Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B

Percentage Accurate: 96.4% → 99.1%

Time: 16.2s

Alternatives: 11

Speedup: 1.5×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(z + b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (- (* y (- (log z) t)) (* a (+ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) - (a * (z + b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) - (a * (z + b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) - (a * (z + b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) - (a * (z + b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) - Float64(a * Float64(z + b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) - (a * (z + b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] - N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

2. Taylor expanded in z around 0 100.0%

   \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)\right)}} \]
3. Step-by-step derivation

   1. associate-*r* 100.0%

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \left(\color{blue}{\left(-1 \cdot a\right) \cdot b} + -1 \cdot \left(a \cdot z\right)\right)} \]

   2. associate-*r* 100.0%

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \left(\left(-1 \cdot a\right) \cdot b + \color{blue}{\left(-1 \cdot a\right) \cdot z}\right)} \]

   3. distribute-lft-out 100.0%

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(-1 \cdot a\right) \cdot \left(b + z\right)}} \]

   4. neg-mul-1 100.0%

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(-a\right)} \cdot \left(b + z\right)} \]

4. Simplified 100.0%

   \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(-a\right) \cdot \left(b + z\right)}} \]

5. Final simplification 100.0%

   \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(z + b\right)} \]

Alternative 2: 87.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{-26} \lor \neg \left(y \leq 5000000000\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.45e-26) (not (<= y 5000000000.0)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (* a (- (- b) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.45e-26) || !(y <= 5000000000.0)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((a * (-b - z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-2.45d-26)) .or. (.not. (y <= 5000000000.0d0))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp((a * (-b - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.45e-26) || !(y <= 5000000000.0)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((a * (-b - z)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.45e-26) or not (y <= 5000000000.0):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((a * (-b - z)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.45e-26) || !(y <= 5000000000.0))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-b) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.45e-26) || ~((y <= 5000000000.0)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp((a * (-b - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.45e-26], N[Not[LessEqual[y, 5000000000.0]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.45e-26 or 5e9 < y

   1. Initial program 99.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in a around 0 88.9%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   if -2.45e-26 < y < 5e9

   1. Initial program 95.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in y around 0 82.4%

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

   3. Taylor expanded in z around 0 86.8%

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \left(\color{blue}{\left(-1 \cdot a\right) \cdot b} + -1 \cdot \left(a \cdot z\right)\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \left(\left(-1 \cdot a\right) \cdot b + \color{blue}{\left(-1 \cdot a\right) \cdot z}\right)} \]

      3. distribute-lft-out 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(-1 \cdot a\right) \cdot \left(b + z\right)}} \]

      4. neg-mul-1 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(-a\right)} \cdot \left(b + z\right)} \]

   5. Simplified 86.8%

      \[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot \left(b + z\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{-26} \lor \neg \left(y \leq 5000000000\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \]

Alternative 3: 77.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+37} \lor \neg \left(y \leq 8500000000000\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -5.2e+37) (not (<= y 8500000000000.0)))
   (* x (pow z y))
   (* x (exp (* a (- (- b) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.2e+37) || !(y <= 8500000000000.0)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((a * (-b - z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-5.2d+37)) .or. (.not. (y <= 8500000000000.0d0))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((a * (-b - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.2e+37) || !(y <= 8500000000000.0)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((a * (-b - z)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -5.2e+37) or not (y <= 8500000000000.0):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((a * (-b - z)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -5.2e+37) || !(y <= 8500000000000.0))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-b) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -5.2e+37) || ~((y <= 8500000000000.0)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((a * (-b - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -5.2e+37], N[Not[LessEqual[y, 8500000000000.0]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.1999999999999998e37 or 8.5e12 < y

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in a around 0 90.6%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   3. Taylor expanded in t around 0 69.5%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if -5.1999999999999998e37 < y < 8.5e12

   1. Initial program 95.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in y around 0 78.5%

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

   3. Taylor expanded in z around 0 83.1%

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \left(\color{blue}{\left(-1 \cdot a\right) \cdot b} + -1 \cdot \left(a \cdot z\right)\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \left(\left(-1 \cdot a\right) \cdot b + \color{blue}{\left(-1 \cdot a\right) \cdot z}\right)} \]

      3. distribute-lft-out 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(-1 \cdot a\right) \cdot \left(b + z\right)}} \]

      4. neg-mul-1 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(-a\right)} \cdot \left(b + z\right)} \]

