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

Percentage Accurate: 96.5% → 99.5%

Time: 21.5s

Alternatives: 20

Speedup: 1.0×

• 39.3% 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: 96.5% 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: 99.5% 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 97.6%

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

   1. fma-define 98.4%

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

   2. sub-neg 98.4%

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

   3. log1p-define 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 96.5% 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]

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. Add Preprocessing

3. Final simplification 97.6%

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

Alternative 3: 84.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.62 \cdot 10^{+100} \lor \neg \left(a \leq 2.3 \cdot 10^{+70}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.62e+100) (not (<= a 2.3e+70)))
   (* 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 <= -1.62e+100) || !(a <= 2.3e+70)) {
		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 <= -1.62e+100) || !(a <= 2.3e+70)) {
		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 <= -1.62e+100) or not (a <= 2.3e+70):
		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 <= -1.62e+100) || !(a <= 2.3e+70))
		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, -1.62e+100], N[Not[LessEqual[a, 2.3e+70]], $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 < -1.62000000000000003e100 or 2.29999999999999994e70 < a

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0 83.2%

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

      1. sub-neg 83.2%

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

      2. log1p-define 87.7%

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

   5. Simplified 87.7%

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

   if -1.62000000000000003e100 < a < 2.29999999999999994e70

   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 y around inf 90.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.62 \cdot 10^{+100} \lor \neg \left(a \leq 2.3 \cdot 10^{+70}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+110} \lor \neg \left(a \leq 1.5 \cdot 10^{+95}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.25e+110) (not (<= a 1.5e+95)))
   (* x (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 <= -1.25e+110) || !(a <= 1.5e+95)) {
		tmp = x * 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 <= (-1.25d+110)) .or. (.not. (a <= 1.5d+95))) 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 <= -1.25e+110) || !(a <= 1.5e+95)) {
		tmp = x * 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 <= -1.25e+110) or not (a <= 1.5e+95):
		tmp = x * 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 <= -1.25e+110) || !(a <= 1.5e+95))
		tmp = Float64(x * exp(Float64(a * Float64(-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 <= -1.25e+110) || ~((a <= 1.5e+95)))
		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, -1.25e+110], N[Not[LessEqual[a, 1.5e+95]], $MachinePrecision]], N[(x * N[Exp[N[(a * (-b)), $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 < -1.24999999999999995e110 or 1.49999999999999996e95 < a

   1. Initial program 92.8%

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

   3. Taylor expanded in b around inf 81.3%

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

      1. associate-*r* 81.3%

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

      2. mul-1-neg 81.3%

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

   5. Simplified 81.3%

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

   if -1.24999999999999995e110 < a < 1.49999999999999996e95

   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 y around inf 89.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+110} \lor \neg \left(a \leq 1.5 \cdot 10^{+95}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(-b\right)}\\ t_2 := \frac{x}{e^{y \cdot t}}\\ \mathbf{if}\;t \leq -9 \cdot 10^{+26}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-164}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-35}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- b))))) (t_2 (/ x (exp (* y t)))))
   (if (<= t -9e+26)
     t_2
     (if (<= t -5.8e-164)
       t_1
       (if (<= t 2.2e-35) (* x (pow z y)) (if (<= t 2.25e+101) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * -b));
	double t_2 = x / exp((y * t));
	double tmp;
	if (t <= -9e+26) {
		tmp = t_2;
	} else if (t <= -5.8e-164) {
		tmp = t_1;
	} else if (t <= 2.2e-35) {
		tmp = x * pow(z, y);
	} else if (t <= 2.25e+101) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = x * exp((a * -b))
    t_2 = x / exp((y * t))
    if (t <= (-9d+26)) then
        tmp = t_2
    else if (t <= (-5.8d-164)) then
        tmp = t_1
    else if (t <= 2.2d-35) then
        tmp = x * (z ** y)
    else if (t <= 2.25d+101) then
        tmp = t_1
    else
        tmp = t_2
    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 t_2 = x / Math.exp((y * t));
	double tmp;
	if (t <= -9e+26) {
		tmp = t_2;
	} else if (t <= -5.8e-164) {
		tmp = t_1;
	} else if (t <= 2.2e-35) {
		tmp = x * Math.pow(z, y);
	} else if (t <= 2.25e+101) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * -b))
	t_2 = x / math.exp((y * t))
	tmp = 0
	if t <= -9e+26:
		tmp = t_2
	elif t <= -5.8e-164:
		tmp = t_1
	elif t <= 2.2e-35:
		tmp = x * math.pow(z, y)
	elif t <= 2.25e+101:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(-b))))
	t_2 = Float64(x / exp(Float64(y * t)))
	tmp = 0.0
	if (t <= -9e+26)
		tmp = t_2;
	elseif (t <= -5.8e-164)
		tmp = t_1;
	elseif (t <= 2.2e-35)
		tmp = Float64(x * (z ^ y));
	elseif (t <= 2.25e+101)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * -b));
	t_2 = x / exp((y * t));
	tmp = 0.0;
	if (t <= -9e+26)
		tmp = t_2;
	elseif (t <= -5.8e-164)
		tmp = t_1;
	elseif (t <= 2.2e-35)
		tmp = x * (z ^ y);
	elseif (t <= 2.25e+101)
		tmp = t_1;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(x / N[Exp[N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9e+26], t$95$2, If[LessEqual[t, -5.8e-164], t$95$1, If[LessEqual[t, 2.2e-35], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.25e+101], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.99999999999999957e26 or 2.2500000000000001e101 < t

