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

Average Error: 1.9 → 2.6

Time: 38.5s

Precision: binary64

Cost: 33736

Math

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

↓

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -10000000:\\ \;\;\;\;\frac{x}{a \cdot \frac{y}{{a}^{t}}}\\ \mathbf{elif}\;t_1 \leq -80:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(t_1 + y \cdot \log z\right) - b}}{y}\\ \end{array} \]

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 -10000000.0)
     (/ x (* a (/ y (pow a t))))
     (if (<= t_1 -80.0)
       (* x (/ (pow z y) (* a (* y (exp 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 <= -10000000.0) {
		tmp = x / (a * (y / pow(a, t)));
	} else if (t_1 <= -80.0) {
		tmp = x * (pow(z, y) / (a * (y * exp(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 <= (-10000000.0d0)) then
        tmp = x / (a * (y / (a ** t)))
    else if (t_1 <= (-80.0d0)) then
        tmp = x * ((z ** y) / (a * (y * exp(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 <= -10000000.0) {
		tmp = x / (a * (y / Math.pow(a, t)));
	} else if (t_1 <= -80.0) {
		tmp = x * (Math.pow(z, y) / (a * (y * Math.exp(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 <= -10000000.0:
		tmp = x / (a * (y / math.pow(a, t)))
	elif t_1 <= -80.0:
		tmp = x * (math.pow(z, y) / (a * (y * math.exp(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 <= -10000000.0)
		tmp = Float64(x / Float64(a * Float64(y / (a ^ t))));
	elseif (t_1 <= -80.0)
		tmp = Float64(x * Float64((z ^ y) / Float64(a * Float64(y * exp(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 <= -10000000.0)
		tmp = x / (a * (y / (a ^ t)));
	elseif (t_1 <= -80.0)
		tmp = x * ((z ^ y) / (a * (y * exp(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, -10000000.0], N[(x / N[(a * N[(y / N[Power[a, t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -80.0], N[(x * N[(N[Power[z, y], $MachinePrecision] / N[(a * N[(y * N[Exp[b], $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	1.9
Target	10.9
Herbie	2.6

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 0

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

   2. Taylor expanded in b around 0 0.9

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

   3. Simplified 9.8

      \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{\left(t + -1\right)} \cdot {z}^{y}\right)}}{y} \]
      Proof
      (*.f64 (pow.f64 a (+.f64 t -1)) (pow.f64 z y)): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 a (+.f64 t (Rewrite<= metadata-eval (neg.f64 1)))) (pow.f64 z y)): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 a (Rewrite<= sub-neg_binary64 (-.f64 t 1))) (pow.f64 z y)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 a) (-.f64 t 1)))) (pow.f64 z y)): 39 points increase in error, 41 points decrease in error
      (*.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 t 1) (log.f64 a)))) (pow.f64 z y)): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (*.f64 (-.f64 t 1) (log.f64 a))) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 z) y)))): 0 points increase in error, 1 points decrease in error
      (*.f64 (exp.f64 (*.f64 (-.f64 t 1) (log.f64 a))) (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 y (log.f64 z))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= exp-sum_binary64 (exp.f64 (+.f64 (*.f64 (-.f64 t 1) (log.f64 a)) (*.f64 y (log.f64 z))))): 5 points increase in error, 23 points decrease in error
      (exp.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t 1) (log.f64 a))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in y around 0 0

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

   5. Simplified 0

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)}}{y} \cdot x} \]
      Proof
      (*.f64 (/.f64 (pow.f64 a (+.f64 t -1)) y) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 a (+.f64 t (Rewrite<= metadata-eval (neg.f64 1)))) y) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 a (Rewrite<= sub-neg_binary64 (-.f64 t 1))) y) x): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 a (-.f64 t 1)) x) y)): 40 points increase in error, 19 points decrease in error

   6. Applied egg-rr 0.0

      \[\leadsto \frac{\color{blue}{{a}^{t} \cdot \frac{1}{a}}}{y} \cdot x \]