   5. Simplified 83.1%

      \[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot \left(b + z\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+37} \lor \neg \left(y \leq 8500000000000\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \]

Alternative 4: 74.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+34} \lor \neg \left(y \leq 9.5 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.55e+34) (not (<= y 9.5e+14)))
   (* x (pow z y))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.55e+34) || !(y <= 9.5e+14)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.55d+34)) .or. (.not. (y <= 9.5d+14))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.55e+34) || !(y <= 9.5e+14)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.55e+34) or not (y <= 9.5e+14):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.55e+34) || !(y <= 9.5e+14))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.55e+34) || ~((y <= 9.5e+14)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.55e+34], N[Not[LessEqual[y, 9.5e+14]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.54999999999999989e34 or 9.5e14 < y

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in a around 0 90.7%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   3. Taylor expanded in t around 0 69.8%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if -1.54999999999999989e34 < y < 9.5e14

   1. Initial program 95.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in y around 0 78.4%

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

   3. Taylor expanded in z around 0 78.4%

      \[\leadsto x \cdot \color{blue}{e^{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg 78.4%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-lft-neg-out 78.4%

         \[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot b}} \]

      3. *-commutative 78.4%

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}} \]

   5. Simplified 78.4%

      \[\leadsto x \cdot \color{blue}{e^{b \cdot \left(-a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+34} \lor \neg \left(y \leq 9.5 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]

Alternative 5: 56.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.1 \cdot 10^{+109} \lor \neg \left(t \leq 1.55 \cdot 10^{+193}\right):\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right) - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -6.1e+109) (not (<= t 1.55e+193)))
   (* x (- (* 0.5 (* y (* y (* t t)))) (* y t)))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -6.1e+109) || !(t <= 1.55e+193)) {
		tmp = x * ((0.5 * (y * (y * (t * t)))) - (y * t));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-6.1d+109)) .or. (.not. (t <= 1.55d+193))) then
        tmp = x * ((0.5d0 * (y * (y * (t * t)))) - (y * t))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -6.1e+109) || !(t <= 1.55e+193)) {
		tmp = x * ((0.5 * (y * (y * (t * t)))) - (y * t));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -6.1e+109) or not (t <= 1.55e+193):
		tmp = x * ((0.5 * (y * (y * (t * t)))) - (y * t))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -6.1e+109) || !(t <= 1.55e+193))
		tmp = Float64(x * Float64(Float64(0.5 * Float64(y * Float64(y * Float64(t * t)))) - Float64(y * t)));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -6.1e+109) || ~((t <= 1.55e+193)))
		tmp = x * ((0.5 * (y * (y * (t * t)))) - (y * t));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -6.1e+109], N[Not[LessEqual[t, 1.55e+193]], $MachinePrecision]], N[(x * N[(N[(0.5 * N[(y * N[(y * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.10000000000000008e109 or 1.54999999999999993e193 < t

   1. Initial program 95.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in a around 0 88.9%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   3. Taylor expanded in t around inf 88.9%

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-1 \cdot t\right)}} \]
   4. Step-by-step derivation

      1. neg-mul-1 88.9%

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 88.9%

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   6. Taylor expanded in y around 0 47.4%

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(-1 \cdot \left(t \cdot y\right) + 0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 47.4%

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + 0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]

      2. mul-1-neg 47.4%

         \[\leadsto x \cdot \left(\left(1 + \color{blue}{\left(-t \cdot y\right)}\right) + 0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right)\right) \]

      3. *-commutative 47.4%

         \[\leadsto x \cdot \left(\left(1 + \left(-\color{blue}{y \cdot t}\right)\right) + 0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right)\right) \]

      4. unsub-neg 47.4%

         \[\leadsto x \cdot \left(\color{blue}{\left(1 - y \cdot t\right)} + 0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right)\right) \]

      5. associate-*r* 47.4%

         \[\leadsto x \cdot \left(\left(1 - y \cdot t\right) + \color{blue}{\left(0.5 \cdot {t}^{2}\right) \cdot {y}^{2}}\right) \]

      6. *-commutative 47.4%

         \[\leadsto x \cdot \left(\left(1 - y \cdot t\right) + \color{blue}{{y}^{2} \cdot \left(0.5 \cdot {t}^{2}\right)}\right) \]