   1. Initial program 96.3%

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

   3. Taylor expanded in t around inf 83.3%

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

      1. mul-1-neg 83.3%

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

      2. distribute-lft-neg-out 83.3%

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

      3. *-commutative 83.3%

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

   5. Simplified 83.3%

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

      1. distribute-rgt-neg-out 83.3%

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

      2. exp-neg 83.3%

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

   7. Applied egg-rr 83.3%

      \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

   8. Taylor expanded in x around 0 83.4%

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

   if -8.99999999999999957e26 < t < -5.8e-164 or 2.19999999999999994e-35 < t < 2.2500000000000001e101

   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 b around inf 77.4%

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

      1. associate-*r* 77.4%

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

      2. mul-1-neg 77.4%

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

   5. Simplified 77.4%

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

   if -5.8e-164 < t < 2.19999999999999994e-35

   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 y around inf 78.6%

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

   4. Taylor expanded in t around 0 78.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-164}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-35}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+101}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{if}\;t \leq -8.6 \cdot 10^{+26}:\\ \;\;\;\;x \cdot {e}^{\left(t \cdot \left(-y\right)\right)}\\ \mathbf{elif}\;t \leq -5.3 \cdot 10^{-149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-35}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- b))))))
   (if (<= t -8.6e+26)
     (* x (pow E (* t (- y))))
     (if (<= t -5.3e-149)
       t_1
       (if (<= t 2.05e-35)
         (* x (pow z y))
         (if (<= t 2e+101) t_1 (/ x (exp (* y t)))))))))

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 (t <= -8.6e+26) {
		tmp = x * pow(((double) M_E), (t * -y));
	} else if (t <= -5.3e-149) {
		tmp = t_1;
	} else if (t <= 2.05e-35) {
		tmp = x * pow(z, y);
	} else if (t <= 2e+101) {
		tmp = t_1;
	} else {
		tmp = x / exp((y * t));
	}
	return tmp;
}

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 (t <= -8.6e+26) {
		tmp = x * Math.pow(Math.E, (t * -y));
	} else if (t <= -5.3e-149) {
		tmp = t_1;
	} else if (t <= 2.05e-35) {
		tmp = x * Math.pow(z, y);
	} else if (t <= 2e+101) {
		tmp = t_1;
	} else {
		tmp = x / Math.exp((y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * -b))
	tmp = 0
	if t <= -8.6e+26:
		tmp = x * math.pow(math.e, (t * -y))
	elif t <= -5.3e-149:
		tmp = t_1
	elif t <= 2.05e-35:
		tmp = x * math.pow(z, y)
	elif t <= 2e+101:
		tmp = t_1
	else:
		tmp = x / math.exp((y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(-b))))
	tmp = 0.0
	if (t <= -8.6e+26)
		tmp = Float64(x * (exp(1) ^ Float64(t * Float64(-y))));
	elseif (t <= -5.3e-149)
		tmp = t_1;
	elseif (t <= 2.05e-35)
		tmp = Float64(x * (z ^ y));
	elseif (t <= 2e+101)
		tmp = t_1;
	else
		tmp = Float64(x / exp(Float64(y * t)));
	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 (t <= -8.6e+26)
		tmp = x * (2.71828182845904523536 ^ (t * -y));
	elseif (t <= -5.3e-149)
		tmp = t_1;
	elseif (t <= 2.05e-35)
		tmp = x * (z ^ y);
	elseif (t <= 2e+101)
		tmp = t_1;
	else
		tmp = x / exp((y * t));
	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[t, -8.6e+26], N[(x * N[Power[E, N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5.3e-149], t$95$1, If[LessEqual[t, 2.05e-35], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e+101], t$95$1, N[(x / N[Exp[N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -8.5999999999999996e26

