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

Average Accuracy: 95.5% → 99.5%

Time: 10.2s

Precision: binary64

Cost: 13896

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.00000001:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (- x (/ 1.0 x))
   (if (<= (exp z) 1.00000001)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* 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) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (exp(z) <= 1.00000001) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (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) <= 0.0d0) then
        tmp = x - (1.0d0 / x)
    else if (exp(z) <= 1.00000001d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * 1.1283791670955126d0)) - (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) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (Math.exp(z) <= 1.00000001) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (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) <= 0.0:
		tmp = x - (1.0 / x)
	elif math.exp(z) <= 1.00000001:
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (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) <= 0.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (exp(z) <= 1.00000001)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * 1.1283791670955126)) - 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) <= 0.0)
		tmp = x - (1.0 / x);
	elseif (exp(z) <= 1.00000001)
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (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], 0.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.00000001], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Target

Original	95.5%
Target	99.9%
Herbie	99.5%

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 88.8%

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

   2. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)}} \]
      Proof

      [Start] 88.8	\[ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
      --rgt-identity [<=] 88.8	\[ \color{blue}{\left(x - 0\right)} + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
      associate-+l- [=>] 88.8	\[ \color{blue}{x - \left(0 - \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)} \]
      sub-neg [=>] 88.8	\[ \color{blue}{x + \left(-\left(0 - \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)\right)} \]
      +-lft-identity [<=] 88.8	\[ x + \left(-\left(0 - \color{blue}{\left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)}\right)\right) \]
      sub0-neg [=>] 88.8	\[ x + \left(-\color{blue}{\left(-\left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)\right)}\right) \]
      neg-mul-1 [=>] 88.8	\[ x + \left(-\color{blue}{-1 \cdot \left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)}\right) \]
      distribute-lft-neg-in [=>] 88.8	\[ x + \color{blue}{\left(--1\right) \cdot \left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)} \]
      +-lft-identity [=>] 88.8	\[ x + \left(--1\right) \cdot \color{blue}{\frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]
      associate-*r/ [=>] 88.8	\[ x + \color{blue}{\frac{\left(--1\right) \cdot y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]
      sub-neg [=>] 88.8	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{1.1283791670955126 \cdot e^{z} + \left(-x \cdot y\right)}} \]
      +-commutative [=>] 88.8	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{\left(-x \cdot y\right) + 1.1283791670955126 \cdot e^{z}}} \]
      neg-sub0 [=>] 88.8	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{\left(0 - x \cdot y\right)} + 1.1283791670955126 \cdot e^{z}} \]
      associate-+l- [=>] 88.8	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{0 - \left(x \cdot y - 1.1283791670955126 \cdot e^{z}\right)}} \]
      sub0-neg [=>] 88.8	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{-\left(x \cdot y - 1.1283791670955126 \cdot e^{z}\right)}} \]
      neg-mul-1 [=>] 88.8	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{-1 \cdot \left(x \cdot y - 1.1283791670955126 \cdot e^{z}\right)}} \]
      times-frac [=>] 88.8	\[ x + \color{blue}{\frac{--1}{-1} \cdot \frac{y}{x \cdot y - 1.1283791670955126 \cdot e^{z}}} \]

   3. Taylor expanded in x around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1.0000000099999999

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.6%

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

   3. Simplified 99.6%

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

      [Start] 99.6	\[ x + \frac{y}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right) - y \cdot x} \]
      *-commutative [=>] 99.6	\[ x + \frac{y}{\left(1.1283791670955126 + \color{blue}{z \cdot 1.1283791670955126}\right) - y \cdot x} \]

   if 1.0000000099999999 < (exp.f64 z)

   1. Initial program 93.8%

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

   2. Taylor expanded in x around inf 98.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.00000001:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13376

\[x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)} \]

Alternative 2
Accuracy	86.1%
Cost	1104

\[\begin{array}{l} t_0 := x - \frac{1}{x}\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-80}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-9}:\\ \;\;\;\;x + 0.8862269254527579 \cdot \left(y - z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	72.6%
Cost	980

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-156}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-174}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-212}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-113}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	86.0%
Cost	848

\[\begin{array}{l} t_0 := x - \frac{1}{x}\\ t_1 := x + \frac{y}{1.1283791670955126}\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	99.2%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1300000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 10^{-8}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	69.2%
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-156}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-174}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.62 \cdot 10^{-218}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	69.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023138 
(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)))))