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

Average Accuracy: 93.9% → 95.8%

Time: 44.3s

Precision: binary64

Cost: 48132

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{t + a}\\ \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot t_1}{t} + \left(b - c\right) \cdot \left(\left(-0.8333333333333334 - a\right) - \frac{-2}{t \cdot 3}\right)\right)}} \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(b - c, \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right), z \cdot \frac{t_1}{t}\right)\right)}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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 (<=
        (/
         x
         (+
          x
          (*
           y
           (exp
            (*
             2.0
             (+
              (/ (* z t_1) t)
              (*
               (- b c)
               (- (- -0.8333333333333334 a) (/ -2.0 (* t 3.0))))))))))
        2e-147)
     (/
      x
      (fma
       y
       (pow
        (exp 2.0)
        (fma
         (- b c)
         (- (/ 0.6666666666666666 t) (+ a 0.8333333333333334))
         (* z (/ t_1 t))))
       x))
     1.0)))

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 ((x / (x + (y * exp((2.0 * (((z * t_1) / t) + ((b - c) * ((-0.8333333333333334 - a) - (-2.0 / (t * 3.0)))))))))) <= 2e-147) {
		tmp = x / fma(y, pow(exp(2.0), fma((b - c), ((0.6666666666666666 / t) - (a + 0.8333333333333334)), (z * (t_1 / t)))), x);
	} else {
		tmp = 1.0;
	}
	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(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * t_1) / t) + Float64(Float64(b - c) * Float64(Float64(-0.8333333333333334 - a) - Float64(-2.0 / Float64(t * 3.0)))))))))) <= 2e-147)
		tmp = Float64(x / fma(y, (exp(2.0) ^ fma(Float64(b - c), Float64(Float64(0.6666666666666666 / t) - Float64(a + 0.8333333333333334)), Float64(z * Float64(t_1 / t)))), x));
	else
		tmp = 1.0;
	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[N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * t$95$1), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(-0.8333333333333334 - a), $MachinePrecision] - N[(-2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-147], N[(x / N[(y * N[Power[N[Exp[2.0], $MachinePrecision], N[(N[(b - c), $MachinePrecision] * N[(N[(0.6666666666666666 / t), $MachinePrecision] - N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision] + N[(z * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 1.0]]

Target

Original	93.9%
Target	95.4%
Herbie	95.8%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 2 (-.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)))))))))) < 1.9999999999999999e-147

   1. Initial program 98.7%

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

   2. Simplified 98.8%

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

      [Start] 98.7	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      +-commutative [=>] 98.7	\[ \frac{x}{\color{blue}{y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)} + x}} \]
      fma-def [=>] 98.7	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(y, e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}, x\right)}} \]

   if 1.9999999999999999e-147 < (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 2 (-.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 89.6%

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

   2. Taylor expanded in b around inf 70.7%

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

   3. Simplified 70.7%

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

      [Start] 70.7	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\left(0.6666666666666666 \cdot \frac{1}{t} - \left(0.8333333333333334 + a\right)\right) \cdot b\right)}} \]
      *-commutative [=>] 70.7	\[ \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(b \cdot \left(0.6666666666666666 \cdot \frac{1}{t} - \left(0.8333333333333334 + a\right)\right)\right)}}} \]
      associate-*r/ [=>] 70.7	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\color{blue}{\frac{0.6666666666666666 \cdot 1}{t}} - \left(0.8333333333333334 + a\right)\right)\right)}} \]
      metadata-eval [=>] 70.7	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{\color{blue}{0.6666666666666666}}{t} - \left(0.8333333333333334 + a\right)\right)\right)}} \]
      +-commutative [=>] 70.7	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} - \color{blue}{\left(a + 0.8333333333333334\right)}\right)\right)}} \]

   4. Taylor expanded in x around inf 93.2%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\left(-0.8333333333333334 - a\right) - \frac{-2}{t \cdot 3}\right)\right)}} \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(b - c, \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right), z \cdot \frac{\sqrt{t + a}}{t}\right)\right)}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.9%
Cost	35716

\[\begin{array}{l} t_1 := \sqrt{t + a}\\ t_2 := \left(b - c\right) \cdot \left(\left(-0.8333333333333334 - a\right) - \frac{-2}{t \cdot 3}\right)\\ \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot t_1}{t} + t_2\right)}} \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z}{\frac{t}{t_1}} + t_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Accuracy	96.4%
Cost	22468

\[\begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\left(-0.8333333333333334 - a\right) - \frac{-2}{t \cdot 3}\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot t_1}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Accuracy	79.0%
Cost	13700

