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

Average Error: 3.1 → 0.4

Time: 14.3s

Precision: binary64

Cost: 20168

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 5 \cdot 10^{-42}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.0000000000002:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 5e-42)
   (- x (/ 1.0 x))
   (if (<= (exp z) 1.0000000000002)
     (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))
     x)))

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) {
	double tmp;
	if (exp(z) <= 5e-42) {
		tmp = x - (1.0 / x);
	} else if (exp(z) <= 1.0000000000002) {
		tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (exp(z) <= 5d-42) then
        tmp = x - (1.0d0 / x)
    else if (exp(z) <= 1.0000000000002d0) then
        tmp = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
    else
        tmp = x
    end if
    code = tmp
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) {
	double tmp;
	if (Math.exp(z) <= 5e-42) {
		tmp = x - (1.0 / x);
	} else if (Math.exp(z) <= 1.0000000000002) {
		tmp = x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 5e-42:
		tmp = x - (1.0 / x)
	elif math.exp(z) <= 1.0000000000002:
		tmp = x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))
	else:
		tmp = x
	return tmp

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)
	tmp = 0.0
	if (exp(z) <= 5e-42)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (exp(z) <= 1.0000000000002)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 5e-42)
		tmp = x - (1.0 / x);
	elseif (exp(z) <= 1.0000000000002)
		tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
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_] := If[LessEqual[N[Exp[z], $MachinePrecision], 5e-42], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.0000000000002], N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Target

Original	3.1
Target	0.1
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 5.00000000000000003e-42

   1. Initial program 8.0

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

   2. Taylor expanded in x around inf 0.0

      \[\leadsto \color{blue}{x - \frac{1}{x}} \]

   if 5.00000000000000003e-42 < (exp.f64 z) < 1.00000000000020006

   1. Initial program 0.1

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

   if 1.00000000000020006 < (exp.f64 z)

   1. Initial program 3.9

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

   2. Taylor expanded in x around inf 1.4

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 5 \cdot 10^{-42}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.0000000000002:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	13896

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 5 \cdot 10^{-42}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.0000000000002:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	18.0
Cost	980

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+146}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{+77}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -2.19 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-13}:\\ \;\;\;\;0.8862269254527579 \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	18.0
Cost	980

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+147}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+75}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -2.19 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{1.1283791670955126} + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	8.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00024:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{1.1283791670955126 + 1.1283791670955126 \cdot z} + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	0.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -95:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	19.3
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-223}:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-195}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	19.3
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-223}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	8.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00115:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{1.1283791670955126} + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	19.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-238}:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	20.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023077 
(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)))))