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

Average Error: 4.0 → 2.3

Time: 1.0min

Precision: binary64

Cost: 22468

Math

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

↓

\[\begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\left(-0.8333333333333334 - a\right) - \frac{-2}{t \cdot 3}\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\right)\right)}}\\ \end{array} \]

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
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (- -0.8333333333333334 a) (/ -2.0 (* t 3.0)))))))
   (if (<= t_1 INFINITY)
     (/ x (+ x (* y (exp (* 2.0 t_1)))))
     (/
      x
      (+
       x
       (*
        y
        (exp
         (*
          2.0
          (* c (+ (/ -0.6666666666666666 t) (+ a 0.8333333333333334)))))))))))

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 = ((z * sqrt((t + a))) / t) + ((b - c) * ((-0.8333333333333334 - a) - (-2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = x / (x + (y * exp((2.0 * t_1))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (c * ((-0.6666666666666666 / t) + (a + 0.8333333333333334)))))));
	}
	return tmp;
}

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 = ((z * Math.sqrt((t + a))) / t) + ((b - c) * ((-0.8333333333333334 - a) - (-2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x / (x + (y * Math.exp((2.0 * t_1))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * (c * ((-0.6666666666666666 / t) + (a + 0.8333333333333334)))))));
	}
	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 = ((z * math.sqrt((t + a))) / t) + ((b - c) * ((-0.8333333333333334 - a) - (-2.0 / (t * 3.0))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = x / (x + (y * math.exp((2.0 * t_1))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * (c * ((-0.6666666666666666 / t) + (a + 0.8333333333333334)))))))
	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 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(-0.8333333333333334 - a) - Float64(-2.0 / Float64(t * 3.0)))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * t_1)))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(c * Float64(Float64(-0.6666666666666666 / t) + Float64(a + 0.8333333333333334))))))));
	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 = ((z * sqrt((t + a))) / t) + ((b - c) * ((-0.8333333333333334 - a) - (-2.0 / (t * 3.0))));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = x / (x + (y * exp((2.0 * t_1))));
	else
		tmp = x / (x + (y * exp((2.0 * (c * ((-0.6666666666666666 / t) + (a + 0.8333333333333334)))))));
	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[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(-0.8333333333333334 - a), $MachinePrecision] - N[(-2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(c * N[(N[(-0.6666666666666666 / t), $MachinePrecision] + N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	4.0
Target	2.8
Herbie	2.3

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 5 6)) (/.f64 2 (*.f64 t 3))))) < +inf.0

   1. Initial program 0.8

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

   if +inf.0 < (-.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 64.0

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

   2. Taylor expanded in c around inf 30.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - 0.6666666666666666 \cdot \frac{1}{t}\right)\right)}}} \]

   3. Simplified 30.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\frac{-0.6666666666666666}{t} - \left(-0.8333333333333334 - a\right)\right)\right)}}} \]
      Proof
      (*.f64 c (-.f64 (/.f64 -2/3 t) (-.f64 -5/6 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (-.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 2/3)) t) (-.f64 -5/6 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (-.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2/3 t))) (-.f64 -5/6 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (-.f64 (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2/3 1)) t)) (-.f64 -5/6 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (-.f64 (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2/3 (/.f64 1 t)))) (-.f64 -5/6 a))): 11 points increase in error, 7 points decrease in error
      (*.f64 c (-.f64 (neg.f64 (*.f64 2/3 (/.f64 1 t))) (-.f64 (Rewrite<= metadata-eval (neg.f64 5/6)) a))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (-.f64 (neg.f64 (*.f64 2/3 (/.f64 1 t))) (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 5/6) (neg.f64 a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (-.f64 (neg.f64 (*.f64 2/3 (/.f64 1 t))) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 5/6 a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (*.f64 2/3 (/.f64 1 t))) (neg.f64 (neg.f64 (+.f64 5/6 a)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (*.f64 2/3 (/.f64 1 t)) (neg.f64 (+.f64 5/6 a)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 2/3 (/.f64 1 t)) (+.f64 5/6 a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (Rewrite=> neg-sub0_binary64 (-.f64 0 (-.f64 (*.f64 2/3 (/.f64 1 t)) (+.f64 5/6 a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (*.f64 2/3 (/.f64 1 t))) (+.f64 5/6 a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (*.f64 2/3 (/.f64 1 t)))) (+.f64 5/6 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 5/6 a) (neg.f64 (*.f64 2/3 (/.f64 1 t)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 c (Rewrite<= sub-neg_binary64 (-.f64 (+.f64 5/6 a) (*.f64 2/3 (/.f64 1 t))))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\left(-0.8333333333333334 - a\right) - \frac{-2}{t \cdot 3}\right) \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\left(-0.8333333333333334 - a\right) - \frac{-2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\right)\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	19.8
Cost	14556