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-160}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2 \cdot a}\right)}^{\left(c - b\right)}}\\ \mathbf{elif}\;t \leq 10^{-72}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a - \left(\frac{0.6666666666666666}{t} + -0.8333333333333334\right)\right)\right)}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \end{array} \]

Alternative 4
Accuracy	51.1%
Cost	8424

\[\begin{array}{l} t_1 := \frac{x}{x + 1.3333333333333333 \cdot \left(y \cdot \frac{b}{t}\right)}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{y \cdot e^{b \cdot -1.6666666666666667}}\\ \mathbf{elif}\;b \leq 1.56 \cdot 10^{-240}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \frac{y \cdot \left(c \cdot c - b \cdot b\right)}{b + c}\right)\right)}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 5300000:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;b \leq 125000000000:\\ \;\;\;\;\frac{0.5}{y \cdot a} \cdot \frac{x}{c}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+77}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{y \cdot e^{c \cdot 1.6666666666666667}}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{x}{y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+241}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	79.5%
Cost	8148

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{-2 \cdot \left(0.6666666666666666 \cdot \frac{c}{t}\right)}}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-182}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 10^{-120}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \frac{b \cdot -0.6666666666666666}{t}}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	51.4%
Cost	8040

\[\begin{array}{l} t_1 := \frac{x}{x + 1.3333333333333333 \cdot \left(y \cdot \frac{b}{t}\right)}\\ t_2 := \frac{x}{x + \left(y + 2 \cdot \left(a \cdot \frac{y \cdot \left(c \cdot c - b \cdot b\right)}{b + c}\right)\right)}\\ \mathbf{if}\;b \leq -4 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y \cdot e^{b \cdot -1.6666666666666667}}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-240}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 5300000:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;b \leq 125000000000:\\ \;\;\;\;\frac{0.5}{y \cdot a} \cdot \frac{x}{c}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+59}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+139}:\\ \;\;\;\;\log \left(1 + \frac{x}{y}\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+241}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	51.4%
Cost	8040

\[\begin{array}{l} t_1 := \frac{x}{x + 1.3333333333333333 \cdot \left(y \cdot \frac{b}{t}\right)}\\ \mathbf{if}\;b \leq -4 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y \cdot e^{b \cdot -1.6666666666666667}}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-241}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-181}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \frac{y \cdot \left(c \cdot c - b \cdot b\right)}{b + c}\right)\right)}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 5300000:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;b \leq 125000000000:\\ \;\;\;\;\frac{0.5}{y \cdot a} \cdot \frac{x}{c}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+77}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{y \cdot e^{c \cdot 1.6666666666666667}}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+139}:\\ \;\;\;\;\log \left(1 + \frac{x}{y}\right)\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+240}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	67.5%
Cost	7892

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{if}\;a \leq -0.84:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{-227}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \frac{y \cdot \left(c \cdot c - b \cdot b\right)}{b + c}\right)\right)}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-285}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(0.6666666666666666 \cdot \frac{c}{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	79.1%
Cost	7888

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{-2 \cdot \left(0.6666666666666666 \cdot \frac{c}{t}\right)}}\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{-258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-180}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \frac{b \cdot -0.6666666666666666}{t}}}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-12}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	80.4%
Cost	7884

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a - \left(\frac{0.6666666666666666}{t} + -0.8333333333333334\right)\right)\right)}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	72.0%
Cost	7760

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{-2 \cdot \left(0.6666666666666666 \cdot \frac{c}{t}\right)}}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{-257}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-120}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \frac{b \cdot -0.6666666666666666}{t}}}\\ \mathbf{elif}\;t \leq 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \end{array} \]

Alternative 12
Accuracy	67.7%
Cost	7500

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-166}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \frac{y \cdot \left(c \cdot c - b \cdot b\right)}{b + c}\right)\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Accuracy	71.0%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-258} \lor \neg \left(t \leq 6.9 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(0.6666666666666666 \cdot \frac{c}{t}\right)}}\\ \end{array} \]

Alternative 14
Accuracy	47.9%
Cost	1888

\[\begin{array}{l} t_1 := \frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{if}\;c \leq -1.7 \cdot 10^{-65}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -3.6 \cdot 10^{-232}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - \frac{b}{t} \cdot -1.3333333333333333\right)}\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{-263}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-196}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{x}{x + \left(y \cdot \left(a \cdot b\right)\right) \cdot -2}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1.3333333333333333 \cdot \left(y \cdot \frac{b}{t}\right)}\\ \end{array} \]

