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

Percentage Accurate: 95.6% → 99.3%

Time: 9.7s

Precision: binary64

Cost: 19912

Math

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

↓

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

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 (* x y))))
     (+ x (/ (* y 0.8862269254527579) (exp z))))))

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

Target

Original	95.6%
Target	99.9%
Herbie	99.3%

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 85.6%

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

   2. Simplified 100.0%

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

      [Start] 85.6%	\[ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
      *-lft-identity [<=] 85.6%	\[ x + \color{blue}{1 \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]
      metadata-eval [<=] 85.6%	\[ x + \color{blue}{\frac{-1}{-1}} \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
      times-frac [<=] 85.6%	\[ x + \color{blue}{\frac{-1 \cdot y}{-1 \cdot \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]
      neg-mul-1 [<=] 85.6%	\[ x + \frac{-1 \cdot y}{\color{blue}{-\left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]
      sub0-neg [<=] 85.4%	\[ x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]
      associate-+l- [<=] 85.4%	\[ x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]
      neg-sub0 [<=] 85.8%	\[ x + \frac{-1 \cdot y}{\color{blue}{\left(-1.1283791670955126 \cdot e^{z}\right)} + x \cdot y} \]
      +-commutative [<=] 85.8%	\[ x + \frac{-1 \cdot y}{\color{blue}{x \cdot y + \left(-1.1283791670955126 \cdot e^{z}\right)}} \]
      sub-neg [<=] 85.8%	\[ x + \frac{-1 \cdot y}{\color{blue}{x \cdot y - 1.1283791670955126 \cdot e^{z}}} \]
      associate-/l* [=>] 86.0%	\[ x + \color{blue}{\frac{-1}{\frac{x \cdot y - 1.1283791670955126 \cdot e^{z}}{y}}} \]
      div-sub [=>] 85.9%	\[ x + \frac{-1}{\color{blue}{\frac{x \cdot y}{y} - \frac{1.1283791670955126 \cdot e^{z}}{y}}} \]
      associate-*r/ [<=] 100.0%	\[ x + \frac{-1}{\color{blue}{x \cdot \frac{y}{y}} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]
      *-inverses [=>] 100.0%	\[ x + \frac{-1}{x \cdot \color{blue}{1} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]
      *-rgt-identity [=>] 100.0%	\[ x + \frac{-1}{\color{blue}{x} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]
      associate-*l/ [<=] 100.0%	\[ x + \frac{-1}{x - \color{blue}{\frac{1.1283791670955126}{y} \cdot e^{z}}} \]
      cancel-sign-sub-inv [=>] 100.0%	\[ x + \frac{-1}{\color{blue}{x + \left(-\frac{1.1283791670955126}{y}\right) \cdot e^{z}}} \]
      distribute-lft-neg-in [<=] 100.0%	\[ x + \frac{-1}{x + \color{blue}{\left(-\frac{1.1283791670955126}{y} \cdot e^{z}\right)}} \]
      distribute-rgt-neg-in [=>] 100.0%	\[ x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126}{y} \cdot \left(-e^{z}\right)}} \]
      associate-*l/ [=>] 100.0%	\[ x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126 \cdot \left(-e^{z}\right)}{y}}} \]
      distribute-rgt-neg-in [<=] 100.0%	\[ x + \frac{-1}{x + \frac{\color{blue}{-1.1283791670955126 \cdot e^{z}}}{y}} \]

   3. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.9%

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

   if 1 < (exp.f64 z)

