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

Percentage Accurate: 96.4% → 99.4%

Time: 19.7s

Alternatives: 15

Speedup: 0.8×

• 41.5% 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: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (fma y (- (log z) t) (* a (- (log1p (- z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(fma(y, (log(z) - t), (a * (log1p(-z) - b))));
}

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(fma(y, Float64(log(z) - t), Float64(a * Float64(log1p(Float64(-z)) - b)))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

   1. fma-define 97.7%

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

   2. sub-neg 97.7%

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

   3. log1p-define 99.9%

      \[\leadsto x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\color{blue}{\mathsf{log1p}\left(-z\right)} - b\right)\right)} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 70.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{{\left(e^{a}\right)}^{b}}\\ \mathbf{if}\;a \leq -5.5 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{-221}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+29}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (pow (exp a) b))))
   (if (<= a -5.5e+33)
     t_1
     (if (<= a -6.2e-221)
       (* x (pow z y))
       (if (<= a 1.05e+29) (* x (exp (* t (- y)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / pow(exp(a), b);
	double tmp;
	if (a <= -5.5e+33) {
		tmp = t_1;
	} else if (a <= -6.2e-221) {
		tmp = x * pow(z, y);
	} else if (a <= 1.05e+29) {
		tmp = x * exp((t * -y));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x / (exp(a) ** b)
    if (a <= (-5.5d+33)) then
        tmp = t_1
    else if (a <= (-6.2d-221)) then
        tmp = x * (z ** y)
    else if (a <= 1.05d+29) then
        tmp = x * exp((t * -y))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / Math.pow(Math.exp(a), b);
	double tmp;
	if (a <= -5.5e+33) {
		tmp = t_1;
	} else if (a <= -6.2e-221) {
		tmp = x * Math.pow(z, y);
	} else if (a <= 1.05e+29) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / math.pow(math.exp(a), b)
	tmp = 0
	if a <= -5.5e+33:
		tmp = t_1
	elif a <= -6.2e-221:
		tmp = x * math.pow(z, y)
	elif a <= 1.05e+29:
		tmp = x * math.exp((t * -y))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / (exp(a) ^ b))
	tmp = 0.0
	if (a <= -5.5e+33)
		tmp = t_1;
	elseif (a <= -6.2e-221)
		tmp = Float64(x * (z ^ y));
	elseif (a <= 1.05e+29)
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (exp(a) ^ b);
	tmp = 0.0;
	if (a <= -5.5e+33)
		tmp = t_1;
	elseif (a <= -6.2e-221)
		tmp = x * (z ^ y);
	elseif (a <= 1.05e+29)
		tmp = x * exp((t * -y));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[Power[N[Exp[a], $MachinePrecision], b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5.5e+33], t$95$1, If[LessEqual[a, -6.2e-221], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.05e+29], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.5000000000000006e33 or 1.0500000000000001e29 < a

   1. Initial program 94.4%

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

   3. Taylor expanded in z around 0 94.4%

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

   4. Taylor expanded in y around 0 78.9%

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

      1. *-commutative 78.9%

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

      2. mul-1-neg 78.9%

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

   6. Simplified 78.9%

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

   7. Taylor expanded in a around inf 78.9%

      \[\leadsto \color{blue}{x \cdot e^{-a \cdot b}} \]
   8. Step-by-step derivation

      1. exp-neg 78.9%

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{a \cdot b}}} \]

      2. *-commutative 78.9%

         \[\leadsto x \cdot \frac{1}{e^{\color{blue}{b \cdot a}}} \]

      3. exp-prod 58.2%

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

      4. associate-*r/ 58.2%

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

      5. *-rgt-identity 58.2%

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

      6. exp-prod 78.9%

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

      7. *-commutative 78.9%

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

      8. exp-prod 84.4%

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

   9. Simplified 84.4%

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

   if -5.5000000000000006e33 < a < -6.1999999999999998e-221

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 87.3%

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

   4. Taylor expanded in t around 0 76.2%

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

   if -6.1999999999999998e-221 < a < 1.0500000000000001e29

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf 79.9%

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

      1. mul-1-neg 79.9%

         \[\leadsto x \cdot e^{\color{blue}{-t \cdot y}} \]

