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

Average Error: 4.0 → 2.2

Time: 1.0min

Precision: binary64

Cost: 33408

Math

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

↓

\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, \left(0.8333333333333334 + \left(a + \frac{-0.6666666666666666}{t}\right)\right) \cdot \left(c - b\right)\right)\right)}, x\right)} \]

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
 (/
  x
  (fma
   y
   (pow
    (exp 2.0)
    (fma
     z
     (/ (sqrt (+ t a)) t)
     (* (+ 0.8333333333333334 (+ a (/ -0.6666666666666666 t))) (- c b))))
   x)))

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) {
	return x / fma(y, pow(exp(2.0), fma(z, (sqrt((t + a)) / t), ((0.8333333333333334 + (a + (-0.6666666666666666 / t))) * (c - b)))), x);
}

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)
	return Float64(x / fma(y, (exp(2.0) ^ fma(z, Float64(sqrt(Float64(t + a)) / t), Float64(Float64(0.8333333333333334 + Float64(a + Float64(-0.6666666666666666 / t))) * Float64(c - b)))), x))
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_] := N[(x / N[(y * N[Power[N[Exp[2.0], $MachinePrecision], N[(z * N[(N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision] + N[(N[(0.8333333333333334 + N[(a + N[(-0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Target

Original	4.0
Target	3.0
Herbie	2.2

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

Derivation

1. Initial program 4.0

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

2. Simplified 2.2

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, \left(0.8333333333333334 + \left(a + \frac{-0.6666666666666666}{t}\right)\right) \cdot \left(c - b\right)\right)\right)}, x\right)}} \]
   Proof
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (+.f64 5/6 (+.f64 a (/.f64 -2/3 t))) (-.f64 c b)))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (+.f64 (Rewrite<= metadata-eval (/.f64 5 6)) (+.f64 a (/.f64 -2/3 t))) (-.f64 c b)))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (+.f64 (/.f64 5 6) (+.f64 a (/.f64 (Rewrite<= metadata-eval (neg.f64 2/3)) t))) (-.f64 c b)))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (+.f64 (/.f64 5 6) (+.f64 a (/.f64 (neg.f64 (Rewrite<= metadata-eval (/.f64 2 3))) t))) (-.f64 c b)))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (+.f64 (/.f64 5 6) (+.f64 a (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (/.f64 2 3) t))))) (-.f64 c b)))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (+.f64 (/.f64 5 6) (+.f64 a (neg.f64 (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 3 t)))))) (-.f64 c b)))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (+.f64 (/.f64 5 6) (+.f64 a (neg.f64 (/.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 t 3)))))) (-.f64 c b)))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (+.f64 (/.f64 5 6) (Rewrite=> unsub-neg_binary64 (-.f64 a (/.f64 2 (*.f64 t 3))))) (-.f64 c b)))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 5 6) a) (/.f64 2 (*.f64 t 3)))) (-.f64 c b)))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 a (/.f64 5 6))) (/.f64 2 (*.f64 t 3))) (-.f64 c b)))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (-.f64 (+.f64 a (/.f64 5 6)) (/.f64 2 (*.f64 t 3))) (Rewrite<= unsub-neg_binary64 (+.f64 c (neg.f64 b)))))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (-.f64 (+.f64 a (/.f64 5 6)) (/.f64 2 (*.f64 t 3))) (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) c))))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (-.f64 (+.f64 a (/.f64 5 6)) (/.f64 2 (*.f64 t 3))) (+.f64 (neg.f64 b) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 c))))))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (-.f64 (+.f64 a (/.f64 5 6)) (/.f64 2 (*.f64 t 3))) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 b (neg.f64 c))))))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (*.f64 (-.f64 (+.f64 a (/.f64 5 6)) (/.f64 2 (*.f64 t 3))) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 b c)))))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (-.f64 (+.f64 a (/.f64 5 6)) (/.f64 2 (*.f64 t 3))) (-.f64 b c)))))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (fma.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 5 6)) (/.f64 2 (*.f64 t 3)))))))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 z (/.f64 (sqrt.f64 (+.f64 t a)) t)) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 5 6)) (/.f64 2 (*.f64 t 3))))))) x)): 7 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (pow.f64 (exp.f64 2) (-.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t)) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 5 6)) (/.f64 2 (*.f64 t 3)))))) x)): 4 points increase in error, 0 points decrease in error
   (/.f64 x (fma.f64 y (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 2 (-.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)))))))) x)): 0 points increase in error, 0 points decrease in error
   (/.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (exp.f64 (*.f64 2 (-.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)))))))) x))): 0 points increase in error, 0 points decrease in error
   (/.f64 x (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 y (exp.f64 (*.f64 2 (-.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))))))))))): 0 points increase in error, 0 points decrease in error