   1. Initial program 91.3%

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

   2. Simplified 100.0%

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

      [Start] 91.3%	\[ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
      *-lft-identity [<=] 91.3%	\[ x + \color{blue}{1 \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]
      metadata-eval [<=] 91.3%	\[ x + \color{blue}{\frac{-1}{-1}} \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
      times-frac [<=] 91.3%	\[ x + \color{blue}{\frac{-1 \cdot y}{-1 \cdot \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]
      neg-mul-1 [<=] 91.3%	\[ x + \frac{-1 \cdot y}{\color{blue}{-\left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]
      sub0-neg [<=] 91.3%	\[ x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]
      associate-+l- [<=] 91.3%	\[ x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]
      neg-sub0 [<=] 91.3%	\[ x + \frac{-1 \cdot y}{\color{blue}{\left(-1.1283791670955126 \cdot e^{z}\right)} + x \cdot y} \]
      +-commutative [<=] 91.3%	\[ x + \frac{-1 \cdot y}{\color{blue}{x \cdot y + \left(-1.1283791670955126 \cdot e^{z}\right)}} \]
      sub-neg [<=] 91.3%	\[ x + \frac{-1 \cdot y}{\color{blue}{x \cdot y - 1.1283791670955126 \cdot e^{z}}} \]
      associate-/l* [=>] 91.3%	\[ x + \color{blue}{\frac{-1}{\frac{x \cdot y - 1.1283791670955126 \cdot e^{z}}{y}}} \]
      div-sub [=>] 91.3%	\[ x + \frac{-1}{\color{blue}{\frac{x \cdot y}{y} - \frac{1.1283791670955126 \cdot e^{z}}{y}}} \]
      associate-*r/ [<=] 100.0%	\[ x + \frac{-1}{\color{blue}{x \cdot \frac{y}{y}} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]
      *-inverses [=>] 100.0%	\[ x + \frac{-1}{x \cdot \color{blue}{1} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]
      *-rgt-identity [=>] 100.0%	\[ x + \frac{-1}{\color{blue}{x} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]
      associate-*l/ [<=] 100.0%	\[ x + \frac{-1}{x - \color{blue}{\frac{1.1283791670955126}{y} \cdot e^{z}}} \]
      cancel-sign-sub-inv [=>] 100.0%	\[ x + \frac{-1}{\color{blue}{x + \left(-\frac{1.1283791670955126}{y}\right) \cdot e^{z}}} \]
      distribute-lft-neg-in [<=] 100.0%	\[ x + \frac{-1}{x + \color{blue}{\left(-\frac{1.1283791670955126}{y} \cdot e^{z}\right)}} \]
      distribute-rgt-neg-in [=>] 100.0%	\[ x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126}{y} \cdot \left(-e^{z}\right)}} \]
      associate-*l/ [=>] 100.0%	\[ x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126 \cdot \left(-e^{z}\right)}{y}}} \]
      distribute-rgt-neg-in [<=] 100.0%	\[ x + \frac{-1}{x + \frac{\color{blue}{-1.1283791670955126 \cdot e^{z}}}{y}} \]

   3. Taylor expanded in y around 0 100.0%

      \[\leadsto x + \color{blue}{0.8862269254527579 \cdot \frac{y}{e^{z}}} \]

   4. Simplified 100.0%

      \[\leadsto x + \color{blue}{\frac{y \cdot 0.8862269254527579}{e^{z}}} \]
      Step-by-step derivation

      [Start] 100.0%	\[ x + 0.8862269254527579 \cdot \frac{y}{e^{z}} \]
      *-commutative [=>] 100.0%	\[ x + \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]
      associate-*l/ [=>] 100.0%	\[ x + \color{blue}{\frac{y \cdot 0.8862269254527579}{e^{z}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	19912

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

Alternative 2
Accuracy	99.9%
Cost	13376

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

Alternative 3
Accuracy	96.2%
Cost	1476

\[\begin{array}{l} \mathbf{if}\;z \leq -1600000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\left(1.2732395447351628 - 1.2732395447351628 \cdot \left(z \cdot z\right)\right) \cdot \left(0.8862269254527579 + z \cdot 0.8862269254527579\right) - x \cdot y}\\ \end{array} \]

Alternative 4
Accuracy	95.4%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;z \leq -1600000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{0.8862269254527579 \cdot \left(1.2732395447351628 - 1.2732395447351628 \cdot \left(z \cdot z\right)\right) - x \cdot y}\\ \end{array} \]

Alternative 5
Accuracy	93.4%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \leq -1600000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \end{array} \]

Alternative 6
Accuracy	92.4%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1600000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+167}:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + 0.8862269254527579 \cdot \frac{y}{z}\\ \end{array} \]

Alternative 7
Accuracy	92.4%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1600000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+165}:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \end{array} \]

Alternative 8
Accuracy	92.4%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1600000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+167}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \end{array} \]

Alternative 9
Accuracy	74.4%
Cost	717

\[\begin{array}{l} \mathbf{if}\;z \leq -1600000 \lor \neg \left(z \leq 1.35 \cdot 10^{+25}\right) \land z \leq 3.6 \cdot 10^{+130}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \end{array} \]

Alternative 10
Accuracy	79.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1600000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x + 0.8862269254527579 \cdot \frac{y}{z}\\ \end{array} \]

Alternative 11
Accuracy	71.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-193} \lor \neg \left(x \leq 9.2 \cdot 10^{-263}\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \end{array} \]

Alternative 12
Accuracy	75.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1600000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{x}\\ \end{array} \]

Alternative 13
Accuracy	14.4%
Cost	192

\[y \cdot 0.8862269254527579 \]

Reproduce

herbie shell --seed 2023165 
(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)))))