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

Average Error: 1.8 → 1.8

Time: 26.9s

Precision: binary64

Cost: 33737

Math

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

↓

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -650 \lor \neg \left(t_1 \leq -225\right):\\ \;\;\;\;\frac{x \cdot e^{\left(t_1 + y \cdot \log z\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

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 (or (<= t_1 -650.0) (not (<= t_1 -225.0)))
     (/ (* x (exp (- (+ t_1 (* y (log z))) b))) y)
     (* x (/ (pow z y) (* a (* y (exp b))))))))

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 <= -650.0) || !(t_1 <= -225.0)) {
		tmp = (x * exp(((t_1 + (y * log(z))) - b))) / y;
	} else {
		tmp = x * (pow(z, y) / (a * (y * exp(b))));
	}
	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 <= (-650.0d0)) .or. (.not. (t_1 <= (-225.0d0)))) then
        tmp = (x * exp(((t_1 + (y * log(z))) - b))) / y
    else
        tmp = x * ((z ** y) / (a * (y * exp(b))))
    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 <= -650.0) || !(t_1 <= -225.0)) {
		tmp = (x * Math.exp(((t_1 + (y * Math.log(z))) - b))) / y;
	} else {
		tmp = x * (Math.pow(z, y) / (a * (y * Math.exp(b))));
	}
	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 <= -650.0) or not (t_1 <= -225.0):
		tmp = (x * math.exp(((t_1 + (y * math.log(z))) - b))) / y
	else:
		tmp = x * (math.pow(z, y) / (a * (y * math.exp(b))))
	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 <= -650.0) || !(t_1 <= -225.0))
		tmp = Float64(Float64(x * exp(Float64(Float64(t_1 + Float64(y * log(z))) - b))) / y);
	else
		tmp = Float64(x * Float64((z ^ y) / Float64(a * Float64(y * exp(b)))));
	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 <= -650.0) || ~((t_1 <= -225.0)))
		tmp = (x * exp(((t_1 + (y * log(z))) - b))) / y;
	else
		tmp = x * ((z ^ y) / (a * (y * exp(b))));
	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[Or[LessEqual[t$95$1, -650.0], N[Not[LessEqual[t$95$1, -225.0]], $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], N[(x * N[(N[Power[z, y], $MachinePrecision] / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	1.8
Target	11.3
Herbie	1.8

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t 1) (log.f64 a)) < -650 or -225 < (*.f64 (-.f64 t 1) (log.f64 a))

   1. Initial program 0.8

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

   if -650 < (*.f64 (-.f64 t 1) (log.f64 a)) < -225

   1. Initial program 6.3

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

   2. Simplified 6.0

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{{a}^{\left(1 - t\right)}}}{y \cdot e^{b}}} \]
      Proof
      (*.f64 x (/.f64 (/.f64 (pow.f64 z y) (pow.f64 a (-.f64 1 t))) (*.f64 y (exp.f64 b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 z) y))) (pow.f64 a (-.f64 1 t))) (*.f64 y (exp.f64 b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 y (log.f64 z)))) (pow.f64 a (-.f64 1 t))) (*.f64 y (exp.f64 b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 (exp.f64 (*.f64 y (log.f64 z))) (pow.f64 a (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 t))))) (*.f64 y (exp.f64 b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 (exp.f64 (*.f64 y (log.f64 z))) (pow.f64 a (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 t) 1)))) (*.f64 y (exp.f64 b)))): 15 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 (exp.f64 (*.f64 y (log.f64 z))) (pow.f64 a (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 t)) 1))) (*.f64 y (exp.f64 b)))): 0 points increase in error, 15 points decrease in error
      (*.f64 x (/.f64 (/.f64 (exp.f64 (*.f64 y (log.f64 z))) (pow.f64 a (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 t 1))))) (*.f64 y (exp.f64 b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 (exp.f64 (*.f64 y (log.f64 z))) (pow.f64 a (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 t 1))))) (*.f64 y (exp.f64 b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 (exp.f64 (*.f64 y (log.f64 z))) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 a) (neg.f64 (-.f64 t 1)))))) (*.f64 y (exp.f64 b)))): 15 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 (exp.f64 (*.f64 y (log.f64 z))) (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (-.f64 t 1)) (log.f64 a))))) (*.f64 y (exp.f64 b)))): 0 points increase in error, 15 points decrease in error
      (*.f64 x (/.f64 (/.f64 (exp.f64 (*.f64 y (log.f64 z))) (exp.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (-.f64 t 1) (log.f64 a)))))) (*.f64 y (exp.f64 b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= exp-diff_binary64 (exp.f64 (-.f64 (*.f64 y (log.f64 z)) (neg.f64 (*.f64 (-.f64 t 1) (log.f64 a)))))) (*.f64 y (exp.f64 b)))): 17 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (exp.f64 (-.f64 (*.f64 y (log.f64 z)) (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 (-.f64 t 1)) (log.f64 a))))) (*.f64 y (exp.f64 b)))): 15 points increase in error, 2 points decrease in error
      (*.f64 x (/.f64 (exp.f64 (Rewrite=> cancel-sign-sub_binary64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t 1) (log.f64 a))))) (*.f64 y (exp.f64 b)))): 0 points increase in error, 15 points decrease in error
      (*.f64 x (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (exp.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t 1) (log.f64 a)))) (exp.f64 b)) y))): 15 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= exp-diff_binary64 (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t 1) (log.f64 a))) b))) y)): 0 points increase in error, 15 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t 1) (log.f64 a))) b))) y)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around 0 6.6