3. Final simplification 2.2

   \[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, \left(0.8333333333333334 + \left(a + \frac{-0.6666666666666666}{t}\right)\right) \cdot \left(c - b\right)\right)\right)}, x\right)} \]

Alternatives

Alternative 1
Error	2.3
Cost	22468

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

Alternative 2
Error	12.8
Cost	14292

\[\begin{array}{l} t_1 := \frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot t_1\right)}}\\ \mathbf{if}\;t \leq 8.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-191}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{-0.6666666666666666}{t} \cdot c\right)}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-128}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(b \cdot t_1\right)}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(z \cdot \sqrt{\frac{1}{t}}\right)}}\\ \mathbf{elif}\;t \leq 0.175:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \end{array} \]

Alternative 3
Error	15.1
Cost	14156

\[\begin{array}{l} t_1 := \frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\\ t_2 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot t_1\right)}}\\ t_3 := \frac{x}{x + y \cdot e^{-2 \cdot \left(b \cdot t_1\right)}}\\ \mathbf{if}\;c \leq -2.069190729966668 \cdot 10^{-108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -8.837680362065031 \cdot 10^{-281}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 1.038970714445344 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{t + a} \cdot \frac{z}{t}\right)}}\\ \mathbf{elif}\;c \leq 2.9605180360542122 \cdot 10^{+47}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	28.4
Cost	8164

\[\begin{array}{l} t_1 := \frac{x}{x - y \cdot \left(-1 + \frac{b}{t} \cdot \left(-1.3333333333333333 - \frac{b}{t} \cdot \left(0.8888888888888888 + \frac{b}{t} \cdot 0.3950617283950617\right)\right)\right)}\\ \mathbf{if}\;c \leq -1.2913673752204182 \cdot 10^{+247}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6.0105886446826214 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{x + \left(y + -2 \cdot \left(y \cdot \frac{c \cdot 0.6666666666666666}{t}\right)\right)}\\ \mathbf{elif}\;c \leq -5.6250383738736 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -2.2821864932278078 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.2664462598146077 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -4.885790761625717 \cdot 10^{-263}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - 2 \cdot \left(b \cdot \left(\left(a + 0.8333333333333334\right) + -0.6666666666666666 \cdot \frac{1}{t}\right)\right)\right)}\\ \mathbf{elif}\;c \leq 3.694720243434357 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 5.420356048694009 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 7.134478262757847 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(c - b\right) \cdot 1.6666666666666667}}\\ \end{array} \]

Alternative 5
Error	13.8
Cost	8152

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{-0.6666666666666666}{t} \cdot c\right)}}\\ \mathbf{if}\;t \leq 8.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-128}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{1.3333333333333333 \cdot \frac{b}{t}}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{-17}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{elif}\;t \leq 0.175:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \end{array} \]

Alternative 6
Error	12.9
Cost	8016

\[\begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{-195}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{-0.6666666666666666}{t} \cdot c\right)}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-128}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{1.3333333333333333 \cdot \frac{b}{t}}}\\ \mathbf{elif}\;t \leq 0.175:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \end{array} \]