      7. unpow2 47.4%

         \[\leadsto x \cdot \left(\left(1 - y \cdot t\right) + \color{blue}{\left(y \cdot y\right)} \cdot \left(0.5 \cdot {t}^{2}\right)\right) \]

      8. unpow2 47.4%

         \[\leadsto x \cdot \left(\left(1 - y \cdot t\right) + \left(y \cdot y\right) \cdot \left(0.5 \cdot \color{blue}{\left(t \cdot t\right)}\right)\right) \]

   8. Simplified 47.4%

      \[\leadsto x \cdot \color{blue}{\left(\left(1 - y \cdot t\right) + \left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)} \]

   9. Taylor expanded in y around inf 46.1%

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(t \cdot y\right) + 0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative 46.1%

          \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right) + -1 \cdot \left(t \cdot y\right)\right)} \]

       2. associate-*r* 46.1%

          \[\leadsto x \cdot \left(0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right) + \color{blue}{\left(-1 \cdot t\right) \cdot y}\right) \]

       3. neg-mul-1 46.1%

          \[\leadsto x \cdot \left(0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right) + \color{blue}{\left(-t\right)} \cdot y\right) \]

       4. cancel-sign-sub-inv 46.1%

          \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right) - t \cdot y\right)} \]

       5. unpow2 46.1%

          \[\leadsto x \cdot \left(0.5 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {y}^{2}\right) - t \cdot y\right) \]

       6. unpow2 46.1%

          \[\leadsto x \cdot \left(0.5 \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) - t \cdot y\right) \]

       7. *-commutative 46.1%

          \[\leadsto x \cdot \left(0.5 \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(t \cdot t\right)\right)} - t \cdot y\right) \]

       8. associate-*l* 54.9%

          \[\leadsto x \cdot \left(0.5 \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)} - t \cdot y\right) \]

   11. Simplified 54.9%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right) - t \cdot y\right)} \]

   if -6.10000000000000008e109 < t < 1.54999999999999993e193

   1. Initial program 98.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in a around 0 65.5%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   3. Taylor expanded in t around 0 60.1%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.1 \cdot 10^{+109} \lor \neg \left(t \leq 1.55 \cdot 10^{+193}\right):\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right) - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]

Alternative 6: 39.4% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-49} \lor \neg \left(y \leq 2.2 \cdot 10^{-13}\right):\\ \;\;\;\;0.5 \cdot \left(\left(x \cdot \left(t \cdot t\right)\right) \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(z + b\right) \cdot \left(x \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7.6e-49) (not (<= y 2.2e-13)))
   (* 0.5 (* (* x (* t t)) (* y y)))
   (- x (* (+ z b) (* x a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7.6e-49) || !(y <= 2.2e-13)) {
		tmp = 0.5 * ((x * (t * t)) * (y * y));
	} else {
		tmp = x - ((z + b) * (x * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-7.6d-49)) .or. (.not. (y <= 2.2d-13))) then
        tmp = 0.5d0 * ((x * (t * t)) * (y * y))
    else
        tmp = x - ((z + b) * (x * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7.6e-49) || !(y <= 2.2e-13)) {
		tmp = 0.5 * ((x * (t * t)) * (y * y));
	} else {
		tmp = x - ((z + b) * (x * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -7.6e-49) or not (y <= 2.2e-13):
		tmp = 0.5 * ((x * (t * t)) * (y * y))
	else:
		tmp = x - ((z + b) * (x * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -7.6e-49) || !(y <= 2.2e-13))
		tmp = Float64(0.5 * Float64(Float64(x * Float64(t * t)) * Float64(y * y)));
	else
		tmp = Float64(x - Float64(Float64(z + b) * Float64(x * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -7.6e-49) || ~((y <= 2.2e-13)))
		tmp = 0.5 * ((x * (t * t)) * (y * y));
	else
		tmp = x - ((z + b) * (x * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -7.6e-49], N[Not[LessEqual[y, 2.2e-13]], $MachinePrecision]], N[(0.5 * N[(N[(x * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z + b), $MachinePrecision] * N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5999999999999994e-49 or 2.19999999999999997e-13 < y

   1. Initial program 99.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in a around 0 85.3%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   3. Taylor expanded in t around inf 58.2%

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-1 \cdot t\right)}} \]
   4. Step-by-step derivation

      1. neg-mul-1 58.2%

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 58.2%

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   6. Taylor expanded in y around 0 30.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(-1 \cdot \left(t \cdot y\right) + 0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 30.0%