   1. Initial program 96.7%

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

   3. Taylor expanded in t around inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. distribute-lft-neg-out 81.6%

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

      3. *-commutative 81.6%

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

   5. Simplified 81.6%

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

      1. *-un-lft-identity 81.6%

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

      2. exp-prod 81.7%

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

      3. exp-1-e 81.7%

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

   7. Applied egg-rr 81.7%

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

   if -8.5999999999999996e26 < t < -5.30000000000000013e-149 or 2.05000000000000013e-35 < t < 2e101

   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 b around inf 77.4%

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

      1. associate-*r* 77.4%

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

      2. mul-1-neg 77.4%

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

   5. Simplified 77.4%

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

   if -5.30000000000000013e-149 < t < 2.05000000000000013e-35

   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 y around inf 78.6%

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

   4. Taylor expanded in t around 0 78.7%

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

   if 2e101 < 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. Add Preprocessing

   3. Taylor expanded in t around inf 85.6%

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

      1. mul-1-neg 85.6%

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

      2. distribute-lft-neg-out 85.6%

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

      3. *-commutative 85.6%

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

   5. Simplified 85.6%

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

      1. distribute-rgt-neg-out 85.6%

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

      2. exp-neg 85.6%

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

   7. Applied egg-rr 85.6%

      \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

   8. Taylor expanded in x around 0 85.7%

      \[\leadsto \color{blue}{\frac{x}{e^{t \cdot y}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+26}:\\ \;\;\;\;x \cdot {e}^{\left(t \cdot \left(-y\right)\right)}\\ \mathbf{elif}\;t \leq -5.3 \cdot 10^{-149}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-35}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+101}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-11} \lor \neg \left(t \leq 5 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -2e-11) (not (<= t 5e-33)))
   (/ x (exp (* 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 <= -2e-11) || !(t <= 5e-33)) {
		tmp = x / exp((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 <= (-2d-11)) .or. (.not. (t <= 5d-33))) then
        tmp = x / exp((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 <= -2e-11) || !(t <= 5e-33)) {
		tmp = x / Math.exp((y * t));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -2e-11) or not (t <= 5e-33):
		tmp = x / math.exp((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 <= -2e-11) || !(t <= 5e-33))
		tmp = Float64(x / exp(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 <= -2e-11) || ~((t <= 5e-33)))
		tmp = x / exp((y * t));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -2e-11], N[Not[LessEqual[t, 5e-33]], $MachinePrecision]], N[(x / N[Exp[N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.99999999999999988e-11 or 5.00000000000000028e-33 < t

