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

Average Error: 2.0 → 2.7

Time: 1.1min

Precision: binary64

Cost: 40396

Math

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

↓

\[\begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := x \cdot \left(e^{y \cdot \log z - b} \cdot \frac{1}{a \cdot y}\right)\\ \mathbf{if}\;t_1 \leq -685:\\ \;\;\;\;\left({a}^{\left(t - 1\right)} \cdot \frac{1}{y}\right) \cdot x\\ \mathbf{elif}\;t_1 \leq 20:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 690:\\ \;\;\;\;\frac{x \cdot \left({z}^{y} \cdot \frac{{a}^{\left(t + -1\right)}}{e^{b}}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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)))
        (t_2 (* x (* (exp (- (* y (log z)) b)) (/ 1.0 (* a y))))))
   (if (<= t_1 -685.0)
     (* (* (pow a (- t 1.0)) (/ 1.0 y)) x)
     (if (<= t_1 20.0)
       t_2
       (if (<= t_1 690.0)
         (/ (* x (* (pow z y) (/ (pow a (+ t -1.0)) (exp b)))) y)
         t_2)))))

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 t_2 = x * (exp(((y * log(z)) - b)) * (1.0 / (a * y)));
	double tmp;
	if (t_1 <= -685.0) {
		tmp = (pow(a, (t - 1.0)) * (1.0 / y)) * x;
	} else if (t_1 <= 20.0) {
		tmp = t_2;
	} else if (t_1 <= 690.0) {
		tmp = (x * (pow(z, y) * (pow(a, (t + -1.0)) / exp(b)))) / y;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (t - 1.0d0) * log(a)
    t_2 = x * (exp(((y * log(z)) - b)) * (1.0d0 / (a * y)))
    if (t_1 <= (-685.0d0)) then
        tmp = ((a ** (t - 1.0d0)) * (1.0d0 / y)) * x
    else if (t_1 <= 20.0d0) then
        tmp = t_2
    else if (t_1 <= 690.0d0) then
        tmp = (x * ((z ** y) * ((a ** (t + (-1.0d0))) / exp(b)))) / y
    else
        tmp = t_2
    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 t_2 = x * (Math.exp(((y * Math.log(z)) - b)) * (1.0 / (a * y)));
	double tmp;
	if (t_1 <= -685.0) {
		tmp = (Math.pow(a, (t - 1.0)) * (1.0 / y)) * x;
	} else if (t_1 <= 20.0) {
		tmp = t_2;
	} else if (t_1 <= 690.0) {
		tmp = (x * (Math.pow(z, y) * (Math.pow(a, (t + -1.0)) / Math.exp(b)))) / y;
	} else {
		tmp = t_2;
	}
	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)
	t_2 = x * (math.exp(((y * math.log(z)) - b)) * (1.0 / (a * y)))
	tmp = 0
	if t_1 <= -685.0:
		tmp = (math.pow(a, (t - 1.0)) * (1.0 / y)) * x
	elif t_1 <= 20.0:
		tmp = t_2
	elif t_1 <= 690.0:
		tmp = (x * (math.pow(z, y) * (math.pow(a, (t + -1.0)) / math.exp(b)))) / y
	else:
		tmp = t_2
	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))
	t_2 = Float64(x * Float64(exp(Float64(Float64(y * log(z)) - b)) * Float64(1.0 / Float64(a * y))))
	tmp = 0.0
	if (t_1 <= -685.0)
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) * Float64(1.0 / y)) * x);
	elseif (t_1 <= 20.0)
		tmp = t_2;
	elseif (t_1 <= 690.0)
		tmp = Float64(Float64(x * Float64((z ^ y) * Float64((a ^ Float64(t + -1.0)) / exp(b)))) / y);
	else
		tmp = t_2;
	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);
	t_2 = x * (exp(((y * log(z)) - b)) * (1.0 / (a * y)));
	tmp = 0.0;
	if (t_1 <= -685.0)
		tmp = ((a ^ (t - 1.0)) * (1.0 / y)) * x;
	elseif (t_1 <= 20.0)
		tmp = t_2;
	elseif (t_1 <= 690.0)
		tmp = (x * ((z ^ y) * ((a ^ (t + -1.0)) / exp(b)))) / y;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(x * N[(N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -685.0], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 20.0], t$95$2, If[LessEqual[t$95$1, 690.0], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] * N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] / N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]]

Target

Original	2.0
Target	11.0
Herbie	2.7

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t 1) (log.f64 a)) < -685

   1. Initial program 0.3

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

   2. Applied egg-rr 17.7

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

   3. Taylor expanded in b around 0 10.1

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

   4. Taylor expanded in y around 0 0.4

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

   if -685 < (*.f64 (-.f64 t 1) (log.f64 a)) < 20 or 690 < (*.f64 (-.f64 t 1) (log.f64 a))

