Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Average Accuracy: 96.9% → 97.0%

Time: 41.6s

Precision: binary64

Cost: 33736

Math

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]

↓

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -20000000:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{elif}\;t_1 \leq -440:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(t_1 + y \cdot \log z\right) - b}}{y}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))))
   (if (<= t_1 -20000000.0)
     (/ (* (pow a (+ t -1.0)) x) y)
     (if (<= t_1 -440.0)
       (/ x (* a (* y (+ 1.0 (+ b (* 0.5 (* b b)))))))
       (/ (* x (exp (- (+ t_1 (* y (log z))) b))) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double tmp;
	if (t_1 <= -20000000.0) {
		tmp = (pow(a, (t + -1.0)) * x) / y;
	} else if (t_1 <= -440.0) {
		tmp = x / (a * (y * (1.0 + (b + (0.5 * (b * b))))));
	} else {
		tmp = (x * exp(((t_1 + (y * log(z))) - b))) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    code = (x * exp((((y * log(z)) + ((t - 1.0d0) * log(a))) - b))) / y
end function

↓

real(8) function code(x, y, z, t, a, b)
    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) :: t_1
    real(8) :: tmp
    t_1 = (t + (-1.0d0)) * log(a)
    if (t_1 <= (-20000000.0d0)) then
        tmp = ((a ** (t + (-1.0d0))) * x) / y
    else if (t_1 <= (-440.0d0)) then
        tmp = x / (a * (y * (1.0d0 + (b + (0.5d0 * (b * b))))))
    else
        tmp = (x * exp(((t_1 + (y * log(z))) - b))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.exp((((y * Math.log(z)) + ((t - 1.0) * Math.log(a))) - b))) / y;
}

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double tmp;
	if (t_1 <= -20000000.0) {
		tmp = (Math.pow(a, (t + -1.0)) * x) / y;
	} else if (t_1 <= -440.0) {
		tmp = x / (a * (y * (1.0 + (b + (0.5 * (b * b))))));
	} else {
		tmp = (x * Math.exp(((t_1 + (y * Math.log(z))) - b))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	return (x * math.exp((((y * math.log(z)) + ((t - 1.0) * math.log(a))) - b))) / y

↓

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	tmp = 0
	if t_1 <= -20000000.0:
		tmp = (math.pow(a, (t + -1.0)) * x) / y
	elif t_1 <= -440.0:
		tmp = x / (a * (y * (1.0 + (b + (0.5 * (b * b))))))
	else:
		tmp = (x * math.exp(((t_1 + (y * math.log(z))) - b))) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
end

↓

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + -1.0) * log(a))
	tmp = 0.0
	if (t_1 <= -20000000.0)
		tmp = Float64(Float64((a ^ Float64(t + -1.0)) * x) / y);
	elseif (t_1 <= -440.0)
		tmp = Float64(x / Float64(a * Float64(y * Float64(1.0 + Float64(b + Float64(0.5 * Float64(b * b)))))));
	else
		tmp = Float64(Float64(x * exp(Float64(Float64(t_1 + Float64(y * log(z))) - b))) / y);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
end

↓

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	tmp = 0.0;
	if (t_1 <= -20000000.0)
		tmp = ((a ^ (t + -1.0)) * x) / y;
	elseif (t_1 <= -440.0)
		tmp = x / (a * (y * (1.0 + (b + (0.5 * (b * b))))));
	else
		tmp = (x * exp(((t_1 + (y * log(z))) - b))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000.0], N[(N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, -440.0], N[(x / N[(a * N[(y * N[(1.0 + N[(b + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Exp[N[(N[(t$95$1 + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Target

Original	96.9%
Target	82.6%
Herbie	97.0%

\[\begin{array}{l} \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t 1) (log.f64 a)) < -2e7

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]

   2. Taylor expanded in y around 0 98.8%

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

   3. Taylor expanded in b around 0 100.0%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t - 1\right)} \cdot x}{y}} \]

   if -2e7 < (*.f64 (-.f64 t 1) (log.f64 a)) < -440

   1. Initial program 88.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]

   2. Simplified 91.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{\frac{{a}^{t}}{a}}{\frac{y}{{z}^{y}}}}{e^{b}}} \]
      Proof