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

   4. Simplified 6.6

      \[\leadsto \color{blue}{\frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)} \cdot x} \]
      Proof
      (*.f64 (/.f64 (pow.f64 z y) (*.f64 a (*.f64 y (exp.f64 b)))) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 z y) (/.f64 (*.f64 a (*.f64 y (exp.f64 b))) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 z y) x) (*.f64 a (*.f64 y (exp.f64 b))))): 6 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 z y) x) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 y (exp.f64 b)) a))): 0 points increase in error, 6 points decrease in error
      (/.f64 (*.f64 (pow.f64 z y) x) (Rewrite=> associate-*l*_binary64 (*.f64 y (*.f64 (exp.f64 b) a)))): 6 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 z y) x) (*.f64 y (Rewrite<= *-commutative_binary64 (*.f64 a (exp.f64 b))))): 0 points increase in error, 6 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -650 \lor \neg \left(\left(t + -1\right) \cdot \log a \leq -225\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\left(t + -1\right) \cdot \log a + y \cdot \log z\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.3
Cost	33480

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -650:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;t_1 \leq -225:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \end{array} \]

Alternative 2
Error	12.3
Cost	13968

\[\begin{array}{l} t_1 := {a}^{\left(t + -1\right)}\\ t_2 := x \cdot \frac{{z}^{y}}{a \cdot y}\\ \mathbf{if}\;b \leq -2.75 \cdot 10^{-83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-255}:\\ \;\;\;\;\frac{t_1}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq -9.2 \cdot 10^{-303}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-228}:\\ \;\;\;\;\frac{{a}^{t}}{a \cdot e^{b}} \cdot \frac{x}{y}\\ \mathbf{elif}\;b \leq 0.023:\\ \;\;\;\;\frac{x \cdot t_1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y \cdot e^{b}}}{a}\\ \end{array} \]

Alternative 3
Error	12.0
Cost	13968

\[\begin{array}{l} t_1 := y \cdot e^{b}\\ t_2 := x \cdot \frac{{z}^{y}}{a \cdot y}\\ \mathbf{if}\;b \leq -1.8 \cdot 10^{-83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -5.8 \cdot 10^{-254}:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)}}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq -9.2 \cdot 10^{-303}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{t_1 \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t_1}}{a}\\ \end{array} \]

Alternative 4
Error	10.0
Cost	13836

\[\begin{array}{l} t_1 := x \cdot \frac{{z}^{y}}{a \cdot y}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-129}:\\ \;\;\;\;\frac{x \cdot e^{\left(t + -1\right) \cdot \log a - b}}{y}\\ \mathbf{elif}\;y \leq 7000000000:\\ \;\;\;\;\frac{x}{\left(y \cdot e^{b}\right) \cdot {a}^{\left(1 - t\right)}}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+168} \lor \neg \left(y \leq 9.5 \cdot 10^{+237}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + y \cdot b}}{a}\\ \end{array} \]

Alternative 5
Error	12.3
Cost	7572

\[\begin{array}{l} t_1 := {a}^{\left(t + -1\right)}\\ t_2 := \frac{t_1}{\frac{y}{x}}\\ t_3 := x \cdot \frac{{z}^{y}}{a \cdot y}\\ \mathbf{if}\;b \leq -4.1 \cdot 10^{-85}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -9.2 \cdot 10^{-303}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-210}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 0.115:\\ \;\;\;\;\frac{x \cdot t_1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y \cdot e^{b}}}{a}\\ \end{array} \]

Alternative 6
Error	12.9
Cost	7440

\[\begin{array}{l} t_1 := \frac{x \cdot {a}^{t}}{a \cdot y}\\ t_2 := x \cdot \frac{{z}^{y}}{a \cdot y}\\ \mathbf{if}\;b \leq -5.9 \cdot 10^{-83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -9.2 \cdot 10^{-303}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y \cdot e^{b}}}{a}\\ \end{array} \]

Alternative 7
Error	11.8
Cost	7440

\[\begin{array}{l} t_1 := \frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ t_2 := x \cdot \frac{{z}^{y}}{a \cdot y}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{-84}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.7 \cdot 10^{-303}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 0.1:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y \cdot e^{b}}}{a}\\ \end{array} \]

Alternative 8
Error	25.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot \left(1 + b\right)\right)}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-153}:\\ \;\;\;\;{\left(a \cdot \frac{y}{x}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{b}}}{a \cdot y}\\ \end{array} \]

Alternative 9
Error	13.7
Cost	7044

\[\begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y \cdot e^{b}}}{a}\\ \end{array} \]

Alternative 10
Error	22.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;b \leq -3.25 \cdot 10^{-149}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot \left(1 + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y \cdot e^{b}}}{a}\\ \end{array} \]

Alternative 11
Error	35.3
Cost	972

\[\begin{array}{l} t_1 := \frac{x}{y \cdot \left(a \cdot \left(1 + b\right)\right)}\\ \mathbf{if}\;b \leq -3.6 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-154}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{1}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y \cdot b}}{a}\\ \end{array} \]

Alternative 12
Error	35.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-29} \lor \neg \left(b \leq 47\right):\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \end{array} \]

Alternative 13
Error	36.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq 6200:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 14
Error	35.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq 9000:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 15
Error	35.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq 1:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y \cdot b}}{a}\\ \end{array} \]

Alternative 16
Error	30.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq 7.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{x}{y + y \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot \left(1 + b\right)\right)}\\ \end{array} \]

Alternative 17
Error	39.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \]

Alternative 18
Error	43.0
Cost	320

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

Reproduce

herbie shell --seed 2022343 
(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))