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

Average Error: 2.8 → 0.3

Time: 13.4s

Precision: binary64

Cost: 19912

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 4 \cdot 10^{-262}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.0000002:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 4e-262)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.0000002)
     (+ x (/ y (- 1.1283791670955126 (* x y))))
     (+ x (/ y (* (exp z) 1.1283791670955126))))))

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) <= 4e-262) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.0000002) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x + (y / (exp(z) * 1.1283791670955126));
	}
	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) <= 4d-262) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 1.0000002d0) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = x + (y / (exp(z) * 1.1283791670955126d0))
    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) <= 4e-262) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 1.0000002) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x + (y / (Math.exp(z) * 1.1283791670955126));
	}
	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) <= 4e-262:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 1.0000002:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x + (y / (math.exp(z) * 1.1283791670955126))
	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) <= 4e-262)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.0000002)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = Float64(x + Float64(y / Float64(exp(z) * 1.1283791670955126)));
	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) <= 4e-262)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 1.0000002)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x + (y / (exp(z) * 1.1283791670955126));
	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], 4e-262], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.0000002], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	2.8
Target	0.0
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) < 4.00000000000000005e-262

   1. Initial program 7.4

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

   2. Simplified 0.0

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

      [Start] 7.4	\[ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
      --rgt-identity [<=] 7.4	\[ \color{blue}{\left(x - 0\right)} + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
      associate-+l- [=>] 7.4	\[ \color{blue}{x - \left(0 - \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)} \]
      sub-neg [=>] 7.4	\[ \color{blue}{x + \left(-\left(0 - \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)\right)} \]
      +-lft-identity [<=] 7.4	\[ x + \left(-\left(0 - \color{blue}{\left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)}\right)\right) \]
      sub0-neg [=>] 7.4	\[ x + \left(-\color{blue}{\left(-\left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)\right)}\right) \]
      neg-mul-1 [=>] 7.4	\[ x + \left(-\color{blue}{-1 \cdot \left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)}\right) \]
      distribute-lft-neg-in [=>] 7.4	\[ x + \color{blue}{\left(--1\right) \cdot \left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)} \]
      +-lft-identity [=>] 7.4	\[ x + \left(--1\right) \cdot \color{blue}{\frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]
      associate-*r/ [=>] 7.4	\[ x + \color{blue}{\frac{\left(--1\right) \cdot y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]
      sub-neg [=>] 7.4	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{1.1283791670955126 \cdot e^{z} + \left(-x \cdot y\right)}} \]
      +-commutative [=>] 7.4	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{\left(-x \cdot y\right) + 1.1283791670955126 \cdot e^{z}}} \]
      neg-sub0 [=>] 7.4	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{\left(0 - x \cdot y\right)} + 1.1283791670955126 \cdot e^{z}} \]
      associate-+l- [=>] 7.4	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{0 - \left(x \cdot y - 1.1283791670955126 \cdot e^{z}\right)}} \]
      sub0-neg [=>] 7.4	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{-\left(x \cdot y - 1.1283791670955126 \cdot e^{z}\right)}} \]
      neg-mul-1 [=>] 7.4	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{-1 \cdot \left(x \cdot y - 1.1283791670955126 \cdot e^{z}\right)}} \]
      times-frac [=>] 7.4	\[ x + \color{blue}{\frac{--1}{-1} \cdot \frac{y}{x \cdot y - 1.1283791670955126 \cdot e^{z}}} \]

   3. Taylor expanded in x around inf 0.0

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

   if 4.00000000000000005e-262 < (exp.f64 z) < 1.00000019999999989

   1. Initial program 0.1

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

   2. Taylor expanded in z around 0 0.4

      \[\leadsto \color{blue}{\frac{y}{1.1283791670955126 - y \cdot x} + x} \]

   if 1.00000019999999989 < (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 0 0.4

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

   3. Simplified 0.4

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

      [Start] 0.4	\[ x + \frac{y}{1.1283791670955126 \cdot e^{z}} \]
      *-commutative [=>] 0.4	\[ x + \frac{y}{\color{blue}{e^{z} \cdot 1.1283791670955126}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 4 \cdot 10^{-262}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.0000002:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	13896

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 4 \cdot 10^{-262}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.005:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	13376

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

Alternative 3
Error	19.0
Cost	984

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-156}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-290}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{-16}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	16.6
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+198}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -600:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-44}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	0.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -245:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.0082:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	8.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -15.5:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-38}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	18.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-194}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-249}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	19.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023016 
(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)))))