      2. distribute-lft-neg-out 79.9%

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

      3. *-commutative 79.9%

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

   5. Simplified 79.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{x}{{\left(e^{a}\right)}^{b}}\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{-221}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+29}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{\left(e^{a}\right)}^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+33} \lor \neg \left(a \leq 3.6 \cdot 10^{+28}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -8.2e+33) (not (<= a 3.6e+28)))
   (* x (exp (* a (- (log1p (- z)) b))))
   (* x (exp (* y (- (log z) t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -8.2e+33) || !(a <= 3.6e+28)) {
		tmp = x * exp((a * (log1p(-z) - b)));
	} else {
		tmp = x * exp((y * (log(z) - t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -8.2e+33) || !(a <= 3.6e+28)) {
		tmp = x * Math.exp((a * (Math.log1p(-z) - b)));
	} else {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -8.2e+33) or not (a <= 3.6e+28):
		tmp = x * math.exp((a * (math.log1p(-z) - b)))
	else:
		tmp = x * math.exp((y * (math.log(z) - t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -8.2e+33) || !(a <= 3.6e+28))
		tmp = Float64(x * exp(Float64(a * Float64(log1p(Float64(-z)) - b))));
	else
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -8.2e+33], N[Not[LessEqual[a, 3.6e+28]], $MachinePrecision]], N[(x * N[Exp[N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.1999999999999999e33 or 3.5999999999999999e28 < a

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0 78.9%

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

      1. sub-neg 78.9%

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

      2. log1p-define 85.3%

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(-z\right)} - b\right)} \]

   5. Simplified 85.3%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   if -8.1999999999999999e33 < a < 3.5999999999999999e28

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 91.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+33} \lor \neg \left(a \leq 3.6 \cdot 10^{+28}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+35} \lor \neg \left(a \leq 1.1 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x}{{\left(e^{a}\right)}^{b}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -2.1e+35) (not (<= a 1.1e+27)))
   (/ x (pow (exp a) b))
   (* x (exp (* y (- (log z) t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -2.1e+35) || !(a <= 1.1e+27)) {
		tmp = x / pow(exp(a), b);
	} else {
		tmp = x * exp((y * (log(z) - 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 ((a <= (-2.1d+35)) .or. (.not. (a <= 1.1d+27))) then
        tmp = x / (exp(a) ** b)
    else
        tmp = x * exp((y * (log(z) - 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 ((a <= -2.1e+35) || !(a <= 1.1e+27)) {
		tmp = x / Math.pow(Math.exp(a), b);
	} else {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -2.1e+35) or not (a <= 1.1e+27):
		tmp = x / math.pow(math.exp(a), b)
	else:
		tmp = x * math.exp((y * (math.log(z) - t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -2.1e+35) || !(a <= 1.1e+27))
		tmp = Float64(x / (exp(a) ^ b));
	else
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -2.1e+35) || ~((a <= 1.1e+27)))
		tmp = x / (exp(a) ^ b);
	else
		tmp = x * exp((y * (log(z) - t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -2.1e+35], N[Not[LessEqual[a, 1.1e+27]], $MachinePrecision]], N[(x / N[Power[N[Exp[a], $MachinePrecision], b], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.0999999999999999e35 or 1.0999999999999999e27 < a

   1. Initial program 94.4%

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

   3. Taylor expanded in z around 0 94.4%

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

   4. Taylor expanded in y around 0 78.9%

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

      1. *-commutative 78.9%

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

      2. mul-1-neg 78.9%

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

   6. Simplified 78.9%

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

   7. Taylor expanded in a around inf 78.9%

      \[\leadsto \color{blue}{x \cdot e^{-a \cdot b}} \]
   8. Step-by-step derivation

      1. exp-neg 78.9%

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{a \cdot b}}} \]

      2. *-commutative 78.9%

         \[\leadsto x \cdot \frac{1}{e^{\color{blue}{b \cdot a}}} \]