   1. Initial program 96.7%

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

   3. Taylor expanded in t around inf 76.1%

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

      1. mul-1-neg 76.1%

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

      2. distribute-lft-neg-out 76.1%

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

      3. *-commutative 76.1%

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

   5. Simplified 76.1%

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

      1. distribute-rgt-neg-out 76.1%

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

      2. exp-neg 76.1%

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

   7. Applied egg-rr 76.1%

      \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

   8. Taylor expanded in x around 0 76.1%

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

   if -1.99999999999999988e-11 < t < 5.00000000000000028e-33

   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. Add Preprocessing

   3. Taylor expanded in y around inf 74.2%

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

   4. Taylor expanded in t around 0 74.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-11} \lor \neg \left(t \leq 5 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{+233}:\\ \;\;\;\;t \cdot \frac{x - x \cdot \left(y \cdot t\right)}{t}\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{+47}:\\ \;\;\;\;x \cdot \frac{1}{1 + y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.66e+233)
   (* t (/ (- x (* x (* y t))) t))
   (if (<= t -4.8e+47) (* x (/ 1.0 (+ 1.0 (* 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 <= -1.66e+233) {
		tmp = t * ((x - (x * (y * t))) / t);
	} else if (t <= -4.8e+47) {
		tmp = x * (1.0 / (1.0 + (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 <= (-1.66d+233)) then
        tmp = t * ((x - (x * (y * t))) / t)
    else if (t <= (-4.8d+47)) then
        tmp = x * (1.0d0 / (1.0d0 + (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 <= -1.66e+233) {
		tmp = t * ((x - (x * (y * t))) / t);
	} else if (t <= -4.8e+47) {
		tmp = x * (1.0 / (1.0 + (y * t)));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.66e+233:
		tmp = t * ((x - (x * (y * t))) / t)
	elif t <= -4.8e+47:
		tmp = x * (1.0 / (1.0 + (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 <= -1.66e+233)
		tmp = Float64(t * Float64(Float64(x - Float64(x * Float64(y * t))) / t));
	elseif (t <= -4.8e+47)
		tmp = Float64(x * Float64(1.0 / Float64(1.0 + 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 <= -1.66e+233)
		tmp = t * ((x - (x * (y * t))) / t);
	elseif (t <= -4.8e+47)
		tmp = x * (1.0 / (1.0 + (y * t)));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.66e+233], N[(t * N[(N[(x - N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.8e+47], N[(x * N[(1.0 / N[(1.0 + N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.66e233

   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. Add Preprocessing

   3. Taylor expanded in t around inf 78.5%

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

      1. mul-1-neg 78.5%

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

      2. distribute-lft-neg-out 78.5%

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

      3. *-commutative 78.5%

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

   5. Simplified 78.5%

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

   6. Taylor expanded in y around 0 24.5%

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

      1. mul-1-neg 24.5%

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

      2. unsub-neg 24.5%

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

      3. associate-*r* 31.2%

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

   8. Simplified 31.2%

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

   9. Taylor expanded in t around inf 38.0%

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

       1. *-commutative 38.0%

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

   11. Simplified 38.0%

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

   12. Taylor expanded in t around 0 38.0%

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

       1. associate-*r* 44.8%

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

       2. associate-*r* 44.8%

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

       3. *-commutative 44.8%

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

       4. mul-1-neg 44.8%

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

       5. distribute-rgt-neg-out 44.8%

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

       6. associate-*r* 44.9%

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

       7. *-commutative 44.9%

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

   14. Simplified 44.9%

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

   if -1.66e233 < t < -4.80000000000000037e47

   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 t around inf 86.1%

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

      1. mul-1-neg 86.1%

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

      2. distribute-lft-neg-out 86.1%

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

      3. *-commutative 86.1%

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

   5. Simplified 86.1%

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

      1. distribute-rgt-neg-out 86.1%

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

      2. exp-neg 86.1%

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

   7. Applied egg-rr 86.1%

      \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0 44.6%

      \[\leadsto x \cdot \frac{1}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. *-commutative 44.6%

         \[\leadsto x \cdot \frac{1}{1 + \color{blue}{y \cdot t}} \]

   10. Simplified 44.6%

       \[\leadsto x \cdot \frac{1}{\color{blue}{1 + y \cdot t}} \]