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + -1 \cdot \left(t \cdot y\right)\right) + 0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]

      2. mul-1-neg 30.0%

         \[\leadsto x \cdot \left(\left(1 + \color{blue}{\left(-t \cdot y\right)}\right) + 0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right)\right) \]

      3. *-commutative 30.0%

         \[\leadsto x \cdot \left(\left(1 + \left(-\color{blue}{y \cdot t}\right)\right) + 0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right)\right) \]

      4. unsub-neg 30.0%

         \[\leadsto x \cdot \left(\color{blue}{\left(1 - y \cdot t\right)} + 0.5 \cdot \left({t}^{2} \cdot {y}^{2}\right)\right) \]

      5. associate-*r* 30.0%

         \[\leadsto x \cdot \left(\left(1 - y \cdot t\right) + \color{blue}{\left(0.5 \cdot {t}^{2}\right) \cdot {y}^{2}}\right) \]

      6. *-commutative 30.0%

         \[\leadsto x \cdot \left(\left(1 - y \cdot t\right) + \color{blue}{{y}^{2} \cdot \left(0.5 \cdot {t}^{2}\right)}\right) \]

      7. unpow2 30.0%

         \[\leadsto x \cdot \left(\left(1 - y \cdot t\right) + \color{blue}{\left(y \cdot y\right)} \cdot \left(0.5 \cdot {t}^{2}\right)\right) \]

      8. unpow2 30.0%

         \[\leadsto x \cdot \left(\left(1 - y \cdot t\right) + \left(y \cdot y\right) \cdot \left(0.5 \cdot \color{blue}{\left(t \cdot t\right)}\right)\right) \]

   8. Simplified 30.0%

      \[\leadsto x \cdot \color{blue}{\left(\left(1 - y \cdot t\right) + \left(y \cdot y\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)} \]

   9. Taylor expanded in y around inf 35.6%

      \[\leadsto \color{blue}{0.5 \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 35.6%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(x \cdot {y}^{2}\right)\right) \]

       2. unpow2 35.6%

          \[\leadsto 0.5 \cdot \left(\left(t \cdot t\right) \cdot \left(x \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       3. associate-*r* 37.6%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(y \cdot y\right)\right)} \]

   11. Simplified 37.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(y \cdot y\right)\right)} \]

   if -7.5999999999999994e-49 < y < 2.19999999999999997e-13

   1. Initial program 95.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in y around 0 81.2%

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

   3. Taylor expanded in z around 0 86.1%

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right) + -1 \cdot \left(a \cdot z\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \left(\color{blue}{\left(-1 \cdot a\right) \cdot b} + -1 \cdot \left(a \cdot z\right)\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \left(\left(-1 \cdot a\right) \cdot b + \color{blue}{\left(-1 \cdot a\right) \cdot z}\right)} \]

      3. distribute-lft-out 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(-1 \cdot a\right) \cdot \left(b + z\right)}} \]

      4. neg-mul-1 100.0%

         \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(-a\right)} \cdot \left(b + z\right)} \]

   5. Simplified 86.1%

      \[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot \left(b + z\right)}} \]

   6. Taylor expanded in a around 0 38.5%

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 38.5%

         \[\leadsto x + \color{blue}{\left(-a \cdot \left(x \cdot \left(b + z\right)\right)\right)} \]

      2. unsub-neg 38.5%

         \[\leadsto \color{blue}{x - a \cdot \left(x \cdot \left(b + z\right)\right)} \]

      3. associate-*r* 43.5%

         \[\leadsto x - \color{blue}{\left(a \cdot x\right) \cdot \left(b + z\right)} \]

      4. *-commutative 43.5%

         \[\leadsto x - \color{blue}{\left(b + z\right) \cdot \left(a \cdot x\right)} \]

   8. Simplified 43.5%

      \[\leadsto \color{blue}{x - \left(b + z\right) \cdot \left(a \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-49} \lor \neg \left(y \leq 2.2 \cdot 10^{-13}\right):\\ \;\;\;\;0.5 \cdot \left(\left(x \cdot \left(t \cdot t\right)\right) \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(z + b\right) \cdot \left(x \cdot a\right)\\ \end{array} \]