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + \frac{0.6666666666666666}{t}\right)\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{if}\;z \leq -1.3587598816432714 \cdot 10^{+271}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.1568536623323058 \cdot 10^{-82}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \mathbf{elif}\;z \leq -9.346585558194183 \cdot 10^{-257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.736690788139636 \cdot 10^{-256}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.0028871000913284 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.7915836445813044 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.926080433246508 \cdot 10^{+172}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(z \cdot \sqrt{\frac{1}{t}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	15.0
Cost	14288

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t}\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\right)\right)}}\\ t_3 := \frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + \frac{0.6666666666666666}{t}\right)\right)}}\\ \mathbf{if}\;c \leq -4.1860684855800244 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -2.172329575732174 \cdot 10^{-222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.161860398982181 \cdot 10^{-291}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 4.317998476977275 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 4.111574321632286 \cdot 10^{+89}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	29.5
Cost	8296

\[\begin{array}{l} t_1 := \frac{x}{x - y \cdot \left(-1 - 2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + -0.6666666666666666 \cdot \frac{-1}{t}\right)\right)\right)}\\ \mathbf{if}\;c \leq -1.415692792391186 \cdot 10^{+115}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -0.01113076800408496:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1.3333333333333333 \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;c \leq -1.6214249570254963 \cdot 10^{-198}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{b \cdot -1.6666666666666667}}\\ \mathbf{elif}\;c \leq -1.0830955490488113 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.2291995512871668 \cdot 10^{-243}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 1.5515092979839378 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 6.753503170724154 \cdot 10^{-129}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 2.601858717370227 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.887679779318286 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 2.8783487460356612 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \end{array} \]

Alternative 4
Error	23.7
Cost	8160

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{if}\;a \leq -9.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(a \cdot b\right)}}\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{-245}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-300}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-272}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{0.6666666666666666}{\frac{t}{b}}}}\\ \mathbf{elif}\;a \leq 2.364772291672499 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.241408041332132 \cdot 10^{+63}:\\ \;\;\;\;1\\ \mathbf{elif}\;a \leq 1.4769680897907026 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	19.1
Cost	8156

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \frac{0.6666666666666666}{\frac{t}{b}}}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \frac{c}{\frac{t}{-0.6666666666666666}}}}\\ t_3 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{-278}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.028:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2.5625640114086796 \cdot 10^{+261}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \end{array} \]

Alternative 6
Error	15.2
Cost	8152

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \frac{0.6666666666666666}{\frac{t}{b}}}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \frac{c}{\frac{t}{-0.6666666666666666}}}}\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{-278}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.028:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \end{array} \]

Alternative 7
Error	31.1
Cost	8032

\[\begin{array}{l} t_1 := \frac{x}{x - y \cdot \left(-1 - 2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + -0.6666666666666666 \cdot \frac{-1}{t}\right)\right)\right)}\\ \mathbf{if}\;c \leq -2.8758451980699706 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -1.0830955490488113 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.2291995512871668 \cdot 10^{-243}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 1.5515092979839378 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 6.753503170724154 \cdot 10^{-129}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 2.601858717370227 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.887679779318286 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 2.8783487460356612 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \end{array} \]

Alternative 8
Error	27.0
Cost	8028

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{if}\;y \leq -2.8378415245697904 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.257283015643677 \cdot 10^{-247}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9.177163888242033 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.940591713250296 \cdot 10^{+120}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(a \cdot b\right)}}\\ \mathbf{elif}\;y \leq 1.5750906217394215 \cdot 10^{+179}:\\ \;\;\;\;\frac{x}{x - y \cdot \left(-1 - 2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + -0.6666666666666666 \cdot \frac{-1}{t}\right)\right)\right)}\\ \mathbf{elif}\;y \leq 2.0490124920017267 \cdot 10^{+188}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{elif}\;y \leq 1.4805398428805562 \cdot 10^{+221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.0708060870498955 \cdot 10^{+253}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{elif}\;y \leq 5.918483819732073 \cdot 10^{+291}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 1.3333333333333333 \cdot \frac{b}{t}\right)}\\ \end{array} \]