Alternative 7
Error	12.7
Cost	8016

\[\begin{array}{l} t_1 := \frac{-0.6666666666666666}{t} + \left(a + 0.8333333333333334\right)\\ \mathbf{if}\;t \leq 8.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-191}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{-0.6666666666666666}{t} \cdot c\right)}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-128}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{-2 \cdot \left(b \cdot t_1\right)}}\\ \mathbf{elif}\;t \leq 0.175:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot t_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(c - b\right) \cdot \left(a + 0.8333333333333334\right)\right)}}\\ \end{array} \]

Alternative 8
Error	18.0
Cost	7896

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{-0.6666666666666666}{t} \cdot c\right)}}\\ t_2 := \frac{x}{x + y \cdot e^{\left(c - b\right) \cdot 1.6666666666666667}}\\ \mathbf{if}\;t \leq 8.5 \cdot 10^{-308}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-128}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{1.3333333333333333 \cdot \frac{b}{t}}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{-17}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{elif}\;t \leq 0.175:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	17.3
Cost	7896

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{-0.6666666666666666}{t} \cdot c\right)}}\\ \mathbf{if}\;t \leq 8.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-128}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{1.3333333333333333 \cdot \frac{b}{t}}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{-17}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{elif}\;t \leq 0.175:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(c - b\right) \cdot 1.6666666666666667}}\\ \end{array} \]

Alternative 10
Error	20.9
Cost	7500

\[\begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{\left(c - b\right) \cdot 1.6666666666666667}}\\ \mathbf{if}\;t \leq 4.3 \cdot 10^{-297}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-144}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	30.4
Cost	2656

\[\begin{array}{l} t_1 := \frac{x}{x - y \cdot \left(-1 + \frac{b}{t} \cdot \left(-1.3333333333333333 - \frac{b}{t} \cdot \left(0.8888888888888888 + \frac{b}{t} \cdot 0.3950617283950617\right)\right)\right)}\\ \mathbf{if}\;c \leq -1.2913673752204182 \cdot 10^{+247}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6.0105886446826214 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{x + \left(y + -2 \cdot \left(y \cdot \frac{c \cdot 0.6666666666666666}{t}\right)\right)}\\ \mathbf{elif}\;c \leq -5.6250383738736 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -2.2821864932278078 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.2664462598146077 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -4.885790761625717 \cdot 10^{-263}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - 2 \cdot \left(b \cdot \left(\left(a + 0.8333333333333334\right) + -0.6666666666666666 \cdot \frac{1}{t}\right)\right)\right)}\\ \mathbf{elif}\;c \leq 3.694720243434357 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 5.420356048694009 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 0.2864218430773139:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 6.9182112393878084 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \end{array} \]

Alternative 12
Error	30.3
Cost	2528

\[\begin{array}{l} \mathbf{if}\;c \leq -1.2913673752204182 \cdot 10^{+247}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6.0105886446826214 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{x + \left(y + -2 \cdot \left(y \cdot \frac{c \cdot 0.6666666666666666}{t}\right)\right)}\\ \mathbf{elif}\;c \leq -5.6250383738736 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -2.2821864932278078 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + \frac{b}{t} \cdot \left(1.3333333333333333 + \frac{b}{t} \cdot 0.8888888888888888\right)\right)}\\ \mathbf{elif}\;c \leq -2.2664462598146077 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -4.885790761625717 \cdot 10^{-263}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - 2 \cdot \left(b \cdot \left(\left(a + 0.8333333333333334\right) + -0.6666666666666666 \cdot \frac{1}{t}\right)\right)\right)}\\ \mathbf{elif}\;c \leq 3.694720243434357 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 2.06 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{x - y \cdot \left(-1 + \left(a \cdot \left(c - b\right)\right) \cdot \left(-2 + \left(c - b\right) \cdot \left(a \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;c \leq 0.2864218430773139:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 6.9182112393878084 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \end{array} \]