   1. Initial program 4.1

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

   2. Taylor expanded in t around 0 4.4

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

   3. Simplified 2.0

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

      [Start] 4.4	\[ \frac{e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b} \cdot x}{y} \]
      rational.json-simplify-49 [=>] 2.0	\[ \color{blue}{x \cdot \frac{e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b}}{y}} \]
      rational.json-simplify-48 [=>] 2.0	\[ x \cdot \frac{e^{\color{blue}{-1 \cdot \log a + \left(y \cdot \log z - b\right)}}}{y} \]
      exponential.json-simplify-1 [=>] 2.0	\[ x \cdot \frac{\color{blue}{e^{-1 \cdot \log a} \cdot e^{y \cdot \log z - b}}}{y} \]
      rational.json-simplify-49 [=>] 3.3	\[ x \cdot \color{blue}{\left(e^{y \cdot \log z - b} \cdot \frac{e^{-1 \cdot \log a}}{y}\right)} \]
      rational.json-simplify-2 [=>] 3.3	\[ x \cdot \left(e^{y \cdot \log z - b} \cdot \frac{e^{\color{blue}{\log a \cdot -1}}}{y}\right) \]
      rational.json-simplify-9 [=>] 3.3	\[ x \cdot \left(e^{y \cdot \log z - b} \cdot \frac{e^{\color{blue}{-\log a}}}{y}\right) \]
      exponential.json-simplify-2 [=>] 3.3	\[ x \cdot \left(e^{y \cdot \log z - b} \cdot \frac{\color{blue}{\frac{1}{e^{\log a}}}}{y}\right) \]
      rational.json-simplify-47 [=>] 3.3	\[ x \cdot \left(e^{y \cdot \log z - b} \cdot \color{blue}{\frac{1}{e^{\log a} \cdot y}}\right) \]
      exponential.json-simplify-7 [=>] 2.0	\[ x \cdot \left(e^{y \cdot \log z - b} \cdot \frac{1}{\color{blue}{a} \cdot y}\right) \]

   if 20 < (*.f64 (-.f64 t 1) (log.f64 a)) < 690

   1. Initial program 1.4

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

   2. Applied egg-rr 7.7

      \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \left({z}^{y} \cdot e^{-b}\right) + 0\right)}}{y} \]

   3. Simplified 7.7

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

      [Start] 7.7	\[ \frac{x \cdot \left({a}^{\left(t + -1\right)} \cdot \left({z}^{y} \cdot e^{-b}\right) + 0\right)}{y} \]
      rational.json-simplify-4 [=>] 7.7	\[ \frac{x \cdot \color{blue}{\left({a}^{\left(t + -1\right)} \cdot \left({z}^{y} \cdot e^{-b}\right)\right)}}{y} \]
      rational.json-simplify-43 [=>] 7.7	\[ \frac{x \cdot \color{blue}{\left({z}^{y} \cdot \left(e^{-b} \cdot {a}^{\left(t + -1\right)}\right)\right)}}{y} \]
      exponential.json-simplify-2 [=>] 7.7	\[ \frac{x \cdot \left({z}^{y} \cdot \left(\color{blue}{\frac{1}{e^{b}}} \cdot {a}^{\left(t + -1\right)}\right)\right)}{y} \]
      rational.json-simplify-7 [<=] 7.7	\[ \frac{x \cdot \left({z}^{y} \cdot \left(\frac{1}{e^{b}} \cdot \color{blue}{\frac{{a}^{\left(t + -1\right)}}{1}}\right)\right)}{y} \]
      rational.json-simplify-55 [=>] 7.7	\[ \frac{x \cdot \left({z}^{y} \cdot \color{blue}{\frac{\frac{{a}^{\left(t + -1\right)}}{1}}{\frac{e^{b}}{1}}}\right)}{y} \]
      rational.json-simplify-7 [=>] 7.7	\[ \frac{x \cdot \left({z}^{y} \cdot \frac{\color{blue}{{a}^{\left(t + -1\right)}}}{\frac{e^{b}}{1}}\right)}{y} \]
      rational.json-simplify-7 [=>] 7.7	\[ \frac{x \cdot \left({z}^{y} \cdot \frac{{a}^{\left(t + -1\right)}}{\color{blue}{e^{b}}}\right)}{y} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -685:\\ \;\;\;\;\left({a}^{\left(t - 1\right)} \cdot \frac{1}{y}\right) \cdot x\\ \mathbf{elif}\;\left(t - 1\right) \cdot \log a \leq 20:\\ \;\;\;\;x \cdot \left(e^{y \cdot \log z - b} \cdot \frac{1}{a \cdot y}\right)\\ \mathbf{elif}\;\left(t - 1\right) \cdot \log a \leq 690:\\ \;\;\;\;\frac{x \cdot \left({z}^{y} \cdot \frac{{a}^{\left(t + -1\right)}}{e^{b}}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{y \cdot \log z - b} \cdot \frac{1}{a \cdot y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	3.7
Cost	40396

\[\begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := x \cdot \left(e^{y \cdot \log z - b} \cdot \frac{1}{a \cdot y}\right)\\ \mathbf{if}\;t_1 \leq -685:\\ \;\;\;\;\left({a}^{\left(t - 1\right)} \cdot \frac{1}{y}\right) \cdot x\\ \mathbf{elif}\;t_1 \leq 185:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 690:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{e^{b}}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	4.9
Cost	40268