   if -4.80000000000000037e47 < t

   1. Initial program 98.0%

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

   3. Taylor expanded in y around inf 72.4%

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

   4. Taylor expanded in t around 0 64.4%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{+233}:\\ \;\;\;\;t \cdot \frac{x - x \cdot \left(y \cdot t\right)}{t}\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{+47}:\\ \;\;\;\;x \cdot \frac{1}{1 + y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 27.5% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{x}{t}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (/ x t))))
   (if (<= y -6.2e-56)
     t_1
     (if (<= y -3.2e-116)
       (* y (/ x y))
       (if (<= y 7.2e+154) t_1 (* x (* t (- y))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (x / t);
	double tmp;
	if (y <= -6.2e-56) {
		tmp = t_1;
	} else if (y <= -3.2e-116) {
		tmp = y * (x / y);
	} else if (y <= 7.2e+154) {
		tmp = t_1;
	} else {
		tmp = x * (t * -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) :: t_1
    real(8) :: tmp
    t_1 = t * (x / t)
    if (y <= (-6.2d-56)) then
        tmp = t_1
    else if (y <= (-3.2d-116)) then
        tmp = y * (x / y)
    else if (y <= 7.2d+154) then
        tmp = t_1
    else
        tmp = x * (t * -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 t_1 = t * (x / t);
	double tmp;
	if (y <= -6.2e-56) {
		tmp = t_1;
	} else if (y <= -3.2e-116) {
		tmp = y * (x / y);
	} else if (y <= 7.2e+154) {
		tmp = t_1;
	} else {
		tmp = x * (t * -y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t * (x / t)
	tmp = 0
	if y <= -6.2e-56:
		tmp = t_1
	elif y <= -3.2e-116:
		tmp = y * (x / y)
	elif y <= 7.2e+154:
		tmp = t_1
	else:
		tmp = x * (t * -y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(x / t))
	tmp = 0.0
	if (y <= -6.2e-56)
		tmp = t_1;
	elseif (y <= -3.2e-116)
		tmp = Float64(y * Float64(x / y));
	elseif (y <= 7.2e+154)
		tmp = t_1;
	else
		tmp = Float64(x * Float64(t * Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t * (x / t);
	tmp = 0.0;
	if (y <= -6.2e-56)
		tmp = t_1;
	elseif (y <= -3.2e-116)
		tmp = y * (x / y);
	elseif (y <= 7.2e+154)
		tmp = t_1;
	else
		tmp = x * (t * -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e-56], t$95$1, If[LessEqual[y, -3.2e-116], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+154], t$95$1, N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.19999999999999975e-56 or -3.20000000000000009e-116 < y < 7.2000000000000001e154

   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. Add Preprocessing

   3. Taylor expanded in t around inf 58.4%

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

      1. mul-1-neg 58.4%

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

      2. distribute-lft-neg-out 58.4%

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

      3. *-commutative 58.4%

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

   5. Simplified 58.4%

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

   6. Taylor expanded in y around 0 28.9%

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

      1. mul-1-neg 28.9%

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

      2. unsub-neg 28.9%

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

      3. associate-*r* 28.3%

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

   8. Simplified 28.3%

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

   9. Taylor expanded in t around inf 34.1%

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

       1. *-commutative 34.1%

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

   11. Simplified 34.1%

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

   12. Taylor expanded in t around 0 34.2%

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]