      [Start] 88.7	\[ \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
      associate-*r/ [<=] 96.5	\[ \color{blue}{x \cdot \frac{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \]
      sub-neg [=>] 96.5	\[ x \cdot \frac{e^{\color{blue}{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) + \left(-b\right)}}}{y} \]
      exp-sum [=>] 88.0	\[ x \cdot \frac{\color{blue}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a} \cdot e^{-b}}}{y} \]
      associate-/l* [=>] 88.0	\[ x \cdot \color{blue}{\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{\frac{y}{e^{-b}}}} \]
      associate-/r/ [=>] 88.0	\[ x \cdot \color{blue}{\left(\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{y} \cdot e^{-b}\right)} \]
      exp-neg [=>] 88.0	\[ x \cdot \left(\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{y} \cdot \color{blue}{\frac{1}{e^{b}}}\right) \]
      associate-*r/ [=>] 88.0	\[ x \cdot \color{blue}{\frac{\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{y} \cdot 1}{e^{b}}} \]

   3. Taylor expanded in y around 0 92.3%

      \[\leadsto x \cdot \frac{\color{blue}{\frac{{a}^{t}}{y \cdot a}}}{e^{b}} \]

   4. Simplified 92.3%

      \[\leadsto x \cdot \frac{\color{blue}{\frac{\frac{{a}^{t}}{a}}{y}}}{e^{b}} \]
      Proof

      [Start] 92.3	\[ x \cdot \frac{\frac{{a}^{t}}{y \cdot a}}{e^{b}} \]
      *-commutative [=>] 92.3	\[ x \cdot \frac{\frac{{a}^{t}}{\color{blue}{a \cdot y}}}{e^{b}} \]
      associate-/r* [=>] 92.3	\[ x \cdot \frac{\color{blue}{\frac{\frac{{a}^{t}}{a}}{y}}}{e^{b}} \]

   5. Taylor expanded in t around 0 89.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(a \cdot e^{b}\right)}} \]

   6. Simplified 89.6%

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

      [Start] 89.6	\[ \frac{x}{y \cdot \left(a \cdot e^{b}\right)} \]
      *-commutative [=>] 89.6	\[ \frac{x}{\color{blue}{\left(a \cdot e^{b}\right) \cdot y}} \]
      associate-*r* [<=] 89.6	\[ \frac{x}{\color{blue}{a \cdot \left(e^{b} \cdot y\right)}} \]
      *-commutative [<=] 89.6	\[ \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

   7. Taylor expanded in b around 0 83.3%

      \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \left(y \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot y + a \cdot \left(y \cdot b\right)\right)}} \]

   8. Simplified 87.2%

      \[\leadsto \frac{x}{\color{blue}{a \cdot \left(\left(y + y \cdot b\right) + \left(y \cdot 0.5\right) \cdot \left(b \cdot b\right)\right)}} \]
      Proof

