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

Average Error: 2.8 → 0.3

Time: 17.8s

Precision: binary64

Cost: 13896

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x - \frac{y}{\left(y \cdot x + -1.1283791670955126 \cdot z\right) - 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (- x (/ 1.0 x))
   (if (<= (exp z) 2.0)
     (- x (/ y (- (+ (* y x) (* -1.1283791670955126 z)) 1.1283791670955126)))
     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) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (exp(z) <= 2.0) {
		tmp = x - (y / (((y * x) + (-1.1283791670955126 * z)) - 1.1283791670955126));
	} 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) <= 0.0d0) then
        tmp = x - (1.0d0 / x)
    else if (exp(z) <= 2.0d0) then
        tmp = x - (y / (((y * x) + ((-1.1283791670955126d0) * z)) - 1.1283791670955126d0))
    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) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (Math.exp(z) <= 2.0) {
		tmp = x - (y / (((y * x) + (-1.1283791670955126 * z)) - 1.1283791670955126));
	} 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) <= 0.0:
		tmp = x - (1.0 / x)
	elif math.exp(z) <= 2.0:
		tmp = x - (y / (((y * x) + (-1.1283791670955126 * z)) - 1.1283791670955126))
	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) <= 0.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (exp(z) <= 2.0)
		tmp = Float64(x - Float64(y / Float64(Float64(Float64(y * x) + Float64(-1.1283791670955126 * z)) - 1.1283791670955126)));
	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) <= 0.0)
		tmp = x - (1.0 / x);
	elseif (exp(z) <= 2.0)
		tmp = x - (y / (((y * x) + (-1.1283791670955126 * z)) - 1.1283791670955126));
	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], 0.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(x - N[(y / N[(N[(N[(y * x), $MachinePrecision] + N[(-1.1283791670955126 * z), $MachinePrecision]), $MachinePrecision] - 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Target

Original	2.8
Target	0.1
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 6.6

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

   2. Taylor expanded in x around inf 0.0

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126 \cdot e^{z}\right)}} \]
      Proof

   3. Taylor expanded in z around 0 0.4

      \[\leadsto x - \frac{y}{\color{blue}{\left(y \cdot x + -1.1283791670955126 \cdot z\right) - 1.1283791670955126}} \]

   if 2 < (exp.f64 z)

   1. Initial program 4.3

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

   2. Taylor expanded in x around inf 0.3

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

Alternatives

Alternative 1
Error	0.1
Cost	19908

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

Alternative 2
Error	1.3
Cost	14276

\[\begin{array}{l} t_0 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+198}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array} \]

Alternative 3
Error	9.0
Cost	1368

\[\begin{array}{l} t_0 := x - \frac{1}{x}\\ t_1 := x - \frac{y}{-1.1283791670955126}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.58 \cdot 10^{-282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-240}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-7}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126 \cdot z - 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	9.0
Cost	1112

\[\begin{array}{l} t_0 := x - \frac{1}{x}\\ t_1 := x - \frac{y}{-1.1283791670955126}\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-246}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	0.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+17}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 9:\\ \;\;\;\;x - \frac{y}{y \cdot x - 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	19.1
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-139}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-160}:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	19.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-127}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{-159}:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	11.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{-217}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	19.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023010 
(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)))))