   if -6.19999999999999975e-56 < y < -3.20000000000000009e-116

   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 t around inf 65.7%

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

      1. mul-1-neg 65.7%

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

      2. distribute-lft-neg-out 65.7%

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

      3. *-commutative 65.7%

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

   5. Simplified 65.7%

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

   6. Taylor expanded in y around 0 26.9%

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

      1. mul-1-neg 26.9%

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

      2. unsub-neg 26.9%

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

      3. associate-*r* 42.7%

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

   8. Simplified 42.7%

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

   9. Taylor expanded in y around inf 42.6%

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

   10. Taylor expanded in y around 0 42.8%

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

   if 7.2000000000000001e154 < y

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf 75.6%

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

      1. mul-1-neg 75.6%

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

      2. distribute-lft-neg-out 75.6%

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

      3. *-commutative 75.6%

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

   5. Simplified 75.6%

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

   6. Taylor expanded in y around 0 30.6%

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

      1. mul-1-neg 30.6%

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

      2. unsub-neg 30.6%

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

      3. associate-*r* 30.7%

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

   8. Simplified 30.7%

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

   9. Taylor expanded in t around inf 30.7%

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

       1. mul-1-neg 30.7%

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

       2. *-commutative 30.7%

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

       3. distribute-rgt-neg-in 30.7%

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

       4. associate-*r* 37.4%

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

   11. Simplified 37.4%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-56}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 28.4% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-55} \lor \neg \left(y \leq -2.6 \cdot 10^{-117}\right) \land y \leq 3.3 \cdot 10^{+106}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4.8e-55) (and (not (<= y -2.6e-117)) (<= y 3.3e+106)))
   (* t (/ x t))
   (* y (/ x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.8e-55) || (!(y <= -2.6e-117) && (y <= 3.3e+106))) {
		tmp = t * (x / t);
	} else {
		tmp = y * (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 ((y <= (-4.8d-55)) .or. (.not. (y <= (-2.6d-117))) .and. (y <= 3.3d+106)) then
        tmp = t * (x / t)
    else
        tmp = y * (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 ((y <= -4.8e-55) || (!(y <= -2.6e-117) && (y <= 3.3e+106))) {
		tmp = t * (x / t);
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -4.8e-55) or (not (y <= -2.6e-117) and (y <= 3.3e+106)):
		tmp = t * (x / t)
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -4.8e-55) || (!(y <= -2.6e-117) && (y <= 3.3e+106)))
		tmp = Float64(t * Float64(x / t));
	else
		tmp = Float64(y * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -4.8e-55) || (~((y <= -2.6e-117)) && (y <= 3.3e+106)))
		tmp = t * (x / t);
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -4.8e-55], And[N[Not[LessEqual[y, -2.6e-117]], $MachinePrecision], LessEqual[y, 3.3e+106]]], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.79999999999999983e-55 or -2.59999999999999983e-117 < y < 3.30000000000000008e106

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf 57.7%

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

      1. mul-1-neg 57.7%

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

      2. distribute-lft-neg-out 57.7%

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

      3. *-commutative 57.7%

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

   5. Simplified 57.7%

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

   6. Taylor expanded in y around 0 30.9%

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

      1. mul-1-neg 30.9%

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

      2. unsub-neg 30.9%

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

      3. associate-*r* 30.3%

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

   8. Simplified 30.3%

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

   9. Taylor expanded in t around inf 36.5%

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

       1. *-commutative 36.5%

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

   11. Simplified 36.5%

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

   12. Taylor expanded in t around 0 36.0%

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]

   if -4.79999999999999983e-55 < y < -2.59999999999999983e-117 or 3.30000000000000008e106 < y

   1. Initial program 96.7%

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

   3. Taylor expanded in t around inf 70.7%

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

      1. mul-1-neg 70.7%

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

      2. distribute-lft-neg-out 70.7%

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

      3. *-commutative 70.7%

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

   5. Simplified 70.7%

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

   6. Taylor expanded in y around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. unsub-neg 22.7%

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

      3. associate-*r* 27.2%

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

   8. Simplified 27.2%

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

   9. Taylor expanded in y around inf 27.2%

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

   10. Taylor expanded in y around 0 28.8%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-55} \lor \neg \left(y \leq -2.6 \cdot 10^{-117}\right) \land y \leq 3.3 \cdot 10^{+106}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.1% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.16 \cdot 10^{-126} \lor \neg \left(a \leq 1.45 \cdot 10^{+209}\right):\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.16e-126) (not (<= a 1.45e+209)))
   (* t (/ x t))
   (* x (- 1.0 (* y t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.16e-126) || !(a <= 1.45e+209)) {
		tmp = t * (x / t);
	} else {
		tmp = x * (1.0 - (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 ((a <= (-1.16d-126)) .or. (.not. (a <= 1.45d+209))) then
        tmp = t * (x / t)
    else
        tmp = x * (1.0d0 - (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 ((a <= -1.16e-126) || !(a <= 1.45e+209)) {
		tmp = t * (x / t);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -1.16e-126) or not (a <= 1.45e+209):
		tmp = t * (x / t)
	else:
		tmp = x * (1.0 - (y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.16e-126) || !(a <= 1.45e+209))
		tmp = Float64(t * Float64(x / t));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -1.16e-126) || ~((a <= 1.45e+209)))
		tmp = t * (x / t);
	else
		tmp = x * (1.0 - (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -1.16e-126], N[Not[LessEqual[a, 1.45e+209]], $MachinePrecision]], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.16e-126 or 1.45e209 < a

