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

Average Error: 2.9 → 0.4

Time: 11.3s

Precision: binary64

Cost: 19912

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;0.8862269254527579 \cdot \frac{y}{e^{z}} + x\\ \end{array} \]

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) 1.0)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     (+ (* 0.8862269254527579 (/ y (exp z))) 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) <= 1.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = (0.8862269254527579 * (y / exp(z))) + 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) <= 1.0d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * 1.1283791670955126d0)) - (x * y)))
    else
        tmp = (0.8862269254527579d0 * (y / exp(z))) + 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) <= 1.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = (0.8862269254527579 * (y / Math.exp(z))) + 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) <= 1.0:
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)))
	else:
		tmp = (0.8862269254527579 * (y / math.exp(z))) + 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) <= 1.0)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * 1.1283791670955126)) - Float64(x * y))));
	else
		tmp = Float64(Float64(0.8862269254527579 * Float64(y / exp(z))) + 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) <= 1.0)
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	else
		tmp = (0.8862269254527579 * (y / exp(z))) + 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], 1.0], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.8862269254527579 * N[(y / N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Target

Original	2.9
Target	0.0
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) < 0.0

   1. Initial program 7.9

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

   2. Taylor expanded in x around inf 0.0

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 0.1

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

   2. Taylor expanded in z around 0 0.3

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

   3. Simplified 0.3

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

      [Start] 0.3	\[ x + \frac{y}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right) - x \cdot y} \]
      rational.json-simplify-2 [=>] 0.3	\[ x + \frac{y}{\left(1.1283791670955126 + \color{blue}{z \cdot 1.1283791670955126}\right) - x \cdot y} \]

   if 1 < (exp.f64 z)

   1. Initial program 3.3

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

   2. Taylor expanded in y around 0 1.0

      \[\leadsto \color{blue}{0.8862269254527579 \cdot \frac{y}{e^{z}} + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;0.8862269254527579 \cdot \frac{y}{e^{z}} + x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13896

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

Alternative 2
Error	0.9
Cost	13636

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

Alternative 3
Error	8.6
Cost	848

\[\begin{array}{l} t_0 := x - \frac{1}{x}\\ \mathbf{if}\;z \leq -1.15:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{-122}:\\ \;\;\;\;0.8862269254527579 \cdot y + x\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{1.1283791670955126} + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	8.6
Cost	848

\[\begin{array}{l} t_0 := x - \frac{1}{x}\\ \mathbf{if}\;z \leq -0.028:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-122}:\\ \;\;\;\;\left(-1 + z\right) \cdot \left(y \cdot -0.8862269254527579\right) + x\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{1.1283791670955126} + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	0.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -225:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 300:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	20.0
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-28}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{-17}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	16.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -510:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-27}:\\ \;\;\;\;0.8862269254527579 \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	16.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -800:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{1.1283791670955126} + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	19.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023074 
(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)))))