Alternative 15
Accuracy	48.4%
Cost	1885

\[\begin{array}{l} t_1 := \frac{x}{x - \left(2 \cdot \left(a \cdot \left(y \cdot \left(b - c\right)\right)\right) - y\right)}\\ \mathbf{if}\;a \leq -1.65 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-262}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-220}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{+51}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+62} \lor \neg \left(a \leq 1.85 \cdot 10^{+141}\right) \land a \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{x}{x - y \cdot \left(-1 + 2 \cdot \left(a \cdot \left(b - c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Accuracy	49.0%
Cost	1885

\[\begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \frac{y \cdot \left(c \cdot c - b \cdot b\right)}{b + c}\right)\right)}\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-262}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-220}:\\ \;\;\;\;\frac{x}{x - \left(2 \cdot \left(a \cdot \left(y \cdot \left(b - c\right)\right)\right) - y\right)}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+51}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+60} \lor \neg \left(a \leq 4.5 \cdot 10^{+141}\right) \land a \leq 4.2 \cdot 10^{+232}:\\ \;\;\;\;\frac{x}{x - y \cdot \left(-1 + 2 \cdot \left(a \cdot \left(b - c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Accuracy	48.8%
Cost	1756

\[\begin{array}{l} t_1 := \frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{if}\;c \leq -1.7 \cdot 10^{-98}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -3.35 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{-129}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 2.25 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{x + \left(y \cdot \left(a \cdot b\right)\right) \cdot -2}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1.3333333333333333 \cdot \left(y \cdot \frac{b}{t}\right)}\\ \end{array} \]

Alternative 18
Accuracy	48.6%
Cost	1753

\[\begin{array}{l} t_1 := \frac{x}{x - y \cdot \left(-1 + 2 \cdot \left(a \cdot \left(b - c\right)\right)\right)}\\ \mathbf{if}\;a \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.35 \cdot 10^{-224}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+51}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+62} \lor \neg \left(a \leq 2.1 \cdot 10^{+140}\right) \land a \leq 7.5 \cdot 10^{+233}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Accuracy	47.4%
Cost	1364

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y \cdot \left(a \cdot b\right)\right) \cdot -2}\\ \mathbf{if}\;a \leq -1.35 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-218}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \mathbf{elif}\;a \leq -2.15 \cdot 10^{-223}:\\ \;\;\;\;0.75 \cdot \left(\frac{t}{y} \cdot \frac{x}{b}\right)\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{+141}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+231}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Accuracy	47.4%
Cost	1364

\[\begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{x + \left(y \cdot \left(a \cdot b\right)\right) \cdot -2}\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-218}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \mathbf{elif}\;a \leq -2.15 \cdot 10^{-223}:\\ \;\;\;\;0.75 \cdot \left(\frac{t}{y} \cdot \frac{x}{b}\right)\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+162}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+232}:\\ \;\;\;\;\frac{x \cdot 0.5}{y \cdot \left(a \cdot \left(c - b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 21
Accuracy	46.7%
Cost	1236

\[\begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{0.5}{y \cdot c} \cdot \frac{x}{a}\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-218}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \mathbf{elif}\;a \leq -2.1 \cdot 10^{-223}:\\ \;\;\;\;0.75 \cdot \left(\frac{t}{y} \cdot \frac{x}{b}\right)\\ \mathbf{elif}\;a \leq 1.72 \cdot 10^{+193}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+231}:\\ \;\;\;\;-0.5 \cdot \frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 22
Accuracy	48.4%
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-288}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-276}:\\ \;\;\;\;0.75 \cdot \left(\frac{t}{y} \cdot \frac{x}{b}\right)\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-25}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+86}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 23
Accuracy	47.5%
Cost	972

\[\begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \mathbf{elif}\;a \leq 1.72 \cdot 10^{+193}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+231}:\\ \;\;\;\;-0.5 \cdot \frac{x}{a \cdot \left(y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 24
Accuracy	47.7%
Cost	972

\[\begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \mathbf{elif}\;a \leq 1.72 \cdot 10^{+193}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+232}:\\ \;\;\;\;-0.5 \cdot \frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 25
Accuracy	48.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;a \leq -3.75 \cdot 10^{-227}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 26
Accuracy	49.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-46}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq -1.75 \cdot 10^{-130}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 27
Accuracy	48.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq -5.1 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 28
Accuracy	51.1%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023137 
(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)))))))))))