   1. Initial program 94.9%

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

   3. Taylor expanded in t around inf 37.6%

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

      1. mul-1-neg 37.6%

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

      2. distribute-lft-neg-out 37.6%

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

      3. *-commutative 37.6%

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

   5. Simplified 37.6%

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

   6. Taylor expanded in y around 0 13.4%

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

      1. mul-1-neg 13.4%

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

      2. unsub-neg 13.4%

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

      3. associate-*r* 13.2%

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

   8. Simplified 13.2%

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

   9. Taylor expanded in t around inf 25.0%

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

       1. *-commutative 25.0%

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

   11. Simplified 25.0%

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

   12. Taylor expanded in t around 0 31.5%

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]

   if -1.16e-126 < a < 1.45e209

   1. Initial program 99.2%

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

   3. Taylor expanded in t around inf 74.7%

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

      1. mul-1-neg 74.7%

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

      2. distribute-lft-neg-out 74.7%

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

      3. *-commutative 74.7%

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

   5. Simplified 74.7%

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

   6. Taylor expanded in y around 0 40.0%

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

      1. mul-1-neg 40.0%

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

      2. unsub-neg 40.0%

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

      3. *-commutative 40.0%

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

   8. Simplified 40.0%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.16 \cdot 10^{-126} \lor \neg \left(a \leq 1.45 \cdot 10^{+209}\right):\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 28.5% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+51}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+148}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot t\right) \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.55e+51)
   (* t (* x (- y)))
   (if (<= y 1.75e+148) (* t (/ x t)) (* (* x t) (- y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.55e+51) {
		tmp = t * (x * -y);
	} else if (y <= 1.75e+148) {
		tmp = t * (x / t);
	} else {
		tmp = (x * t) * -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 (y <= (-2.55d+51)) then
        tmp = t * (x * -y)
    else if (y <= 1.75d+148) then
        tmp = t * (x / t)
    else
        tmp = (x * t) * -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 (y <= -2.55e+51) {
		tmp = t * (x * -y);
	} else if (y <= 1.75e+148) {
		tmp = t * (x / t);
	} else {
		tmp = (x * t) * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.55e+51:
		tmp = t * (x * -y)
	elif y <= 1.75e+148:
		tmp = t * (x / t)
	else:
		tmp = (x * t) * -y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.55e+51)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 1.75e+148)
		tmp = Float64(t * Float64(x / t));
	else
		tmp = Float64(Float64(x * t) * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.55e+51)
		tmp = t * (x * -y);
	elseif (y <= 1.75e+148)
		tmp = t * (x / t);
	else
		tmp = (x * t) * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.55e+51], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e+148], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(N[(x * t), $MachinePrecision] * (-y)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.55000000000000005e51

   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 t around inf 63.9%

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

      1. mul-1-neg 63.9%

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

      2. distribute-lft-neg-out 63.9%

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

      3. *-commutative 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in y around 0 27.8%

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

      1. mul-1-neg 27.8%

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

      2. unsub-neg 27.8%

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

      3. associate-*r* 20.8%

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

   8. Simplified 20.8%

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

   9. Taylor expanded in t around inf 27.6%

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

       1. associate-*r* 27.6%

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

       2. neg-mul-1 27.6%

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

       3. *-commutative 27.6%

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

   11. Simplified 27.6%

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

   if -2.55000000000000005e51 < y < 1.7499999999999999e148

   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. Add Preprocessing

   3. Taylor expanded in t around inf 57.4%

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

      1. mul-1-neg 57.4%

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

      2. distribute-lft-neg-out 57.4%

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

      3. *-commutative 57.4%

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

   5. Simplified 57.4%

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

   6. Taylor expanded in y around 0 29.1%

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

      1. mul-1-neg 29.1%

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

      2. unsub-neg 29.1%

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

      3. associate-*r* 32.1%

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

   8. Simplified 32.1%

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

   9. Taylor expanded in t around inf 33.2%

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

       1. *-commutative 33.2%

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

   11. Simplified 33.2%

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

   12. Taylor expanded in t around 0 36.4%

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]