      3. exp-prod 58.2%

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

      4. associate-*r/ 58.2%

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

      5. *-rgt-identity 58.2%

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

      6. exp-prod 78.9%

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

      7. *-commutative 78.9%

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

      8. exp-prod 84.4%

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

   9. Simplified 84.4%

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

   if -2.0999999999999999e35 < a < 1.0999999999999999e27

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 91.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+35} \lor \neg \left(a \leq 1.1 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x}{{\left(e^{a}\right)}^{b}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -47000:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \log z - a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -47000.0)
   (* x (exp (* t (- y))))
   (* x (exp (- (* y (log z)) (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -47000.0) {
		tmp = x * exp((t * -y));
	} else {
		tmp = x * exp(((y * log(z)) - (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 (t <= (-47000.0d0)) then
        tmp = x * exp((t * -y))
    else
        tmp = x * exp(((y * log(z)) - (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 (t <= -47000.0) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = x * Math.exp(((y * Math.log(z)) - (a * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -47000.0:
		tmp = x * math.exp((t * -y))
	else:
		tmp = x * math.exp(((y * math.log(z)) - (a * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -47000.0)
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	else
		tmp = Float64(x * exp(Float64(Float64(y * log(z)) - Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -47000.0)
		tmp = x * exp((t * -y));
	else
		tmp = x * exp(((y * log(z)) - (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -47000.0], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - N[(a * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -47000

   1. Initial program 98.4%

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

   3. Taylor expanded in t around inf 75.7%

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

      1. mul-1-neg 75.7%

         \[\leadsto x \cdot e^{\color{blue}{-t \cdot y}} \]

      2. distribute-lft-neg-out 75.7%

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

      3. *-commutative 75.7%

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

   5. Simplified 75.7%

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

   if -47000 < t

   1. Initial program 96.9%

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

   3. Taylor expanded in z around 0 96.9%

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

   4. Taylor expanded in t around 0 93.8%

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

      1. add-log-exp 85.7%

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

      2. *-commutative 85.7%

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

      3. exp-to-pow 85.7%

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

   6. Applied egg-rr 85.7%

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

   7. Taylor expanded in a around 0 85.7%

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

      1. mul-1-neg 85.7%

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

      2. unsub-neg 85.7%

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

      3. exp-to-pow 85.7%

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

      4. *-commutative 85.7%

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

      5. rem-log-exp 93.8%

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

   9. Simplified 93.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -47000:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \log z - a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) - (a * 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 * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in z around 0 97.3%

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

4. Final simplification 97.3%

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

Alternative 7: 68.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{if}\;a \leq -1.4 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -2.5 \cdot 10^{-222}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+23}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- b))))))
   (if (<= a -1.4e+34)
     t_1
     (if (<= a -2.5e-222)
       (* x (pow z y))
       (if (<= a 2.8e+23) (* x (exp (* t (- y)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * -b));
	double tmp;
	if (a <= -1.4e+34) {
		tmp = t_1;
	} else if (a <= -2.5e-222) {
		tmp = x * pow(z, y);
	} else if (a <= 2.8e+23) {
		tmp = x * exp((t * -y));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x * exp((a * -b))
    if (a <= (-1.4d+34)) then
        tmp = t_1
    else if (a <= (-2.5d-222)) then
        tmp = x * (z ** y)
    else if (a <= 2.8d+23) then
        tmp = x * exp((t * -y))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((a * -b));
	double tmp;
	if (a <= -1.4e+34) {
		tmp = t_1;
	} else if (a <= -2.5e-222) {
		tmp = x * Math.pow(z, y);
	} else if (a <= 2.8e+23) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * -b))
	tmp = 0
	if a <= -1.4e+34:
		tmp = t_1
	elif a <= -2.5e-222:
		tmp = x * math.pow(z, y)
	elif a <= 2.8e+23:
		tmp = x * math.exp((t * -y))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(-b))))
	tmp = 0.0
	if (a <= -1.4e+34)
		tmp = t_1;
	elseif (a <= -2.5e-222)
		tmp = Float64(x * (z ^ y));
	elseif (a <= 2.8e+23)
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * -b));
	tmp = 0.0;
	if (a <= -1.4e+34)
		tmp = t_1;
	elseif (a <= -2.5e-222)
		tmp = x * (z ^ y);
	elseif (a <= 2.8e+23)
		tmp = x * exp((t * -y));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.4e+34], t$95$1, If[LessEqual[a, -2.5e-222], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.8e+23], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.40000000000000004e34 or 2.8e23 < a