\[\begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := x \cdot \left(e^{y \cdot \log z - b} \cdot \frac{1}{a \cdot y}\right)\\ \mathbf{if}\;t_1 \leq -685:\\ \;\;\;\;\left({a}^{\left(t - 1\right)} \cdot \frac{1}{y}\right) \cdot x\\ \mathbf{elif}\;t_1 \leq 150:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 395:\\ \;\;\;\;{a}^{-1} \cdot \left(\frac{{a}^{t}}{y} \cdot \frac{x}{e^{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	4.7
Cost	33996

\[\begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := x \cdot \left(e^{y \cdot \log z - b} \cdot \frac{1}{a \cdot y}\right)\\ \mathbf{if}\;t_1 \leq -685:\\ \;\;\;\;\left({a}^{\left(t - 1\right)} \cdot \frac{1}{y}\right) \cdot x\\ \mathbf{elif}\;t_1 \leq 232:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 395:\\ \;\;\;\;\frac{\frac{\frac{x}{e^{b}}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	5.8
Cost	33868

\[\begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := \frac{\frac{\frac{x}{y}}{a}}{e^{b - y \cdot \log z}}\\ \mathbf{if}\;t_1 \leq -680:\\ \;\;\;\;\left({a}^{\left(t - 1\right)} \cdot \frac{1}{y}\right) \cdot x\\ \mathbf{elif}\;t_1 \leq -520:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -380:\\ \;\;\;\;x \cdot \frac{e^{t_1 - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	2.1
Cost	20160

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

Alternative 6
Error	8.3
Cost	13768

\[\begin{array}{l} t_1 := \frac{{z}^{y} \cdot x}{a \cdot y}\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+167}:\\ \;\;\;\;x \cdot \frac{e^{\left(t - 1\right) \cdot \log a - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	11.1
Cost	13704

\[\begin{array}{l} t_1 := {a}^{\left(t - 1\right)}\\ t_2 := x \cdot \frac{t_1}{y}\\ \mathbf{if}\;b \leq 4.5 \cdot 10^{-242}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{{z}^{y}}{e^{b}}}{y \cdot \frac{a}{x}}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\left(t_1 \cdot \frac{1}{y}\right) \cdot x\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{{z}^{y} \cdot x}{a \cdot y}\\ \mathbf{elif}\;b \leq 2.5:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{e^{b}}}{a}}{y}\\ \end{array} \]

Alternative 8
Error	10.9
Cost	7572

\[\begin{array}{l} t_1 := \frac{{z}^{y} \cdot x}{a \cdot y}\\ t_2 := x \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \mathbf{if}\;b \leq 2.55 \cdot 10^{-241}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 280:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{e^{b}}}{a}}{y}\\ \end{array} \]

Alternative 9
Error	10.9
Cost	7572

\[\begin{array}{l} t_1 := \frac{{z}^{y} \cdot x}{a \cdot y}\\ t_2 := {a}^{\left(t - 1\right)}\\ t_3 := x \cdot \frac{t_2}{y}\\ \mathbf{if}\;b \leq 1.46 \cdot 10^{-241}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-43}:\\ \;\;\;\;\left(t_2 \cdot \frac{1}{y}\right) \cdot x\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 40:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{e^{b}}}{a}}{y}\\ \end{array} \]

Alternative 10
Error	24.5
Cost	7376

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+215}:\\ \;\;\;\;x \cdot \frac{a}{a \cdot \left(a \cdot y\right)}\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+187}:\\ \;\;\;\;\frac{e^{-b}}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -7 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \left(\frac{a}{y} \cdot \frac{1}{a \cdot a}\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+177}:\\ \;\;\;\;\frac{\frac{\frac{x}{e^{b}}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{\left(a \cdot a\right) \cdot \frac{y + y}{a}}\\ \end{array} \]

Alternative 11
Error	10.6
Cost	7044

\[\begin{array}{l} \mathbf{if}\;b \leq 260:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{e^{b}}}{a}}{y}\\ \end{array} \]

Alternative 12
Error	25.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-255}:\\ \;\;\;\;x \cdot \frac{a}{a \cdot \left(a \cdot y\right)}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{e^{b}}\\ \end{array} \]

Alternative 13
Error	22.5
Cost	6984

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-257}:\\ \;\;\;\;x \cdot \frac{a}{a \cdot \left(a \cdot y\right)}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{b}}}{y}\\ \end{array} \]

Alternative 14
Error	41.1
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{y \cdot a}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{a}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	38.9
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{y \cdot a}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	35.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{a}{a \cdot \left(a \cdot y\right)}\\ \end{array} \]

Alternative 17
Error	41.2
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{y \cdot a}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-122}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	41.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-72}:\\ \;\;\;\;x \cdot \frac{1}{a \cdot y}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 19
Error	42.3
Cost	320

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

Alternative 20
Error	54.8
Cost	192

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

Reproduce

herbie shell --seed 2023074 
(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))