Alternative 9
Error	31.5
Cost	7900

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{b \cdot -1.6666666666666667}}\\ \mathbf{if}\;x \leq -3.9561871705938593 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.97357803962773 \cdot 10^{-184}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5.612124522435174 \cdot 10^{-243}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{elif}\;x \leq -1.2600379544376568 \cdot 10^{-287}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.216724517316541 \cdot 10^{-194}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(a \cdot b\right)}}\\ \mathbf{elif}\;x \leq 4.640017171719926 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.719593169931984 \cdot 10^{+183}:\\ \;\;\;\;\frac{x}{x - y \cdot \left(-1 - 2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + -0.6666666666666666 \cdot \frac{-1}{t}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	19.1
Cost	7896

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \frac{0.6666666666666666}{\frac{t}{b}}}}\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \frac{c}{\frac{t}{-0.6666666666666666}}}}\\ t_3 := \frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{-278}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.028:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 11
Error	14.4
Cost	7752

\[\begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-291}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b - c\right) \cdot -1.6666666666666667}}\\ \mathbf{elif}\;t \leq 7.881540533187586 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + \frac{0.6666666666666666}{t}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(b - c\right) \cdot \left(-0.8333333333333334 - a\right)\right)}}\\ \end{array} \]

Alternative 12
Error	13.5
Cost	7752

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{if}\;c \leq -3.445983704584345 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 4.111574321632286 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + \frac{0.6666666666666666}{t}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	33.9
Cost	2400

\[\begin{array}{l} t_1 := \frac{x}{x - y \cdot \left(-1 - 2 \cdot \left(b \cdot \left(\left(-0.8333333333333334 - a\right) + -0.6666666666666666 \cdot \frac{-1}{t}\right)\right)\right)}\\ \mathbf{if}\;z \leq -5.560102286709182 \cdot 10^{+190}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.0719872164362018 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{elif}\;z \leq -1.7746167903223702 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -3.960421876784285 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;z \leq 2.942685750940532 \cdot 10^{-264}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 4.842984899129378 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.1077224311674708 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 3.0872827417063757 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	33.0
Cost	1752

\[\begin{array}{l} t_1 := \frac{x}{x + \left(y + 2 \cdot \left(a \cdot \left(y \cdot \left(c - b\right)\right)\right)\right)}\\ \mathbf{if}\;z \leq -5.560102286709182 \cdot 10^{+190}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.0719872164362018 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.7746167903223702 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -3.960421876784285 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;z \leq 8.669689693620235 \cdot 10^{-215}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 4.842984899129378 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.1334709978530805 \cdot 10^{+103}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 5.926080433246508 \cdot 10^{+172}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Error	33.0
Cost	1632

\[\begin{array}{l} t_1 := \left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{if}\;z \leq -1.363820409253221 \cdot 10^{+235}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -1.0719872164362018 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.7746167903223702 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -3.960421876784285 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{elif}\;z \leq 5.735259243003259 \cdot 10^{-271}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 4.842984899129378 \cdot 10^{-103}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + 1.3333333333333333 \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;z \leq 3.1334709978530805 \cdot 10^{+103}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 5.926080433246508 \cdot 10^{+172}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	31.1
Cost	1500

\[\begin{array}{l} t_1 := \frac{x}{x + y}\\ t_2 := \left(1 + t_1\right) + -1\\ \mathbf{if}\;t \leq 4.9 \cdot 10^{-301}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-223}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1.3333333333333333 \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-159}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 53:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 7.082502641584921 \cdot 10^{+49}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.935365439300556 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.0351249839669863 \cdot 10^{+200}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	31.1
Cost	1500

\[\begin{array}{l} t_1 := 1.3333333333333333 \cdot \frac{b}{t}\\ t_2 := \left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{if}\;c \leq -1.415692792391186 \cdot 10^{+115}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -3.6998399817185366 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{x + y \cdot t_1}\\ \mathbf{elif}\;c \leq 1.3519878417317133 \cdot 10^{-273}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 4.410696063017626 \cdot 10^{-259}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + t_1\right)}\\ \mathbf{elif}\;c \leq 5.476832273819477 \cdot 10^{-175}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 6.753503170724154 \cdot 10^{-129}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 1.6697088279711822 \cdot 10^{+169}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	31.6
Cost	1368

\[\begin{array}{l} t_1 := \left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{if}\;z \leq -1.363820409253221 \cdot 10^{+235}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -3.960421876784285 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.735259243003259 \cdot 10^{-271}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 4.842984899129378 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.1334709978530805 \cdot 10^{+103}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 5.926080433246508 \cdot 10^{+172}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Error	31.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq 1.8543537228861388 \cdot 10^{-196}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 4.842984899129378 \cdot 10^{-103}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Error	31.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022317 
(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)))))))))))