   1. Initial program 94.4%

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

   3. Taylor expanded in b around inf 78.9%

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

      1. associate-*r* 78.9%

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

      2. mul-1-neg 78.9%

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

   5. Simplified 78.9%

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

   if -1.40000000000000004e34 < a < -2.50000000000000004e-222

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 87.3%

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

   4. Taylor expanded in t around 0 76.2%

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

   if -2.50000000000000004e-222 < a < 2.8e23

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf 79.9%

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

      1. mul-1-neg 79.9%

         \[\leadsto x \cdot e^{\color{blue}{-t \cdot y}} \]

      2. distribute-lft-neg-out 79.9%

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

      3. *-commutative 79.9%

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

   5. Simplified 79.9%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+34}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;a \leq -2.5 \cdot 10^{-222}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+23}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -880 \lor \neg \left(t \leq 9.6 \cdot 10^{-55}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -880.0) (not (<= t 9.6e-55)))
   (* x (exp (* t (- y))))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -880.0) || !(t <= 9.6e-55)) {
		tmp = x * exp((t * -y));
	} 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 <= (-880.0d0)) .or. (.not. (t <= 9.6d-55))) then
        tmp = x * exp((t * -y))
    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 <= -880.0) || !(t <= 9.6e-55)) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -880.0) or not (t <= 9.6e-55):
		tmp = x * math.exp((t * -y))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -880.0) || !(t <= 9.6e-55))
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	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 <= -880.0) || ~((t <= 9.6e-55)))
		tmp = x * exp((t * -y));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -880.0], N[Not[LessEqual[t, 9.6e-55]], $MachinePrecision]], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -880 or 9.59999999999999966e-55 < t

   1. Initial program 98.6%

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

   3. Taylor expanded in t around inf 79.6%

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

      1. mul-1-neg 79.6%

         \[\leadsto x \cdot e^{\color{blue}{-t \cdot y}} \]

      2. distribute-lft-neg-out 79.6%

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

      3. *-commutative 79.6%

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

   5. Simplified 79.6%

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

   if -880 < t < 9.59999999999999966e-55

   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. Add Preprocessing

   3. Taylor expanded in y around inf 65.6%

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

   4. Taylor expanded in t around 0 65.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -880 \lor \neg \left(t \leq 9.6 \cdot 10^{-55}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-128} \lor \neg \left(y \leq 4.8 \cdot 10^{+27}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x + b \cdot \left(a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot b\right)\right) - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.5e-128) (not (<= y 4.8e+27)))
   (* x (pow z y))
   (+ x (* b (* a (- (* 0.5 (* a (* x b))) x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.5e-128) || !(y <= 4.8e+27)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - 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 <= (-2.5d-128)) .or. (.not. (y <= 4.8d+27))) then
        tmp = x * (z ** y)
    else
        tmp = x + (b * (a * ((0.5d0 * (a * (x * b))) - 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 <= -2.5e-128) || !(y <= 4.8e+27)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.5e-128) or not (y <= 4.8e+27):
		tmp = x * math.pow(z, y)
	else:
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.5e-128) || !(y <= 4.8e+27))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x + Float64(b * Float64(a * Float64(Float64(0.5 * Float64(a * Float64(x * b))) - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.5e-128) || ~((y <= 4.8e+27)))
		tmp = x * (z ^ y);
	else
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.5e-128], N[Not[LessEqual[y, 4.8e+27]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x + N[(b * N[(a * N[(N[(0.5 * N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5000000000000001e-128 or 4.79999999999999995e27 < y

