), $MachinePrecision] + N[(-2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(x / N[(y * N[Power[N[Exp[2.0], $MachinePrecision], N[(N[(b - c), $MachinePrecision] * t$95$1 + N[(z * N[(t$95$2 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.8
Target	3.2
Herbie	2.1

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

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 0.9

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

   2. Simplified 0.8

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

      [Start] 0.9	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      +-commutative [=>] 0.9	\[ \frac{x}{\color{blue}{y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)} + x}} \]
      fma-def [=>] 0.9	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(y, e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}, x\right)}} \]

   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 27.8

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

   3. Simplified 27.8

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

      [Start] 27.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\left(0.6666666666666666 \cdot \frac{1}{t} - \left(0.8333333333333334 + a\right)\right) \cdot b\right)}} \]
      *-commutative [=>] 27.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(b \cdot \left(0.6666666666666666 \cdot \frac{1}{t} - \left(0.8333333333333334 + a\right)\right)\right)}}} \]
      associate-*r/ [=>] 27.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\color{blue}{\frac{0.6666666666666666 \cdot 1}{t}} - \left(0.8333333333333334 + a\right)\right)\right)}} \]
      metadata-eval [=>] 27.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{\color{blue}{0.6666666666666666}}{t} - \left(0.8333333333333334 + a\right)\right)\right)}} \]
      +-commutative [=>] 27.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} - \color{blue}{\left(a + 0.8333333333333334\right)}\right)\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.1

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

Alternatives

Alternative 1
Error	1.9
Cost	28804

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

Alternative 2
Error	2.1
Cost	22468

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

Alternative 3
Error	19.5
Cost	8156

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(-0.6666666666666666 \cdot \frac{c}{t}\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ t_3 := \frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{if}\;a \leq -0.82:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -7 \cdot 10^{-121}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{-209}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-190}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	19.4
Cost	8156

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(-0.6666666666666666 \cdot \frac{c}{t}\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ t_3 := \frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{if}\;a \leq -0.82:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -7 \cdot 10^{-121}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -5.5 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{0.6666666666666666}{\frac{t}{b}}}}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	31.7
Cost	7776

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y - \frac{\left(c \cdot c - b \cdot b\right) \cdot \left(\left(a \cdot y\right) \cdot -2\right)}{b + c}\right)}\\ \mathbf{if}\;b \leq -1.15 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4.6 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-259}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - 2 \cdot \left(c \cdot \left(-0.8333333333333334 - a\right)\right)\right)}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-124}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.16 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{+52}:\\ \;\;\;\;\left(x - y\right) \cdot \left(x \cdot \frac{-1}{y \cdot y - x \cdot x}\right)\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{+65}:\\ \;\;\;\;\log \left(1 + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	13.0
Cost	7753

\[\begin{array}{l} \mathbf{if}\;c \leq -1300000000 \lor \neg \left(c \leq 0.365\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(c \cdot \left(\left(\frac{0.6666666666666666}{t} + -0.8333333333333334\right) - a\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ \end{array} \]

Alternative 7
Error	20.2
Cost	7632

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{if}\;t \leq -9 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-107}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{y \cdot e^{\frac{c}{t} \cdot -1.3333333333333333}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-37}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	13.7
Cost	7625

\[\begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{-287} \lor \neg \left(t \leq 0.0018\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(-0.6666666666666666 \cdot \frac{c}{t}\right)}}\\ \end{array} \]

Alternative 9
Error	18.4
Cost	7497

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-285} \lor \neg \left(t \leq 0.00156\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(-0.6666666666666666 \cdot \frac{c}{t}\right)}}\\ \end{array} \]

Alternative 10
Error	24.1
Cost	7236

\[\begin{array}{l} \mathbf{if}\;b - c \leq -4 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{-206}:\\ \;\;\;\;1\\ \mathbf{elif}\;b - c \leq 2.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	32.7
Cost	2016

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y - x\right)}{y - x}}{x + y}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{+145}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + -2 \cdot \left(c \cdot \left(-0.8333333333333334 + \left(\frac{0.6666666666666666}{t} - a\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+150}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+224}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{x \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	33.8
Cost	1884

\[\begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+42}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-301}:\\ \;\;\;\;\frac{x}{x + \left(y - \frac{\left(c \cdot c - b \cdot b\right) \cdot \left(\left(a \cdot y\right) \cdot -2\right)}{b + c}\right)}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y - x\right)}{y - x}}{x + y}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+66}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+145}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + -2 \cdot \left(c \cdot \left(-0.8333333333333334 + \left(\frac{0.6666666666666666}{t} - a\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+150}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+213}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{x \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	33.5
Cost	1757

\[\begin{array}{l} \mathbf{if}\;b \leq -2.35 \cdot 10^{+94}:\\ \;\;\;\;-0.5 \cdot \frac{x}{a \cdot \left(b \cdot y\right)}\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -4.6 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y - x\right)}{y - x}}{x + y}\\ \mathbf{elif}\;b \leq -8.6 \cdot 10^{-277}:\\ \;\;\;\;\frac{x}{x + 2 \cdot \left(y \cdot \left(a \cdot c\right)\right)}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-243} \lor \neg \left(b \leq 1.05 \cdot 10^{-128}\right) \land b \leq 1.38 \cdot 10^{-52}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{y \cdot y - x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	32.8
Cost	1752

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{if}\;x \leq -7 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-303}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{x \cdot \left(y - x\right)}{y - x}}{x + y}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-21}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+167}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{y \cdot y - x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	30.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+172} \lor \neg \left(y \leq 5 \cdot 10^{+177}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	30.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+97}:\\ \;\;\;\;-0.5 \cdot \frac{x}{a \cdot \left(b \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	31.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-303}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-221}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	31.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-303}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-221}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Error	30.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023039 
(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)))))))))))