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

Average Error: 2.9 → 0.0

Time: 9.0s

Precision: binary64

Cost: 7104

Math

\[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

↓

\[x + \frac{-1}{x + \frac{-1.1283791670955126}{y} \cdot e^{z}} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

↓

(FPCore (x y z)
 :precision binary64
 (+ x (/ -1.0 (+ x (* (/ -1.1283791670955126 y) (exp z))))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

↓

double code(double x, double y, double z) {
	return x + (-1.0 / (x + ((-1.1283791670955126 / y) * exp(z))));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((-1.0d0) / (x + (((-1.1283791670955126d0) / y) * exp(z))))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

↓

public static double code(double x, double y, double z) {
	return x + (-1.0 / (x + ((-1.1283791670955126 / y) * Math.exp(z))));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

↓

def code(x, y, z):
	return x + (-1.0 / (x + ((-1.1283791670955126 / y) * math.exp(z))))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

↓

function code(x, y, z)
	return Float64(x + Float64(-1.0 / Float64(x + Float64(Float64(-1.1283791670955126 / y) * exp(z)))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
end

↓

function tmp = code(x, y, z)
	tmp = x + (-1.0 / (x + ((-1.1283791670955126 / y) * exp(z))));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(x + N[(-1.0 / N[(x + N[(N[(-1.1283791670955126 / y), $MachinePrecision] * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	2.9
Target	0.0
Herbie	0.0

\[x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x} \]