   1. Initial program 98.8%

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

   3. Taylor expanded in y around inf 83.5%

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

   4. Taylor expanded in t around 0 66.3%

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

   if -2.5000000000000001e-128 < y < 4.79999999999999995e27

   1. Initial program 94.8%

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

   3. Taylor expanded in z around 0 94.8%

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

   4. Taylor expanded in y around 0 84.3%

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

      1. *-commutative 84.3%

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

      2. mul-1-neg 84.3%

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

   6. Simplified 84.3%

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

   7. Taylor expanded in b around 0 58.8%

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

      1. +-commutative 58.8%

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

      2. mul-1-neg 58.8%

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

      3. unsub-neg 58.8%

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

      4. *-commutative 58.8%

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

      5. *-commutative 58.8%

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

   9. Simplified 58.8%

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

   10. Taylor expanded in a around 0 59.1%

       \[\leadsto x + b \cdot \color{blue}{\left(a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot x\right)\right) - x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-128} \lor \neg \left(y \leq 4.8 \cdot 10^{+27}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x + b \cdot \left(a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot b\right)\right) - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.8% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{+35} \lor \neg \left(a \leq 7.2 \cdot 10^{-20}\right):\\ \;\;\;\;x + b \cdot \left(a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot b\right)\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -4.9e+35) (not (<= a 7.2e-20)))
   (+ x (* b (* a (- (* 0.5 (* a (* x b))) x))))
   (- x (* t (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -4.9e+35) || !(a <= 7.2e-20)) {
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	} 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 ((a <= (-4.9d+35)) .or. (.not. (a <= 7.2d-20))) then
        tmp = x + (b * (a * ((0.5d0 * (a * (x * b))) - x)))
    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 ((a <= -4.9e+35) || !(a <= 7.2e-20)) {
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	} else {
		tmp = x - (t * (x * y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -4.9e+35) or not (a <= 7.2e-20):
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)))
	else:
		tmp = x - (t * (x * y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -4.9e+35) || !(a <= 7.2e-20))
		tmp = Float64(x + Float64(b * Float64(a * Float64(Float64(0.5 * Float64(a * Float64(x * b))) - x))));
	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 ((a <= -4.9e+35) || ~((a <= 7.2e-20)))
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	else
		tmp = x - (t * (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -4.9e+35], N[Not[LessEqual[a, 7.2e-20]], $MachinePrecision]], N[(x + N[(b * N[(a * N[(N[(0.5 * N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.90000000000000025e35 or 7.19999999999999948e-20 < a

   1. Initial program 94.7%

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

   3. Taylor expanded in z around 0 94.7%

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

   4. Taylor expanded in y around 0 76.5%

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

      1. *-commutative 76.5%

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

      2. mul-1-neg 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in b around 0 40.9%

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

      1. +-commutative 40.9%

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

      2. mul-1-neg 40.9%

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

      3. unsub-neg 40.9%

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

      4. *-commutative 40.9%

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

      5. *-commutative 40.9%

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

   9. Simplified 40.9%

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

   10. Taylor expanded in a around 0 41.3%

       \[\leadsto x + b \cdot \color{blue}{\left(a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot x\right)\right) - x\right)\right)} \]

