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

Average Accuracy: 94.2% → 96.5%

Time: 48.1s

Precision: binary64

Cost: 35716

Math

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

↓

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

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
             (-
              (* (- b c) (+ (+ a 0.8333333333333334) (/ -2.0 (* t 3.0))))
              (/ (* z t_1) t)))))))
        1.0)
     (/
      x
      (+
       x
       (*
        y
        (pow
         (exp 2.0)
         (+
          (/ z (/ t t_1))
          (* (- b c) (+ (/ 2.0 (* t 3.0)) (- -0.8333333333333334 a))))))))
     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 * (((b - c) * ((a + 0.8333333333333334) + (-2.0 / (t * 3.0)))) - ((z * t_1) / t))))))) <= 1.0) {
		tmp = x / (x + (y * pow(exp(2.0), ((z / (t / t_1)) + ((b - c) * ((2.0 / (t * 3.0)) + (-0.8333333333333334 - a)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = x / (x + (y * exp((2.0d0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))))))
end function

↓

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((t + a))
    if ((x / (x + (y * exp(((-2.0d0) * (((b - c) * ((a + 0.8333333333333334d0) + ((-2.0d0) / (t * 3.0d0)))) - ((z * t_1) / t))))))) <= 1.0d0) then
        tmp = x / (x + (y * (exp(2.0d0) ** ((z / (t / t_1)) + ((b - c) * ((2.0d0 / (t * 3.0d0)) + ((-0.8333333333333334d0) - a)))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

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 ((x / (x + (y * Math.exp((-2.0 * (((b - c) * ((a + 0.8333333333333334) + (-2.0 / (t * 3.0)))) - ((z * t_1) / t))))))) <= 1.0) {
		tmp = x / (x + (y * Math.pow(Math.exp(2.0), ((z / (t / t_1)) + ((b - c) * ((2.0 / (t * 3.0)) + (-0.8333333333333334 - a)))))));
	} else {
		tmp = 1.0;
	}
	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 (x / (x + (y * math.exp((-2.0 * (((b - c) * ((a + 0.8333333333333334) + (-2.0 / (t * 3.0)))) - ((z * t_1) / t))))))) <= 1.0:
		tmp = x / (x + (y * math.pow(math.exp(2.0), ((z / (t / t_1)) + ((b - c) * ((2.0 / (t * 3.0)) + (-0.8333333333333334 - a)))))))
	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(b - c) * Float64(Float64(a + 0.8333333333333334) + Float64(-2.0 / Float64(t * 3.0)))) - Float64(Float64(z * t_1) / t))))))) <= 1.0)
		tmp = Float64(x / Float64(x + Float64(y * (exp(2.0) ^ Float64(Float64(z / Float64(t / t_1)) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) + Float64(-0.8333333333333334 - a))))))));
	else
		tmp = 1.0;
	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 ((x / (x + (y * exp((-2.0 * (((b - c) * ((a + 0.8333333333333334) + (-2.0 / (t * 3.0)))) - ((z * t_1) / t))))))) <= 1.0)
		tmp = x / (x + (y * (exp(2.0) ^ ((z / (t / t_1)) + ((b - c) * ((2.0 / (t * 3.0)) + (-0.8333333333333334 - a)))))));
	else
		tmp = 1.0;
	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[(x / N[(x + N[(y * N[Exp[N[(-2.0 * N[(N[(N[(b - c), $MachinePrecision] * N[(N[(a + 0.8333333333333334), $MachinePrecision] + N[(-2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(z * t$95$1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], 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[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] + N[(-0.8333333333333334 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Target

Original	94.2%
Target	95.1%
Herbie	96.5%

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

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

   1. Initial program 98.8%

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

   2. Simplified 99.0%

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

      [Start] 98.8	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      exp-prod [=>] 98.8	\[ \frac{x}{x + y \cdot \color{blue}{{\left(e^{2}\right)}^{\left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}} \]
      associate-/l* [=>] 99.0	\[ \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\color{blue}{\frac{z}{\frac{t}{\sqrt{t + a}}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
      metadata-eval [=>] 99.0	\[ \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \color{blue}{0.8333333333333334}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]

