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

Percentage Accurate: 95.9% → 95.9%

Time: 1.7s

Alternatives: 5

Speedup: 1.0×

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

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

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
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]

Initial Program: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

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

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
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]

Alternative 1: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

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

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
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]

Derivation

1. Initial program 95.6%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 29.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return y / ((1.1283791670955126 * exp(z)) - (x * y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y / ((1.1283791670955126d0 * exp(z)) - (x * y))
end function

Java

public static double code(double x, double y, double z) {
	return y / ((1.1283791670955126 * Math.exp(z)) - (x * y));
}

Python

def code(x, y, z):
	return y / ((1.1283791670955126 * math.exp(z)) - (x * y))

Julia

function code(x, y, z)
	return Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = y / ((1.1283791670955126 * exp(z)) - (x * y));
end

Wolfram

code[x_, y_, z_] := N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.6%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}} + x \cdot \left(1 - \frac{-25000000000000000000000000000000}{31830988618379068626528276418969} \cdot \frac{{y}^{2}}{{\left(e^{z}\right)}^{2}}\right)} \]

5. Applied rewrites 27.0%

   \[\leadsto \color{blue}{\frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]
6. Add Preprocessing

Alternative 3: 3.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 1.1283791670955126 \cdot e^{z} - x \cdot y \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (1.1283791670955126 * exp(z)) - (x * y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.1283791670955126d0 * exp(z)) - (x * y)
end function

Java

public static double code(double x, double y, double z) {
	return (1.1283791670955126 * Math.exp(z)) - (x * y);
}

Python

def code(x, y, z):
	return (1.1283791670955126 * math.exp(z)) - (x * y)

Julia

function code(x, y, z)
	return Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.1283791670955126 * exp(z)) - (x * y);
end

Wolfram

code[x_, y_, z_] := N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.6%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}} + x \cdot \left(\left(1 + \frac{125000000000000000000000000000000000000000000000}{179587122125166574152748016388400639365587092547} \cdot \frac{x \cdot {y}^{3}}{{\left(e^{z}\right)}^{3}}\right) - \frac{-25000000000000000000000000000000}{31830988618379068626528276418969} \cdot \frac{{y}^{2}}{{\left(e^{z}\right)}^{2}}\right)} \]

5. Applied rewrites 3.2%

   \[\leadsto \color{blue}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
6. Add Preprocessing

Alternative 4: 3.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 1.1283791670955126 \cdot e^{z} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return 1.1283791670955126 * exp(z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.1283791670955126d0 * exp(z)
end function

Java

public static double code(double x, double y, double z) {
	return 1.1283791670955126 * Math.exp(z);
}

Python

def code(x, y, z):
	return 1.1283791670955126 * math.exp(z)

Julia

function code(x, y, z)
	return Float64(1.1283791670955126 * exp(z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.1283791670955126 * exp(z);
end

Wolfram

code[x_, y_, z_] := N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.6%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}} + x \cdot \left(\left(1 + x \cdot \left(\frac{625000000000000000000000000000000000000000000000000000000000000}{1013211836423377808197068612118338110498536634308026710423022961} \cdot \frac{x \cdot {y}^{4}}{{\left(e^{z}\right)}^{4}} - \frac{-125000000000000000000000000000000000000000000000}{179587122125166574152748016388400639365587092547} \cdot \frac{{y}^{3}}{{\left(e^{z}\right)}^{3}}\right)\right) - \frac{-25000000000000000000000000000000}{31830988618379068626528276418969} \cdot \frac{{y}^{2}}{{\left(e^{z}\right)}^{2}}\right)} \]

5. Applied rewrites 2.9%

   \[\leadsto \color{blue}{1.1283791670955126 \cdot e^{z}} \]
6. Add Preprocessing

Alternative 5: 3.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ e^{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp z))

C

double code(double x, double y, double z) {
	return exp(z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(z)
end function

Java

public static double code(double x, double y, double z) {
	return Math.exp(z);
}

Python

def code(x, y, z):
	return math.exp(z)

Julia

function code(x, y, z)
	return exp(z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = exp(z);
end

Wolfram

code[x_, y_, z_] := N[Exp[z], $MachinePrecision]

Derivation

1. Initial program 95.6%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}} + x \cdot \left(\left(1 + x \cdot \left(\frac{625000000000000000000000000000000000000000000000000000000000000}{1013211836423377808197068612118338110498536634308026710423022961} \cdot \frac{x \cdot {y}^{4}}{{\left(e^{z}\right)}^{4}} - \frac{-125000000000000000000000000000000000000000000000}{179587122125166574152748016388400639365587092547} \cdot \frac{{y}^{3}}{{\left(e^{z}\right)}^{3}}\right)\right) - \frac{-25000000000000000000000000000000}{31830988618379068626528276418969} \cdot \frac{{y}^{2}}{{\left(e^{z}\right)}^{2}}\right)} \]

5. Applied rewrites 2.9%

   \[\leadsto \color{blue}{1.1283791670955126 \cdot e^{z}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}} + x \cdot \left(\left(1 + \frac{125000000000000000000000000000000000000000000000}{179587122125166574152748016388400639365587092547} \cdot \frac{x \cdot {y}^{3}}{{\left(e^{z}\right)}^{3}}\right) - \frac{-25000000000000000000000000000000}{31830988618379068626528276418969} \cdot \frac{{y}^{2}}{{\left(e^{z}\right)}^{2}}\right)} \]

8. Applied rewrites 2.9%

   \[\leadsto \color{blue}{e^{z}} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x))))

C

double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
}

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 + (1.0d0 / (((1.1283791670955126d0 / y) * exp(z)) - x))
end function

Java

public static double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * Math.exp(z)) - x));
}

Python

def code(x, y, z):
	return x + (1.0 / (((1.1283791670955126 / y) * math.exp(z)) - x))

Julia

function code(x, y, z)
	return Float64(x + Float64(1.0 / Float64(Float64(Float64(1.1283791670955126 / y) * exp(z)) - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(1.0 / N[(N[(N[(1.1283791670955126 / y), $MachinePrecision] * N[Exp[z], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024321 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64
  :pre (TRUE)

  :alt
  (! :herbie-platform default (+ x (/ 1 (- (* (/ 5641895835477563/5000000000000000 y) (exp z)) x))))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))