   if -4.90000000000000025e35 < a < 7.19999999999999948e-20

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 91.2%

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

   4. Taylor expanded in y around 0 49.1%

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

      1. associate-*r* 47.7%

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

   6. Simplified 47.7%

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

   7. Taylor expanded in t around inf 45.9%

      \[\leadsto x + \color{blue}{-1 \cdot \left(t \cdot \left(x \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{+35} \lor \neg \left(a \leq 7.2 \cdot 10^{-20}\right):\\ \;\;\;\;x + b \cdot \left(a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot b\right)\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.6% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{+35}:\\ \;\;\;\;b \cdot \left(\frac{x}{b} - x \cdot a\right)\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-19}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -6.4e+35)
   (* b (- (/ x b) (* x a)))
   (if (<= a 4.2e-19) (- x (* t (* x y))) (* x (- 1.0 (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -6.4e+35) {
		tmp = b * ((x / b) - (x * a));
	} else if (a <= 4.2e-19) {
		tmp = x - (t * (x * y));
	} else {
		tmp = x * (1.0 - (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 (a <= (-6.4d+35)) then
        tmp = b * ((x / b) - (x * a))
    else if (a <= 4.2d-19) then
        tmp = x - (t * (x * y))
    else
        tmp = x * (1.0d0 - (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 (a <= -6.4e+35) {
		tmp = b * ((x / b) - (x * a));
	} else if (a <= 4.2e-19) {
		tmp = x - (t * (x * y));
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -6.4e+35:
		tmp = b * ((x / b) - (x * a))
	elif a <= 4.2e-19:
		tmp = x - (t * (x * y))
	else:
		tmp = x * (1.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -6.4e+35)
		tmp = Float64(b * Float64(Float64(x / b) - Float64(x * a)));
	elseif (a <= 4.2e-19)
		tmp = Float64(x - Float64(t * Float64(x * y)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -6.4e+35)
		tmp = b * ((x / b) - (x * a));
	elseif (a <= 4.2e-19)
		tmp = x - (t * (x * y));
	else
		tmp = x * (1.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -6.4e+35], N[(b * N[(N[(x / b), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.2e-19], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.39999999999999965e35

   1. Initial program 90.3%

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

   3. Taylor expanded in z around 0 90.3%

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

   4. Taylor expanded in y around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. mul-1-neg 75.7%

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

   6. Simplified 75.7%

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

   7. Taylor expanded in a around 0 22.4%

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

      1. neg-mul-1 22.4%

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

      2. unsub-neg 22.4%

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

   9. Simplified 22.4%

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

   10. Taylor expanded in b around inf 23.9%

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

   if -6.39999999999999965e35 < a < 4.1999999999999998e-19

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 91.2%

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

   4. Taylor expanded in y around 0 49.1%

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

      1. associate-*r* 47.7%

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

   6. Simplified 47.7%

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

   7. Taylor expanded in t around inf 45.9%

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

   if 4.1999999999999998e-19 < a

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0 98.6%

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

   4. Taylor expanded in y around 0 77.1%

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

      1. *-commutative 77.1%

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

      2. mul-1-neg 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in a around 0 28.5%

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

      1. neg-mul-1 28.5%

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

      2. unsub-neg 28.5%

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

   9. Simplified 28.5%

      \[\leadsto \color{blue}{\left(1 - a \cdot b\right)} \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{+35}:\\ \;\;\;\;b \cdot \left(\frac{x}{b} - x \cdot a\right)\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-19}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.9% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+162}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.6e+162)
   (- x (* t (* x y)))
   (if (<= y 1.02e+105) (* x (- 1.0 (* a b))) (* a (* x (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.6e+162) {
		tmp = x - (t * (x * y));
	} else if (y <= 1.02e+105) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = a * (x * -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 <= (-1.6d+162)) then
        tmp = x - (t * (x * y))
    else if (y <= 1.02d+105) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = a * (x * -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 <= -1.6e+162) {
		tmp = x - (t * (x * y));
	} else if (y <= 1.02e+105) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.6e+162:
		tmp = x - (t * (x * y))
	elif y <= 1.02e+105:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = a * (x * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.6e+162)
		tmp = Float64(x - Float64(t * Float64(x * y)));
	elseif (y <= 1.02e+105)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(a * Float64(x * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.6e+162)
		tmp = x - (t * (x * y));
	elseif (y <= 1.02e+105)
		tmp = x * (1.0 - (a * b));
	else
		tmp = a * (x * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.6e+162], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.02e+105], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6000000000000001e162