Derivation

1. Initial program 2.9

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

2. Simplified 0.0

   \[\leadsto \color{blue}{x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)}} \]
   Proof
   (+.f64 x (/.f64 -1 (fma.f64 (exp.f64 z) (/.f64 -5641895835477563/5000000000000000 y) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (exp.f64 z) (/.f64 (Rewrite<= metadata-eval (/.f64 5641895835477563/5000000000000000 -1)) y) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (exp.f64 z) (Rewrite<= associate-/r*_binary64 (/.f64 5641895835477563/5000000000000000 (*.f64 -1 y))) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (exp.f64 z) (/.f64 5641895835477563/5000000000000000 (Rewrite<= neg-mul-1_binary64 (neg.f64 y))) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (exp.f64 z) (/.f64 5641895835477563/5000000000000000 (neg.f64 y))) x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 5641895835477563/5000000000000000 (neg.f64 y)) (exp.f64 z))) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (*.f64 (/.f64 5641895835477563/5000000000000000 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 y))) (exp.f64 z)) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 5641895835477563/5000000000000000 -1) y)) (exp.f64 z)) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (*.f64 (/.f64 (Rewrite=> metadata-eval -5641895835477563/5000000000000000) y) (exp.f64 z)) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (*.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 5641895835477563/5000000000000000)) y) (exp.f64 z)) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (neg.f64 5641895835477563/5000000000000000) (/.f64 y (exp.f64 z)))) x))): 2 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (neg.f64 5641895835477563/5000000000000000) (exp.f64 z)) y)) x))): 0 points increase in error, 1 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (/.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)))) y) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y) (Rewrite=> remove-double-neg_binary64 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y) (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y) (*.f64 x (Rewrite<= *-inverses_binary64 (/.f64 (neg.f64 y) (neg.f64 y))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y) (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 x 1)) (/.f64 (neg.f64 y) (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y) (*.f64 (/.f64 x (Rewrite<= metadata-eval (neg.f64 -1))) (/.f64 (neg.f64 y) (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y) (*.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 x))) (neg.f64 -1)) (/.f64 (neg.f64 y) (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y) (*.f64 (Rewrite=> distribute-frac-neg_binary64 (neg.f64 (/.f64 (neg.f64 x) (neg.f64 -1)))) (/.f64 (neg.f64 y) (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y) (*.f64 (/.f64 (neg.f64 x) (neg.f64 -1)) (/.f64 (neg.f64 y) (neg.f64 y))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 5641895835477563/5000000000000000) (exp.f64 z))) y) (*.f64 (/.f64 (neg.f64 x) (neg.f64 -1)) (/.f64 (neg.f64 y) (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 5641895835477563/5000000000000000) (/.f64 (exp.f64 z) y))) (*.f64 (/.f64 (neg.f64 x) (neg.f64 -1)) (/.f64 (neg.f64 y) (neg.f64 y)))))): 11 points increase in error, 1 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (*.f64 (Rewrite=> metadata-eval -5641895835477563/5000000000000000) (/.f64 (exp.f64 z) y)) (*.f64 (/.f64 (neg.f64 x) (neg.f64 -1)) (/.f64 (neg.f64 y) (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 5641895835477563/5000000000000000 -1)) (/.f64 (exp.f64 z) y)) (*.f64 (/.f64 (neg.f64 x) (neg.f64 -1)) (/.f64 (neg.f64 y) (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 -1 y))) (*.f64 (/.f64 (neg.f64 x) (neg.f64 -1)) (/.f64 (neg.f64 y) (neg.f64 y)))))): 1 points increase in error, 11 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (Rewrite<= neg-mul-1_binary64 (neg.f64 y))) (*.f64 (/.f64 (neg.f64 x) (neg.f64 -1)) (/.f64 (neg.f64 y) (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (neg.f64 y)) (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 (neg.f64 y) (neg.f64 y)) (/.f64 (neg.f64 x) (neg.f64 -1))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (neg.f64 y)) (*.f64 (/.f64 (neg.f64 y) (neg.f64 y)) (/.f64 (neg.f64 x) (Rewrite=> metadata-eval 1)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (neg.f64 y)) (*.f64 (/.f64 (neg.f64 y) (neg.f64 y)) (Rewrite=> /-rgt-identity_binary64 (neg.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (neg.f64 y)) (Rewrite<= associate-/r/_binary64 (/.f64 (neg.f64 y) (/.f64 (neg.f64 y) (neg.f64 x))))))): 22 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (neg.f64 y)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (neg.f64 y) (neg.f64 x)) (neg.f64 y)))))): 15 points increase in error, 22 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (neg.f64 y)) (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 y) (/.f64 (neg.f64 x) (neg.f64 y))))))): 23 points increase in error, 15 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (neg.f64 y)) (*.f64 (neg.f64 y) (Rewrite=> distribute-frac-neg_binary64 (neg.f64 (/.f64 x (neg.f64 y)))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (neg.f64 y)) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (neg.f64 y) (/.f64 x (neg.f64 y)))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (neg.f64 y)) (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 (neg.f64 y)) (/.f64 x (neg.f64 y))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (neg.f64 y)) (*.f64 (Rewrite=> remove-double-neg_binary64 y) (/.f64 x (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (neg.f64 y)) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y x) (neg.f64 y)))))): 15 points increase in error, 23 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (neg.f64 y)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) (neg.f64 y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)) (neg.f64 y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1 (neg.f64 y)) (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))): 13 points increase in error, 7 points decrease in error
   (+.f64 x (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (neg.f64 y) (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (*.f64 -1 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (*.f64 -1 (neg.f64 (Rewrite<= +-lft-identity_binary64 (+.f64 0 (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (*.f64 -1 (Rewrite<= sub0-neg_binary64 (-.f64 0 (+.f64 0 (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (*.f64 -1 (-.f64 0 (Rewrite=> +-lft-identity_binary64 (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 0 (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 x (-.f64 0 (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 x 0) (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> --rgt-identity_binary64 x) (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.0

   \[\leadsto x + \frac{-1}{\color{blue}{\frac{-1.1283791670955126}{y} \cdot e^{z} + x}} \]

4. Final simplification 0.0

   \[\leadsto x + \frac{-1}{x + \frac{-1.1283791670955126}{y} \cdot e^{z}} \]

Alternatives

Alternative 1
Error	0.3
Cost	13896

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.001:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	8.6
Cost	976

\[\begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{-1}{\frac{-1.1283791670955126}{y}}\\ \mathbf{if}\;z \leq -0.00014:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.05 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	8.6
Cost	976

\[\begin{array}{l} t_0 := x + \frac{-1}{\frac{-1.1283791670955126}{y}}\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.86 \cdot 10^{-109}:\\ \;\;\;\;x + \frac{y}{x \cdot \left(-y\right)}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	8.6
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00036:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-76}:\\ \;\;\;\;x + y \cdot \left(0.8862269254527579 + z \cdot -0.8862269254527579\right)\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-109}:\\ \;\;\;\;x + \frac{y}{x \cdot \left(-y\right)}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{-1}{\frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	17.7
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-228} \lor \neg \left(x \leq 8.4 \cdot 10^{-166}\right) \land x \leq 2 \cdot 10^{-27}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	8.6
Cost	848

\[\begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + y \cdot 0.8862269254527579\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	8.6
Cost	848

\[\begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-76}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	0.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -105:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.00186:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	19.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022326 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))