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

Average Error: 6.2% → 2.94%

Time: 33.7s

Precision: binary64

Cost: 33672

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{t + a}\\ \mathbf{if}\;t \leq -225000:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z}{\frac{t}{t_1}} + \left(\left(a + 0.8333333333333334\right) + \frac{-2}{t \cdot 3}\right) \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-280}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{z \cdot \sqrt{a} + \left(b - c\right) \cdot 0.6666666666666666}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(b - c, \left(-0.8333333333333334 - a\right) - \frac{-0.6666666666666666}{t}, z \cdot \frac{t_1}{t}\right)\right)}, x\right)}\\ \end{array} \]

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 (<= t -225000.0)
     (/
      x
      (+
       x
       (*
        y
        (pow
         (exp 2.0)
         (+
          (/ z (/ t t_1))
          (* (+ (+ a 0.8333333333333334) (/ -2.0 (* t 3.0))) (- c b)))))))
     (if (<= t -1.02e-280)
       (/
        x
        (+
         x
         (*
          y
          (exp
           (* 2.0 (/ (+ (* z (sqrt a)) (* (- b c) 0.6666666666666666)) t))))))
       (/
        x
        (fma
         y
         (pow
          (exp 2.0)
          (fma
           (- b c)
           (- (- -0.8333333333333334 a) (/ -0.6666666666666666 t))
           (* z (/ t_1 t))))
         x))))))

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 (t <= -225000.0) {
		tmp = x / (x + (y * pow(exp(2.0), ((z / (t / t_1)) + (((a + 0.8333333333333334) + (-2.0 / (t * 3.0))) * (c - b))))));
	} else if (t <= -1.02e-280) {
		tmp = x / (x + (y * exp((2.0 * (((z * sqrt(a)) + ((b - c) * 0.6666666666666666)) / t)))));
	} else {
		tmp = x / fma(y, pow(exp(2.0), fma((b - c), ((-0.8333333333333334 - a) - (-0.6666666666666666 / t)), (z * (t_1 / t)))), x);
	}
	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 (t <= -225000.0)
		tmp = Float64(x / Float64(x + Float64(y * (exp(2.0) ^ Float64(Float64(z / Float64(t / t_1)) + Float64(Float64(Float64(a + 0.8333333333333334) + Float64(-2.0 / Float64(t * 3.0))) * Float64(c - b)))))));
	elseif (t <= -1.02e-280)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(a)) + Float64(Float64(b - c) * 0.6666666666666666)) / t))))));
	else
		tmp = Float64(x / fma(y, (exp(2.0) ^ fma(Float64(b - c), Float64(Float64(-0.8333333333333334 - a) - Float64(-0.6666666666666666 / t)), Float64(z * Float64(t_1 / t)))), x));
	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[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -225000.0], N[(x / N[(x + N[(y * N[Power[N[Exp[2.0], $MachinePrecision], N[(N[(z / N[(t / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a + 0.8333333333333334), $MachinePrecision] + N[(-2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.02e-280], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[Power[N[Exp[2.0], $MachinePrecision], N[(N[(b - c), $MachinePrecision] * N[(N[(-0.8333333333333334 - a), $MachinePrecision] - N[(-0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision] + N[(z * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.2%
Target	4.97%
Herbie	2.94%

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

Derivation

1. Split input into 3 regimes

2. if t < -225000

   1. Initial program 7.28

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

   2. Simplified 0.99

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

      [Start] 7.28	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      exp-prod [=>] 7.28	\[ \frac{x}{x + y \cdot \color{blue}{{\left(e^{2}\right)}^{\left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}} \]
      associate-/l* [=>] 0.99	\[ \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\color{blue}{\frac{z}{\frac{t}{\sqrt{t + a}}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      metadata-eval [=>] 0.99	\[ \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \color{blue}{0.8333333333333334}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]

   if -225000 < t < -1.02000000000000005e-280

   1. Initial program 7.96

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

   2. Taylor expanded in t around 0 3.62

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

   if -1.02000000000000005e-280 < t

   1. Initial program 5.73

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

   2. Simplified 2.95

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

      [Start] 5.73	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      +-commutative [=>] 5.73	\[ \frac{x}{\color{blue}{y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)} + x}} \]
      fma-def [=>] 5.73	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(y, e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}, x\right)}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.94

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -225000:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z}{\frac{t}{\sqrt{t + a}}} + \left(\left(a + 0.8333333333333334\right) + \frac{-2}{t \cdot 3}\right) \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-280}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{z \cdot \sqrt{a} + \left(b - c\right) \cdot 0.6666666666666666}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(b - c, \left(-0.8333333333333334 - a\right) - \frac{-0.6666666666666666}{t}, z \cdot \frac{\sqrt{t + a}}{t}\right)\right)}, x\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.95%
Cost	22468