      [Start] 83.3	\[ \frac{x}{0.5 \cdot \left(y \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot y + a \cdot \left(y \cdot b\right)\right)} \]
      +-commutative [=>] 83.3	\[ \frac{x}{\color{blue}{\left(a \cdot y + a \cdot \left(y \cdot b\right)\right) + 0.5 \cdot \left(y \cdot \left(a \cdot {b}^{2}\right)\right)}} \]
      distribute-lft-out [=>] 87.5	\[ \frac{x}{\color{blue}{a \cdot \left(y + y \cdot b\right)} + 0.5 \cdot \left(y \cdot \left(a \cdot {b}^{2}\right)\right)} \]
      *-commutative [=>] 87.5	\[ \frac{x}{\color{blue}{\left(y + y \cdot b\right) \cdot a} + 0.5 \cdot \left(y \cdot \left(a \cdot {b}^{2}\right)\right)} \]
      associate-*r* [=>] 87.5	\[ \frac{x}{\left(y + y \cdot b\right) \cdot a + \color{blue}{\left(0.5 \cdot y\right) \cdot \left(a \cdot {b}^{2}\right)}} \]
      *-commutative [=>] 87.5	\[ \frac{x}{\left(y + y \cdot b\right) \cdot a + \left(0.5 \cdot y\right) \cdot \color{blue}{\left({b}^{2} \cdot a\right)}} \]
      associate-*r* [=>] 85.2	\[ \frac{x}{\left(y + y \cdot b\right) \cdot a + \color{blue}{\left(\left(0.5 \cdot y\right) \cdot {b}^{2}\right) \cdot a}} \]
      distribute-rgt-out [=>] 87.2	\[ \frac{x}{\color{blue}{a \cdot \left(\left(y + y \cdot b\right) + \left(0.5 \cdot y\right) \cdot {b}^{2}\right)}} \]
      *-commutative [=>] 87.2	\[ \frac{x}{a \cdot \left(\left(y + y \cdot b\right) + \color{blue}{\left(y \cdot 0.5\right)} \cdot {b}^{2}\right)} \]
      unpow2 [=>] 87.2	\[ \frac{x}{a \cdot \left(\left(y + y \cdot b\right) + \left(y \cdot 0.5\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   9. Taylor expanded in y around 0 89.4%

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot \left(1 + \left(b + 0.5 \cdot {b}^{2}\right)\right)\right)}} \]

   10. Simplified 89.4%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot \left(1 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)}} \]
       Proof

       [Start] 89.4	\[ \frac{x}{a \cdot \left(y \cdot \left(1 + \left(b + 0.5 \cdot {b}^{2}\right)\right)\right)} \]
       unpow2 [=>] 89.4	\[ \frac{x}{a \cdot \left(y \cdot \left(1 + \left(b + 0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)} \]

   if -440 < (*.f64 (-.f64 t 1) (log.f64 a))

   1. Initial program 96.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -20000000:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -440:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(\left(t + -1\right) \cdot \log a + y \cdot \log z\right) - b}}{y}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.7%
Cost	33480

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -20000000:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{elif}\;t_1 \leq -440:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \end{array} \]

Alternative 2
Accuracy	89.9%
Cost	27016

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -20000000:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+37}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y \cdot \left(a \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	82.9%
Cost	13836

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{elif}\;b \leq -4.1 \cdot 10^{-240}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{a}}{y}\\ \mathbf{elif}\;b \leq 100:\\ \;\;\;\;x \cdot \frac{\frac{\frac{{a}^{t}}{a}}{y}}{e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 4
Accuracy	81.9%
Cost	13836

\[\begin{array}{l} \mathbf{if}\;b \leq -190:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-270}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y \cdot \left(a \cdot e^{b}\right)}\\ \mathbf{elif}\;b \leq 700:\\ \;\;\;\;x \cdot \frac{\frac{\frac{{a}^{t}}{a}}{y}}{e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 5
Accuracy	70.5%
Cost	7640

\[\begin{array}{l} t_1 := \left(1 + \frac{x}{a \cdot y}\right) + -1\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)}\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{x}{a \cdot \left(\left(y + y \cdot b\right) + \left(b \cdot b\right) \cdot \left(y \cdot 0.5\right)\right)}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-267}:\\ \;\;\;\;\frac{x \cdot \frac{-b}{y}}{a}\\ \mathbf{elif}\;b \leq 3.45 \cdot 10^{-208}:\\ \;\;\;\;\frac{x}{a \cdot y + a \cdot \left(y \cdot b\right)}\\ \mathbf{elif}\;b \leq 24:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 6
Accuracy	80.6%
Cost	7308