Alternative 13
Error	30.5
Cost	2272

\[\begin{array}{l} t_1 := \frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ t_2 := \frac{x}{x + \left(y + \left(1.3333333333333333 + \frac{b}{t} \cdot 0.8888888888888888\right) \cdot \left(b \cdot \frac{y}{t}\right)\right)}\\ \mathbf{if}\;c \leq -1.2913673752204182 \cdot 10^{+247}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6.0105886446826214 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{x + \left(y + -2 \cdot \left(y \cdot \frac{c \cdot 0.6666666666666666}{t}\right)\right)}\\ \mathbf{elif}\;c \leq -5.6250383738736 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -2.2821864932278078 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.2664462598146077 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -3.763366789764395 \cdot 10^{-258}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3.694720243434357 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 7.108526263582835 \cdot 10^{-89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 0.2864218430773139:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 6.9182112393878084 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \end{array} \]

Alternative 14
Error	30.4
Cost	2272

\[\begin{array}{l} t_1 := 1.3333333333333333 + \frac{b}{t} \cdot 0.8888888888888888\\ t_2 := \frac{x}{x + y \cdot \left(1 + \frac{b}{t} \cdot t_1\right)}\\ \mathbf{if}\;c \leq -1.2913673752204182 \cdot 10^{+247}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6.0105886446826214 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{x + \left(y + -2 \cdot \left(y \cdot \frac{c \cdot 0.6666666666666666}{t}\right)\right)}\\ \mathbf{elif}\;c \leq -5.6250383738736 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -2.2821864932278078 \cdot 10^{-104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -2.2664462598146077 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -3.763366789764395 \cdot 10^{-258}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3.694720243434357 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 7.108526263582835 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{x + \left(y + t_1 \cdot \left(b \cdot \frac{y}{t}\right)\right)}\\ \mathbf{elif}\;c \leq 0.2864218430773139:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 6.9182112393878084 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \end{array} \]

Alternative 15
Error	30.3
Cost	2272

\[\begin{array}{l} t_1 := 1.3333333333333333 + \frac{b}{t} \cdot 0.8888888888888888\\ \mathbf{if}\;c \leq -1.2913673752204182 \cdot 10^{+247}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6.0105886446826214 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{x + \left(y + -2 \cdot \left(y \cdot \frac{c \cdot 0.6666666666666666}{t}\right)\right)}\\ \mathbf{elif}\;c \leq -5.6250383738736 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -2.2821864932278078 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 + \frac{b}{t} \cdot t_1\right)}\\ \mathbf{elif}\;c \leq -2.2664462598146077 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -4.885790761625717 \cdot 10^{-263}:\\ \;\;\;\;\frac{x}{x + y \cdot \left(1 - 2 \cdot \left(b \cdot \left(\left(a + 0.8333333333333334\right) + -0.6666666666666666 \cdot \frac{1}{t}\right)\right)\right)}\\ \mathbf{elif}\;c \leq 3.694720243434357 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 7.108526263582835 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{x + \left(y + t_1 \cdot \left(b \cdot \frac{y}{t}\right)\right)}\\ \mathbf{elif}\;c \leq 0.2864218430773139:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 6.9182112393878084 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \end{array} \]

Alternative 16
Error	30.8
Cost	1888

\[\begin{array}{l} t_1 := \frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{if}\;c \leq -1.0926653498940973 \cdot 10^{+245}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6.0105886446826214 \cdot 10^{+228}:\\ \;\;\;\;\frac{-0.75}{y} \cdot \left(t \cdot \frac{x}{c}\right)\\ \mathbf{elif}\;c \leq -5.6250383738736 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -2.2821864932278078 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.2664462598146077 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -3.763366789764395 \cdot 10^{-258}:\\ \;\;\;\;\frac{x}{x + \left(y - \frac{y \cdot b}{t} \cdot -1.3333333333333333\right)}\\ \mathbf{elif}\;c \leq 0.2864218430773139:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 6.9182112393878084 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \end{array} \]