   if 1 < (/.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 2.1%

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

   2. Taylor expanded in t around inf 64.6%

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

   3. Simplified 64.6%

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

      [Start] 64.6	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(-1 \cdot \left(\left(b - c\right) \cdot \left(0.8333333333333334 + a\right)\right)\right)}} \]
      mul-1-neg [=>] 64.6	\[ \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(-\left(b - c\right) \cdot \left(0.8333333333333334 + a\right)\right)}}} \]
      distribute-rgt-neg-in [=>] 64.6	\[ \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(b - c\right) \cdot \left(-\left(0.8333333333333334 + a\right)\right)\right)}}} \]
      neg-sub0 [=>] 64.6	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \color{blue}{\left(0 - \left(0.8333333333333334 + a\right)\right)}\right)}} \]
      associate--r+ [=>] 64.6	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \color{blue}{\left(\left(0 - 0.8333333333333334\right) - a\right)}\right)}} \]
      metadata-eval [=>] 64.6	\[ \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(\color{blue}{-0.8333333333333334} - a\right)\right)}} \]

   4. Taylor expanded in x around inf 46.2%

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

4. Final simplification 96.5%

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

Alternatives

Alternative 1
Accuracy	96.7%
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
Accuracy	96.4%
Cost	22468

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

Alternative 3
Accuracy	68.7%
Cost	15972

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{-2 \cdot \left(c \cdot \left(\left(\frac{0.6666666666666666}{t} + -0.8333333333333334\right) - a\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(z \cdot \sqrt{\frac{1}{t}}\right)}}\\ t_3 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ t_4 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{if}\;b - c \leq -5 \cdot 10^{+291}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b - c \leq -2 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq -2 \cdot 10^{+110}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b - c \leq -1 \cdot 10^{+47}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b - c \leq -4 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{t} \cdot \sqrt{a}\right)}}\\ \mathbf{elif}\;b - c \leq -2 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq 10^{-63}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t}\right)\right)}\\ \mathbf{elif}\;b - c \leq 4 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 4
Accuracy	71.6%
Cost	15580

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{-2 \cdot \left(c \cdot \left(\left(\frac{0.6666666666666666}{t} + -0.8333333333333334\right) - a\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t}\right)}}\\ t_3 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ t_4 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{if}\;b - c \leq -5 \cdot 10^{+291}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b - c \leq -2 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq -2 \cdot 10^{+110}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b - c \leq -1 \cdot 10^{+47}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b - c \leq 4 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq 2 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{+156}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 5
Accuracy	66.2%
Cost	14932

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{-2 \cdot \left(c \cdot \left(\left(\frac{0.6666666666666666}{t} + -0.8333333333333334\right) - a\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ t_3 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{if}\;b - c \leq -5 \cdot 10^{+291}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq -2 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq -2 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq -1 \cdot 10^{+47}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b - c \leq -4 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{t} \cdot \sqrt{a}\right)}}\\ \mathbf{elif}\;b - c \leq -5 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;b - c \leq 10^{-75}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t}\right)\right)}\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{+216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{+280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq 10^{+285}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Accuracy	66.8%
Cost	11128

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{-2 \cdot \left(c \cdot \left(\left(\frac{0.6666666666666666}{t} + -0.8333333333333334\right) - a\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ t_3 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{if}\;b - c \leq -5 \cdot 10^{+291}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq -2 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq -2 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq -5 \cdot 10^{+66}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b - c \leq -4 \cdot 10^{+23}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq -0.002:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq -4 \cdot 10^{-147}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{-204}:\\ \;\;\;\;1\\ \mathbf{elif}\;b - c \leq 10^{-75}:\\ \;\;\;\;\frac{x}{x + \left(y + -1.3333333333333333 \cdot \frac{c}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;b - c \leq 2 \cdot 10^{-39}:\\ \;\;\;\;1\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{+216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{+280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq 10^{+285}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Accuracy	66.7%
Cost	11128

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{-2 \cdot \left(c \cdot \left(\left(\frac{0.6666666666666666}{t} + -0.8333333333333334\right) - a\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ t_3 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{if}\;b - c \leq -5 \cdot 10^{+291}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq -2 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq -2 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq -5 \cdot 10^{+66}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b - c \leq -4 \cdot 10^{+23}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq -0.002:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq -4 \cdot 10^{-147}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b - c \leq 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;b - c \leq 10^{-75}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t}\right)\right)}\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{+216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{+280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b - c \leq 10^{+285}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 8
Accuracy	67.2%
Cost	8412

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} + \left(-0.8333333333333334 - a\right)\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ t_3 := y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-259}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.38 \cdot 10^{-226}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-219}:\\ \;\;\;\;\frac{x}{t_3}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+115} \lor \neg \left(x \leq 10^{+187}\right):\\ \;\;\;\;\frac{x}{x + t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-1.3333333333333333 \cdot \frac{c}{t}}}\\ \end{array} \]