   7. Applied egg-rr 0

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

   if -1e7 < (*.f64 (-.f64 t 1) (log.f64 a)) < -80

   1. Initial program 5.8

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

   2. Simplified 7.3

      \[\leadsto \color{blue}{x \cdot \left({z}^{y} \cdot \frac{{a}^{t}}{y \cdot \left(e^{b} \cdot a\right)}\right)} \]
      Proof
      (*.f64 x (*.f64 (pow.f64 z y) (/.f64 (pow.f64 a t) (*.f64 y (*.f64 (exp.f64 b) a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 z) y))) (/.f64 (pow.f64 a t) (*.f64 y (*.f64 (exp.f64 b) a))))): 1 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 y (log.f64 z)))) (/.f64 (pow.f64 a t) (*.f64 y (*.f64 (exp.f64 b) a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 a) t))) (*.f64 y (*.f64 (exp.f64 b) a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (exp.f64 (*.f64 (log.f64 a) t)) (*.f64 y (*.f64 (exp.f64 b) (Rewrite<= rem-exp-log_binary64 (exp.f64 (log.f64 a)))))))): 35 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (exp.f64 (*.f64 (log.f64 a) t)) (*.f64 y (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (exp.f64 b) 1)) (exp.f64 (log.f64 a))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (exp.f64 (*.f64 (log.f64 a) t)) (*.f64 y (Rewrite<= associate-/r/_binary64 (/.f64 (exp.f64 b) (/.f64 1 (exp.f64 (log.f64 a))))))))): 3 points increase in error, 2 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (exp.f64 (*.f64 (log.f64 a) t)) (*.f64 y (/.f64 (exp.f64 b) (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 (log.f64 a))))))))): 3 points increase in error, 2 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (exp.f64 (*.f64 (log.f64 a) t)) (*.f64 y (/.f64 (exp.f64 b) (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 a))))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (exp.f64 (*.f64 (log.f64 a) t)) (*.f64 y (/.f64 (exp.f64 b) (exp.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 1)) (log.f64 a)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (exp.f64 (*.f64 (log.f64 a) t)) (/.f64 (exp.f64 b) (exp.f64 (*.f64 (neg.f64 1) (log.f64 a))))) y)))): 8 points increase in error, 7 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (exp.f64 (*.f64 (log.f64 a) t)) (exp.f64 (*.f64 (neg.f64 1) (log.f64 a)))) (exp.f64 b))) y))): 2 points increase in error, 3 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (/.f64 (*.f64 (exp.f64 (Rewrite=> *-commutative_binary64 (*.f64 t (log.f64 a)))) (exp.f64 (*.f64 (neg.f64 1) (log.f64 a)))) (exp.f64 b)) y))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (/.f64 (Rewrite<= exp-sum_binary64 (exp.f64 (+.f64 (*.f64 t (log.f64 a)) (*.f64 (neg.f64 1) (log.f64 a))))) (exp.f64 b)) y))): 0 points increase in error, 1 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (/.f64 (exp.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 (log.f64 a) (+.f64 t (neg.f64 1))))) (exp.f64 b)) y))): 1 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (/.f64 (exp.f64 (*.f64 (log.f64 a) (Rewrite<= sub-neg_binary64 (-.f64 t 1)))) (exp.f64 b)) y))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 t 1) (log.f64 a)))) (exp.f64 b)) y))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (exp.f64 (*.f64 y (log.f64 z))) (/.f64 (Rewrite<= exp-diff_binary64 (exp.f64 (-.f64 (*.f64 (-.f64 t 1) (log.f64 a)) b))) y))): 1 points increase in error, 29 points decrease in error
      (*.f64 x (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (exp.f64 (-.f64 (*.f64 (-.f64 t 1) (log.f64 a)) b)) y) (exp.f64 (*.f64 y (log.f64 z)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/r/_binary64 (/.f64 (exp.f64 (-.f64 (*.f64 (-.f64 t 1) (log.f64 a)) b)) (/.f64 y (exp.f64 (*.f64 y (log.f64 z))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (exp.f64 (-.f64 (*.f64 (-.f64 t 1) (log.f64 a)) b)) (exp.f64 (*.f64 y (log.f64 z)))) y))): 1 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (Rewrite<= exp-sum_binary64 (exp.f64 (+.f64 (-.f64 (*.f64 (-.f64 t 1) (log.f64 a)) b) (*.f64 y (log.f64 z))))) y)): 3 points increase in error, 37 points decrease in error
      (*.f64 x (/.f64 (exp.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 y (log.f64 z)) (-.f64 (*.f64 (-.f64 t 1) (log.f64 a)) b)))) y)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (exp.f64 (Rewrite<= associate--l+_binary64 (-.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
      (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)): 18 points increase in error, 12 points decrease in error