Alternative 7: 31.6% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+153} \lor \neg \left(b \leq 9 \cdot 10^{-51}\right):\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -5.2e+153) (not (<= b 9e-51)))
   (* x (- 1.0 (* a b)))
   (- x (* x (* y t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -5.2e+153) || !(b <= 9e-51)) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x - (x * (y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-5.2d+153)) .or. (.not. (b <= 9d-51))) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = x - (x * (y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -5.2e+153) || !(b <= 9e-51)) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x - (x * (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -5.2e+153) or not (b <= 9e-51):
		tmp = x * (1.0 - (a * b))
	else:
		tmp = x - (x * (y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -5.2e+153) || !(b <= 9e-51))
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(x - Float64(x * Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -5.2e+153) || ~((b <= 9e-51)))
		tmp = x * (1.0 - (a * b));
	else
		tmp = x - (x * (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -5.2e+153], N[Not[LessEqual[b, 9e-51]], $MachinePrecision]], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.1999999999999998e153 or 8.99999999999999948e-51 < b

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in y around 0 74.4%

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

   3. Taylor expanded in z around 0 74.4%

      \[\leadsto x \cdot \color{blue}{e^{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg 74.4%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-lft-neg-out 74.4%

         \[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot b}} \]

      3. *-commutative 74.4%

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}} \]

   5. Simplified 74.4%

      \[\leadsto x \cdot \color{blue}{e^{b \cdot \left(-a\right)}} \]

   6. Taylor expanded in b around 0 29.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 29.0%

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

      2. unsub-neg 29.0%

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   8. Simplified 29.0%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   if -5.1999999999999998e153 < b < 8.99999999999999948e-51

   1. Initial program 95.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in a around 0 79.6%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   3. Taylor expanded in t around inf 63.1%

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-1 \cdot t\right)}} \]
   4. Step-by-step derivation

      1. neg-mul-1 63.1%

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 63.1%

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   6. Taylor expanded in y around 0 34.1%

      \[\leadsto \color{blue}{x + -1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 34.1%

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

      2. unsub-neg 34.1%

         \[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)} \]

      3. *-commutative 34.1%

         \[\leadsto x - \color{blue}{\left(x \cdot y\right) \cdot t} \]

      4. associate-*l* 34.1%

         \[\leadsto x - \color{blue}{x \cdot \left(y \cdot t\right)} \]

   8. Simplified 34.1%

      \[\leadsto \color{blue}{x - x \cdot \left(y \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+153} \lor \neg \left(b \leq 9 \cdot 10^{-51}\right):\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 8: 31.2% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+153} \lor \neg \left(b \leq 9.2 \cdot 10^{-51}\right):\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -2.3e+153) (not (<= b 9.2e-51)))
   (* x (- 1.0 (* a b)))
   (- x (* t (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -2.3e+153) || !(b <= 9.2e-51)) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x - (t * (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-2.3d+153)) .or. (.not. (b <= 9.2d-51))) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = x - (t * (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -2.3e+153) || !(b <= 9.2e-51)) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x - (t * (x * y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -2.3e+153) or not (b <= 9.2e-51):
		tmp = x * (1.0 - (a * b))
	else:
		tmp = x - (t * (x * y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -2.3e+153) || !(b <= 9.2e-51))
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(x - Float64(t * Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -2.3e+153) || ~((b <= 9.2e-51)))
		tmp = x * (1.0 - (a * b));
	else
		tmp = x - (t * (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -2.3e+153], N[Not[LessEqual[b, 9.2e-51]], $MachinePrecision]], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.3000000000000001e153 or 9.20000000000000007e-51 < b

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in y around 0 74.4%

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

   3. Taylor expanded in z around 0 74.4%

      \[\leadsto x \cdot \color{blue}{e^{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg 74.4%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-lft-neg-out 74.4%

         \[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot b}} \]

      3. *-commutative 74.4%

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}} \]

   5. Simplified 74.4%

      \[\leadsto x \cdot \color{blue}{e^{b \cdot \left(-a\right)}} \]

   6. Taylor expanded in b around 0 29.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 29.0%

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

      2. unsub-neg 29.0%

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   8. Simplified 29.0%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   if -2.3000000000000001e153 < b < 9.20000000000000007e-51

   1. Initial program 95.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in a around 0 79.6%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   3. Taylor expanded in y around 0 37.0%

      \[\leadsto \color{blue}{x + x \cdot \left(y \cdot \left(\log z - t\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 35.1%