   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. Add Preprocessing

   3. Taylor expanded in y around inf 86.4%

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

   4. Taylor expanded in y around 0 43.3%

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

      1. associate-*r* 33.6%

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

   6. Simplified 33.6%

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

   7. Taylor expanded in t around inf 22.8%

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

   if -1.6000000000000001e162 < y < 1.02e105

   1. Initial program 96.6%

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

   3. Taylor expanded in z around 0 96.6%

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

   4. Taylor expanded in y around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

   6. Simplified 71.7%

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

   7. Taylor expanded in a around 0 40.0%

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

      1. neg-mul-1 40.0%

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

      2. unsub-neg 40.0%

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

   9. Simplified 40.0%

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

   if 1.02e105 < y

   1. Initial program 98.1%

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

   3. Taylor expanded in z around 0 98.1%

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

   4. Taylor expanded in y around 0 43.5%

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

      1. *-commutative 43.5%

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

      2. mul-1-neg 43.5%

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

   6. Simplified 43.5%

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

   7. Taylor expanded in a around 0 14.2%

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

      1. neg-mul-1 14.2%

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

      2. unsub-neg 14.2%

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

   9. Simplified 14.2%

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

   10. Taylor expanded in a around inf 30.4%

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

       1. associate-*r* 30.4%

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

       2. neg-mul-1 30.4%

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

   12. Simplified 30.4%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+162}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.1% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-23} \lor \neg \left(y \leq 3.95 \cdot 10^{-16}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4.9e-23) (not (<= y 3.95e-16))) (* a (* x (- b))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.9e-23) || !(y <= 3.95e-16)) {
		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 <= (-4.9d-23)) .or. (.not. (y <= 3.95d-16))) 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 <= -4.9e-23) || !(y <= 3.95e-16)) {
		tmp = a * (x * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -4.9e-23) or not (y <= 3.95e-16):
		tmp = a * (x * -b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -4.9e-23) || !(y <= 3.95e-16))
		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 <= -4.9e-23) || ~((y <= 3.95e-16)))
		tmp = a * (x * -b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -4.9e-23], N[Not[LessEqual[y, 3.95e-16]], $MachinePrecision]], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -4.8999999999999998e-23 or 3.9500000000000001e-16 < y

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0 98.7%

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

   4. Taylor expanded in y around 0 45.3%

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

      1. *-commutative 45.3%

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

      2. mul-1-neg 45.3%

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

   6. Simplified 45.3%

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

   7. Taylor expanded in a around 0 15.7%

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

      1. neg-mul-1 15.7%

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

      2. unsub-neg 15.7%

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

   9. Simplified 15.7%

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

   10. Taylor expanded in a around inf 19.3%

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

       1. associate-*r* 19.3%

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

       2. neg-mul-1 19.3%

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

   12. Simplified 19.3%

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

   if -4.8999999999999998e-23 < y < 3.9500000000000001e-16

   1. Initial program 95.2%

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

   3. Taylor expanded in y around inf 59.2%

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

   4. Taylor expanded in y around 0 44.8%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-23} \lor \neg \left(y \leq 3.95 \cdot 10^{-16}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.2% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 1.02e+105) (* x (- 1.0 (* a b))) (* a (* x (- b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 1.02e+105) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = a * (x * -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 <= 1.02d+105) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = a * (x * -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 <= 1.02e+105) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 1.02e+105:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = a * (x * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 1.02e+105)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(a * Float64(x * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 1.02e+105)
		tmp = x * (1.0 - (a * b));
	else
		tmp = a * (x * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 1.02e+105], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.02e105

   1. Initial program 97.0%

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

   3. Taylor expanded in z around 0 97.0%

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

   4. Taylor expanded in y around 0 66.6%

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

      1. *-commutative 66.6%

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

      2. mul-1-neg 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in a around 0 35.1%

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

      1. neg-mul-1 35.1%

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

      2. unsub-neg 35.1%

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

   9. Simplified 35.1%

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

   if 1.02e105 < y

   1. Initial program 98.1%

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

   3. Taylor expanded in z around 0 98.1%

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

   4. Taylor expanded in y around 0 43.5%

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

      1. *-commutative 43.5%

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

      2. mul-1-neg 43.5%

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

   6. Simplified 43.5%

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

   7. Taylor expanded in a around 0 14.2%

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

      1. neg-mul-1 14.2%

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

      2. unsub-neg 14.2%

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

   9. Simplified 14.2%

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

   10. Taylor expanded in a around inf 30.4%

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

       1. associate-*r* 30.4%

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

       2. neg-mul-1 30.4%

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

   12. Simplified 30.4%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 18.9% 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 97.3%

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

3. Taylor expanded in y around inf 73.2%

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

4. Taylor expanded in y around 0 20.6%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024102 
(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))))))