   3. Taylor expanded in t around 0 8.5

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

   if -80 < (*.f64 (-.f64 t 1) (log.f64 a))

   1. Initial program 1.1

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

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -10000000:\\ \;\;\;\;\frac{x}{a \cdot \frac{y}{{a}^{t}}}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -80:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(\left(t + -1\right) \cdot \log a + y \cdot \log z\right) - b}}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.4
Cost	33868

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := x \cdot \frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{if}\;t_1 \leq -10000000:\\ \;\;\;\;\frac{x}{a \cdot \frac{y}{{a}^{t}}}\\ \mathbf{elif}\;t_1 \leq 380:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 560:\\ \;\;\;\;\frac{x \cdot \left({z}^{y} \cdot {a}^{\left(t + -1\right)}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	7.4
Cost	33804

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := x \cdot \frac{{z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{if}\;t_1 \leq -10000000:\\ \;\;\;\;\frac{x}{a \cdot \frac{y}{{a}^{t}}}\\ \mathbf{elif}\;t_1 \leq 380:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 560:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	2.6
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -519.5:\\ \;\;\;\;\frac{x}{a \cdot \frac{y}{{a}^{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \end{array} \]

Alternative 4
Error	11.7
Cost	13704

\[\begin{array}{l} t_1 := \frac{{z}^{y}}{a}\\ \mathbf{if}\;b \leq -0.00085:\\ \;\;\;\;\frac{x}{a \cdot \frac{y}{{a}^{t}}}\\ \mathbf{elif}\;b \leq -2.7 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{t_1}{e^{b}}\\ \mathbf{elif}\;b \leq -1.85 \cdot 10^{-184}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;b \leq 0.082:\\ \;\;\;\;t_1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 5
Error	11.9
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.52 \cdot 10^{-207}:\\ \;\;\;\;\frac{x}{a \cdot \frac{y}{{a}^{t}}}\\ \mathbf{elif}\;b \leq 0.082:\\ \;\;\;\;\frac{{z}^{y}}{y \cdot \frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 6
Error	11.0
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-182}:\\ \;\;\;\;\frac{x}{a \cdot \frac{y}{{a}^{t}}}\\ \mathbf{elif}\;b \leq 0.082:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 7
Error	11.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-184}:\\ \;\;\;\;\frac{x}{a \cdot \frac{y}{{a}^{t}}}\\ \mathbf{elif}\;b \leq 0.082:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 8
Error	10.4
Cost	7044

\[\begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{a \cdot \frac{y}{{a}^{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 9
Error	16.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;b \leq 0.082:\\ \;\;\;\;x \cdot \left(-1 + \left(1 + \frac{1}{a \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 10
Error	31.1
Cost	1104

\[\begin{array}{l} t_1 := -1 + \left(1 + \frac{x}{a \cdot y}\right)\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	29.3
Cost	1100

\[\begin{array}{l} t_1 := x \cdot \left(-1 + \left(1 + \frac{1}{a \cdot y}\right)\right)\\ \mathbf{if}\;x \leq -1.2508922246572483 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.512416134700596 \cdot 10^{-234}:\\ \;\;\;\;\frac{\frac{x}{y}}{a} \cdot \left(1 - b\right)\\ \mathbf{elif}\;x \leq 455673114.4690578:\\ \;\;\;\;\frac{\frac{x}{a + a \cdot b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	29.9
Cost	972

\[\begin{array}{l} t_1 := -1 + \left(1 + \frac{x}{a \cdot y}\right)\\ \mathbf{if}\;b \leq -2.15 \cdot 10^{-249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq 3.2409093916637476 \cdot 10^{+168}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a + a \cdot b}}{y}\\ \end{array} \]

Alternative 13
Error	41.9
Cost	584

\[\begin{array}{l} t_1 := \frac{\frac{x}{y}}{a}\\ \mathbf{if}\;y \leq -2.95 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	38.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 10^{-12}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \]

Alternative 15
Error	42.8
Cost	320

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

Reproduce

herbie shell --seed 2022300 
(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))