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

Alternative 2
Error	2.59%
Cost	21257

\[\begin{array}{l} \mathbf{if}\;t \leq -170000 \lor \neg \left(t \leq 9.6 \cdot 10^{-261}\right):\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z}{\frac{t}{\sqrt{t + a}}} + \left(\left(a + 0.8333333333333334\right) + \frac{-2}{t \cdot 3}\right) \cdot \left(c - b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{z \cdot \sqrt{a} + \left(b - c\right) \cdot 0.6666666666666666}{t}}}\\ \end{array} \]

Alternative 3
Error	15.39%
Cost	14544

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \frac{z \cdot \sqrt{a} + \left(b - c\right) \cdot 0.6666666666666666}{t}}}\\ \mathbf{if}\;t \leq -4 \cdot 10^{+155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{-130}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right)\right)}}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(z \cdot \sqrt{\frac{1}{t}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \end{array} \]

Alternative 4
Error	22.07%
Cost	14288

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right)\right)}}\\ \mathbf{if}\;c \leq -4 \cdot 10^{+251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -1.65 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{b}{\frac{1}{\left(-0.8333333333333334 - a\right) - \frac{-0.6666666666666666}{t}}}}}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-179}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t}\right)}}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(b \cdot \left(\left(a + 0.8333333333333334\right) + \frac{-0.6666666666666666}{t}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	19%
Cost	14028

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right)\right)}}\\ \mathbf{elif}\;t \leq 0.048:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(z \cdot \sqrt{\frac{1}{t}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	40.76%
Cost	7899

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+65} \lor \neg \left(z \leq -1.35 \cdot 10^{-49}\right) \land \left(z \leq -7 \cdot 10^{-171} \lor \neg \left(z \leq 2 \cdot 10^{-285} \lor \neg \left(z \leq 4 \cdot 10^{+149}\right) \land z \leq 10^{+184}\right)\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	41.31%
Cost	7897

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{if}\;z \leq -5 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-47}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-173}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(a \cdot b\right)}}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+143} \lor \neg \left(z \leq 2.2 \cdot 10^{+190}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	29.93%
Cost	7892

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{if}\;a \leq -0.82:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(-0.6666666666666666 \cdot \frac{c}{t}\right)}}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - \left(a \cdot c + \left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	29.85%
Cost	7760

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(-0.6666666666666666 \cdot \frac{c}{t}\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{-286}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	21.25%
Cost	7756

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(-0.6666666666666666 \cdot \frac{c}{t}\right)}}\\ \mathbf{elif}\;t \leq 620:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	18.34%
Cost	7753

\[\begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-280} \lor \neg \left(t \leq 0.8\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right)\right)}}\\ \end{array} \]

Alternative 12
Error	36.52%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;b - c \leq -50000000000000:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{-129}:\\ \;\;\;\;\frac{x}{x - y \cdot \left(-1 + 2 \cdot \left(a \cdot \left(b - c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	50.82%
Cost	2016

\[\begin{array}{l} t_1 := \frac{x}{x - y \cdot \left(-1 + 2 \cdot \left(a \cdot \left(b - c\right)\right)\right)}\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{-213}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq -2.25 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 1.3333333333333333 \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+162}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+195}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	50.65%
Cost	2016

\[\begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-213}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 1.3333333333333333 \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.25 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+164}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+195}:\\ \;\;\;\;\frac{x}{x - y \cdot \left(-1 + 2 \cdot \left(a \cdot \left(b - c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Error	50.33%
Cost	1624

\[\begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-213}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 1.3333333333333333 \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{x + \left(y - -2 \cdot \left(c \cdot \left(y \cdot a\right)\right)\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	48.18%
Cost	1364

\[\begin{array}{l} t_1 := \frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{if}\;c \leq -6.5 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -1.16 \cdot 10^{-229}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 0.00185:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{+223}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + c \cdot \left(a \cdot \left(y \cdot 2\right)\right)}\\ \end{array} \]

Alternative 17
Error	49.15%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+205}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.45 \cdot 10^{+257} \lor \neg \left(y \leq 3.4 \cdot 10^{+289}\right):\\ \;\;\;\;\frac{x}{y - -2 \cdot \left(y \cdot \left(a \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	49.64%
Cost	968

\[\begin{array}{l} \mathbf{if}\;a \leq 2.15 \cdot 10^{+175}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+264}:\\ \;\;\;\;\frac{x}{x + c \cdot \left(a \cdot \left(y \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Error	48.28%
Cost	840

\[\begin{array}{l} \mathbf{if}\;c \leq 3.5 \cdot 10^{+237}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{+272}:\\ \;\;\;\;0.5 \cdot \frac{x}{c \cdot \left(y \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Error	48.13%
Cost	840

\[\begin{array}{l} \mathbf{if}\;c \leq 2.05 \cdot 10^{+237}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+277}:\\ \;\;\;\;0.5 \cdot \frac{x}{y \cdot \left(a \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 21
Error	48.22%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023121 
(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)))))))))))