Alternative 17
Error	30.6
Cost	1888

\[\begin{array}{l} t_1 := \frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)\\ \mathbf{if}\;c \leq -1.2913673752204182 \cdot 10^{+247}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6.0105886446826214 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{x + \left(y + -2 \cdot \left(y \cdot \frac{c \cdot 0.6666666666666666}{t}\right)\right)}\\ \mathbf{elif}\;c \leq -5.6250383738736 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -2.2821864932278078 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.2664462598146077 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -3.763366789764395 \cdot 10^{-258}:\\ \;\;\;\;\frac{x}{x + \left(y - \frac{y \cdot b}{t} \cdot -1.3333333333333333\right)}\\ \mathbf{elif}\;c \leq 0.2864218430773139:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq 6.9182112393878084 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \end{array} \]

Alternative 18
Error	29.7
Cost	1360

\[\begin{array}{l} \mathbf{if}\;c \leq -1.0926653498940973 \cdot 10^{+245}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6.0105886446826214 \cdot 10^{+228}:\\ \;\;\;\;\frac{-0.75}{y} \cdot \left(t \cdot \frac{x}{c}\right)\\ \mathbf{elif}\;c \leq -2.2664462598146077 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -3.763366789764395 \cdot 10^{-258}:\\ \;\;\;\;\frac{x}{x + \left(y - \frac{y \cdot b}{t} \cdot -1.3333333333333333\right)}\\ \mathbf{elif}\;c \leq 7.134478262757847 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{x + y}\right) + -1\\ \end{array} \]

Alternative 19
Error	31.9
Cost	1236

\[\begin{array}{l} t_1 := \frac{-0.75}{y} \cdot \left(t \cdot \frac{x}{c}\right)\\ \mathbf{if}\;c \leq -1.0926653498940973 \cdot 10^{+245}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6.0105886446826214 \cdot 10^{+228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -5.729091785212044 \cdot 10^{-148}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -3.384830113832022 \cdot 10^{-192}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;c \leq 9.66097151676074 \cdot 10^{+179}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	32.3
Cost	1236

\[\begin{array}{l} \mathbf{if}\;c \leq -1.0926653498940973 \cdot 10^{+245}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6.0105886446826214 \cdot 10^{+228}:\\ \;\;\;\;\frac{-0.75}{y} \cdot \left(t \cdot \frac{x}{c}\right)\\ \mathbf{elif}\;c \leq -5.729091785212044 \cdot 10^{-148}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -3.384830113832022 \cdot 10^{-192}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;c \leq 1.2372633676933432 \cdot 10^{+67}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{c \cdot \left(y \cdot a\right)}\\ \end{array} \]

Alternative 21
Error	29.5
Cost	1236

\[\begin{array}{l} t_1 := \left(1 + \frac{x}{x + y}\right) + -1\\ \mathbf{if}\;c \leq -1.0926653498940973 \cdot 10^{+245}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -6.0105886446826214 \cdot 10^{+228}:\\ \;\;\;\;\frac{-0.75}{y} \cdot \left(t \cdot \frac{x}{c}\right)\\ \mathbf{elif}\;c \leq -2.2664462598146077 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -8.534585543015326 \cdot 10^{-244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 7.134478262757847 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 22
Error	31.1
Cost	840

\[\begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -3.5160026330361374 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.090949660127913 \cdot 10^{+210}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 23
Error	32.0
Cost	716

\[\begin{array}{l} t_1 := \frac{x}{x + y}\\ \mathbf{if}\;c \leq -2.2664462598146077 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;c \leq -8.534585543015326 \cdot 10^{-244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 4.7190591668407655 \cdot 10^{+117}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 24
Error	30.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022316 
(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)))))))))))