Alternative 9
Accuracy	67.9%
Cost	7892

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot a\right)}}\\ \mathbf{if}\;a \leq -0.82:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.52 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-1.3333333333333333 \cdot \frac{c}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	69.2%
Cost	7892

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot a\right)}}\\ \mathbf{if}\;a \leq -0.82:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-173}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-1.3333333333333333 \cdot \frac{c}{t}}}\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \frac{b \cdot -0.6666666666666666}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	79.0%
Cost	7756

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{if}\;t \leq -6.4 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-1.3333333333333333 \cdot \frac{c}{t}}}\\ \mathbf{elif}\;t \leq 0.48:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \frac{b \cdot -0.6666666666666666}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	46.7%
Cost	7636

\[\begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 2 \cdot \left(b \cdot \left(-0.8333333333333334 - a\right)\right)\right)}\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-260}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-172}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+74}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 9 \cdot 10^{+179}:\\ \;\;\;\;\frac{x}{y \cdot e^{-2 \cdot \left(b \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Accuracy	61.6%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;b - c \leq -1 \cdot 10^{+47}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{elif}\;b - c \leq -5 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + -2 \cdot \left(b \cdot \left(y \cdot a\right)\right)}\\ \mathbf{elif}\;b - c \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + -2 \cdot \left(\left(b - c\right) \cdot a\right)\right)}\\ \mathbf{elif}\;b - c \leq 5 \cdot 10^{-204}:\\ \;\;\;\;1\\ \mathbf{elif}\;b - c \leq 10^{-75}:\\ \;\;\;\;\frac{x}{x + \left(y + -1.3333333333333333 \cdot \frac{c}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;b - c \leq 10^{+259}:\\ \;\;\;\;1\\ \mathbf{elif}\;b - c \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{x}{x + 1.3333333333333333 \cdot \left(b \cdot \frac{y}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Accuracy	50.7%
Cost	1889

\[\begin{array}{l} t_1 := \frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{if}\;c \leq -1.06 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-250}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{+44}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{x}{x + 1.3333333333333333 \cdot \left(b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{x + -2 \cdot \left(b \cdot \left(y \cdot a\right)\right)}\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+232} \lor \neg \left(c \leq 1.75 \cdot 10^{+281}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(1 + -2 \cdot \left(\left(b - c\right) \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Accuracy	50.4%
Cost	1628

\[\begin{array}{l} t_1 := \frac{x}{x + -2 \cdot \left(b \cdot \left(y \cdot a\right)\right)}\\ t_2 := \frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{if}\;c \leq -5.4 \cdot 10^{-139}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-251}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3.35 \cdot 10^{+44}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+92}:\\ \;\;\;\;\frac{x}{x + 1.3333333333333333 \cdot \left(b \cdot \frac{y}{t}\right)}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+181}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{+271}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Accuracy	49.3%
Cost	1620

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot \left(1 + -2 \cdot \left(\left(b - c\right) \cdot a\right)\right)}\\ \mathbf{if}\;a \leq -2.9 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-260}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-173}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+77}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{+179}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Accuracy	53.3%
Cost	1620

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot \left(1 + 2 \cdot \left(b \cdot \left(-0.8333333333333334 - a\right)\right)\right)}\\ \mathbf{if}\;b \leq -7.2 \cdot 10^{+222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.2 \cdot 10^{+192}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -9.2 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-136}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + -2 \cdot \left(\left(b - c\right) \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Accuracy	54.1%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;b \leq -9.6 \cdot 10^{+222}:\\ \;\;\;\;0.5 \cdot \frac{x}{y \cdot \left(b \cdot \left(-0.8333333333333334 - a\right)\right)}\\ \mathbf{elif}\;b \leq -1.1 \cdot 10^{+182}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{x + \left(y \cdot b\right) \cdot \left(a \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Accuracy	50.8%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-121}:\\ \;\;\;\;-0.5 \cdot \frac{x}{y \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Accuracy	50.9%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-121}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{x}{y}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 21
Accuracy	50.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+202}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{y \cdot \left(b \cdot \left(-0.8333333333333334 - a\right)\right)}\\ \end{array} \]

Alternative 22
Accuracy	51.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{-121}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 23
Accuracy	52.0%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023138 
(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)))))))))))