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)}\\ \mathbf{elif}\;b \leq -2.55 \cdot 10^{-286}:\\ \;\;\;\;\frac{{z}^{y}}{y} \cdot \frac{x}{a}\\ \mathbf{elif}\;b \leq 100:\\ \;\;\;\;x \cdot \frac{{a}^{t}}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 7
Accuracy	80.2%
Cost	7308

\[\begin{array}{l} \mathbf{if}\;b \leq -13000000000000:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{elif}\;b \leq -2.15 \cdot 10^{-289}:\\ \;\;\;\;\frac{{z}^{y}}{y} \cdot \frac{x}{a}\\ \mathbf{elif}\;b \leq 98:\\ \;\;\;\;x \cdot \frac{{a}^{t}}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 8
Accuracy	81.2%
Cost	7308

\[\begin{array}{l} \mathbf{if}\;b \leq -15200000000000:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{elif}\;b \leq -4.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{x \cdot {z}^{y}}{a}}{y}\\ \mathbf{elif}\;b \leq 125:\\ \;\;\;\;x \cdot \frac{{a}^{t}}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 9
Accuracy	80.9%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;b \leq 240:\\ \;\;\;\;x \cdot \frac{{a}^{t}}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 10
Accuracy	62.5%
Cost	1496

\[\begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ t_2 := \left(1 + t_1\right) + -1\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)}\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-267}:\\ \;\;\;\;\frac{x \cdot \frac{-b}{y}}{a}\\ \mathbf{elif}\;b \leq 3.25 \cdot 10^{-208}:\\ \;\;\;\;\frac{x}{a \cdot y + a \cdot \left(y \cdot b\right)}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+44}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.5 \cdot \left(y \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	62.5%
Cost	1496

\[\begin{array}{l} t_1 := \left(1 + \frac{x}{a \cdot y}\right) + -1\\ \mathbf{if}\;b \leq -1.36 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)}\\ \mathbf{elif}\;b \leq -9.8 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-198}:\\ \;\;\;\;\frac{x}{a \cdot \left(\left(y + y \cdot b\right) + \left(b \cdot b\right) \cdot \left(y \cdot 0.5\right)\right)}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-267}:\\ \;\;\;\;\frac{x \cdot \frac{-b}{y}}{a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-209}:\\ \;\;\;\;\frac{x}{a \cdot y + a \cdot \left(y \cdot b\right)}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.5 \cdot \left(y \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	59.4%
Cost	1364

\[\begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ t_2 := \left(1 + t_1\right) + -1\\ \mathbf{if}\;b \leq -8 \cdot 10^{-26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-300}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-209}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y} \cdot \frac{x}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 13
Accuracy	61.9%
Cost	1364

\[\begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ t_2 := \left(1 + t_1\right) + -1\\ \mathbf{if}\;b \leq -7.5 \cdot 10^{-26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-301}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+44}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.5 \cdot \left(y \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	61.9%
Cost	1364

\[\begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ t_2 := \left(1 + t_1\right) + -1\\ \mathbf{if}\;b \leq -7.8 \cdot 10^{-26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -3.7 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-299}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-209}:\\ \;\;\;\;\frac{x}{a \cdot y + a \cdot \left(y \cdot b\right)}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+44}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.5 \cdot \left(y \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}\\ \end{array} \]

Alternative 15
Accuracy	51.5%
Cost	1236

\[\begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ t_2 := \left(1 + t_1\right) + -1\\ \mathbf{if}\;b \leq -7.8 \cdot 10^{-26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-301}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.82 \cdot 10^{-209}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y + y \cdot b\right)}\\ \end{array} \]

Alternative 16
Accuracy	50.1%
Cost	1106

\[\begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ \mathbf{if}\;b \leq -8 \cdot 10^{-26} \lor \neg \left(b \leq -4.5 \cdot 10^{-151}\right) \land \left(b \leq 4.6 \cdot 10^{-298} \lor \neg \left(b \leq 1.8 \cdot 10^{-199}\right)\right):\\ \;\;\;\;\left(1 + t_1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Accuracy	39.1%
Cost	908

\[\begin{array}{l} \mathbf{if}\;x \leq -500000:\\ \;\;\;\;x \cdot \frac{1}{a \cdot y}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{x \cdot \frac{-b}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \]

Alternative 18
Accuracy	55.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-16} \lor \neg \left(y \leq 7.5 \cdot 10^{-8}\right):\\ \;\;\;\;\left(1 + \frac{x}{a \cdot y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \end{array} \]

Alternative 19
Accuracy	40.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 200000000:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \]

Alternative 20
Accuracy	34.9%
Cost	320

\[\frac{x}{a \cdot y} \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))