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

   5. Simplified 35.1%

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

   6. Taylor expanded in t around inf 34.1%

      \[\leadsto x + \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 34.1%

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

      2. *-commutative 34.1%

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

      3. distribute-rgt-neg-in 34.1%

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

   8. Simplified 34.1%

      \[\leadsto x + \color{blue}{\left(x \cdot y\right) \cdot \left(-t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+153} \lor \neg \left(b \leq 9.2 \cdot 10^{-51}\right):\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 9: 24.6% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{-53} \lor \neg \left(y \leq 4.7 \cdot 10^{-183}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.02e-53) (not (<= y 4.7e-183))) (* a (* x (- b))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.02e-53) || !(y <= 4.7e-183)) {
		tmp = a * (x * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.02d-53)) .or. (.not. (y <= 4.7d-183))) then
        tmp = a * (x * -b)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.02e-53) || !(y <= 4.7e-183)) {
		tmp = a * (x * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.02e-53) or not (y <= 4.7e-183):
		tmp = a * (x * -b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.02e-53) || !(y <= 4.7e-183))
		tmp = Float64(a * Float64(x * Float64(-b)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.02e-53) || ~((y <= 4.7e-183)))
		tmp = a * (x * -b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.02e-53], N[Not[LessEqual[y, 4.7e-183]], $MachinePrecision]], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.02000000000000002e-53 or 4.7000000000000001e-183 < y

   1. Initial program 98.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in y around 0 49.6%

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

   3. Taylor expanded in z around 0 48.5%

      \[\leadsto x \cdot \color{blue}{e^{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg 48.5%

         \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

      2. distribute-lft-neg-out 48.5%

         \[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot b}} \]

      3. *-commutative 48.5%

         \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}} \]

   5. Simplified 48.5%

      \[\leadsto x \cdot \color{blue}{e^{b \cdot \left(-a\right)}} \]

   6. Taylor expanded in b around 0 16.9%

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 16.9%

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

      2. unsub-neg 16.9%

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   8. Simplified 16.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

   9. Taylor expanded in a around inf 16.6%

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 16.6%

          \[\leadsto \color{blue}{-a \cdot \left(b \cdot x\right)} \]

       2. distribute-rgt-neg-in 16.6%

          \[\leadsto \color{blue}{a \cdot \left(-b \cdot x\right)} \]

   11. Simplified 16.6%

       \[\leadsto \color{blue}{a \cdot \left(-b \cdot x\right)} \]

   if -1.02000000000000002e-53 < y < 4.7000000000000001e-183

   1. Initial program 94.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

   2. Taylor expanded in a around 0 54.3%

      \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

   3. Taylor expanded in y around 0 40.3%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{-53} \lor \neg \left(y \leq 4.7 \cdot 10^{-183}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 28.3% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - a \cdot b\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* x (- 1.0 (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * (1.0 - (a * b));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * (1.0d0 - (a * b))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * (1.0 - (a * b));
}

Python

def code(x, y, z, t, a, b):
	return x * (1.0 - (a * b))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * Float64(1.0 - Float64(a * b)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * (1.0 - (a * b));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

2. Taylor expanded in y around 0 59.8%

   \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

3. Taylor expanded in z around 0 59.0%

   \[\leadsto x \cdot \color{blue}{e^{-1 \cdot \left(a \cdot b\right)}} \]
4. Step-by-step derivation

   1. mul-1-neg 59.0%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   2. distribute-lft-neg-out 59.0%

      \[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot b}} \]

   3. *-commutative 59.0%

      \[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}} \]

5. Simplified 59.0%

   \[\leadsto x \cdot \color{blue}{e^{b \cdot \left(-a\right)}} \]

6. Taylor expanded in b around 0 24.6%

   \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 24.6%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right) \]

   2. unsub-neg 24.6%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

8. Simplified 24.6%

   \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

9. Final simplification 24.6%

   \[\leadsto x \cdot \left(1 - a \cdot b\right) \]

Alternative 11: 19.6% accurate, 315.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

def code(x, y, z, t, a, b):
	return x

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 97.6%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

2. Taylor expanded in a around 0 71.9%

   \[\leadsto \color{blue}{x \cdot e^{y \cdot \left(\log z - t\right)}} \]

3. Taylor expanded in y around 0 15.9%

   \[\leadsto \color{blue}{x} \]

4. Final simplification 15.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))