   if 1.7499999999999999e148 < y

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf 75.6%

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

      1. mul-1-neg 75.6%

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

      2. distribute-lft-neg-out 75.6%

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

      3. *-commutative 75.6%

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

   5. Simplified 75.6%

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

   6. Taylor expanded in y around 0 30.6%

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

      1. mul-1-neg 30.6%

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

      2. unsub-neg 30.6%

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

      3. associate-*r* 30.7%

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

   8. Simplified 30.7%

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

   9. Taylor expanded in y around inf 30.7%

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

   10. Taylor expanded in y around inf 37.6%

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

       1. associate-*r* 37.6%

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

       2. mul-1-neg 37.6%

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

   12. Simplified 37.6%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+51}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+148}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot t\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 31.7% accurate, 19.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \frac{1}{1 + y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x - x \cdot \left(y \cdot t\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 1.35e+42)
   (* x (/ 1.0 (+ 1.0 (* y t))))
   (* t (/ (- x (* x (* y t))) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 1.35e+42) {
		tmp = x * (1.0 / (1.0 + (y * t)));
	} else {
		tmp = t * ((x - (x * (y * t))) / 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 (x <= 1.35d+42) then
        tmp = x * (1.0d0 / (1.0d0 + (y * t)))
    else
        tmp = t * ((x - (x * (y * t))) / 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 (x <= 1.35e+42) {
		tmp = x * (1.0 / (1.0 + (y * t)));
	} else {
		tmp = t * ((x - (x * (y * t))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 1.35e+42:
		tmp = x * (1.0 / (1.0 + (y * t)))
	else:
		tmp = t * ((x - (x * (y * t))) / t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 1.35e+42)
		tmp = Float64(x * Float64(1.0 / Float64(1.0 + Float64(y * t))));
	else
		tmp = Float64(t * Float64(Float64(x - Float64(x * Float64(y * t))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 1.35e+42)
		tmp = x * (1.0 / (1.0 + (y * t)));
	else
		tmp = t * ((x - (x * (y * t))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 1.35e+42], N[(x * N[(1.0 / N[(1.0 + N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(x - N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e42

   1. Initial program 98.0%

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

   3. Taylor expanded in t around inf 62.4%

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

      1. mul-1-neg 62.4%

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

      2. distribute-lft-neg-out 62.4%

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

      3. *-commutative 62.4%

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

   5. Simplified 62.4%

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

      1. distribute-rgt-neg-out 62.4%

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

      2. exp-neg 62.4%

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

   7. Applied egg-rr 62.4%

      \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0 30.1%

      \[\leadsto x \cdot \frac{1}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. *-commutative 30.1%

         \[\leadsto x \cdot \frac{1}{1 + \color{blue}{y \cdot t}} \]

   10. Simplified 30.1%

       \[\leadsto x \cdot \frac{1}{\color{blue}{1 + y \cdot t}} \]

   if 1.35e42 < x

   1. Initial program 96.2%

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

   3. Taylor expanded in t around inf 54.1%

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

      1. mul-1-neg 54.1%

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

      2. distribute-lft-neg-out 54.1%

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

      3. *-commutative 54.1%

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

   5. Simplified 54.1%

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

   6. Taylor expanded in y around 0 40.3%

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

      1. mul-1-neg 40.3%

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

      2. unsub-neg 40.3%

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

      3. associate-*r* 40.1%

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

   8. Simplified 40.1%

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

   9. Taylor expanded in t around inf 40.2%

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

       1. *-commutative 40.2%

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

   11. Simplified 40.2%

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

   12. Taylor expanded in t around 0 40.2%

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

       1. associate-*r* 49.3%

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

       2. associate-*r* 49.3%

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

       3. *-commutative 49.3%

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

       4. mul-1-neg 49.3%

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

       5. distribute-rgt-neg-out 49.3%

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

       6. associate-*r* 42.2%

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

       7. *-commutative 42.2%

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

   14. Simplified 42.2%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \frac{1}{1 + y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x - x \cdot \left(y \cdot t\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.7% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-77}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot \left(\frac{1}{t} - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 1.5e-77) (* t (/ x t)) (* t (* x (- (/ 1.0 t) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 1.5e-77) {
		tmp = t * (x / t);
	} else {
		tmp = t * (x * ((1.0 / t) - 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 (x <= 1.5d-77) then
        tmp = t * (x / t)
    else
        tmp = t * (x * ((1.0d0 / t) - 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 (x <= 1.5e-77) {
		tmp = t * (x / t);
	} else {
		tmp = t * (x * ((1.0 / t) - y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 1.5e-77:
		tmp = t * (x / t)
	else:
		tmp = t * (x * ((1.0 / t) - y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 1.5e-77)
		tmp = Float64(t * Float64(x / t));
	else
		tmp = Float64(t * Float64(x * Float64(Float64(1.0 / t) - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 1.5e-77)
		tmp = t * (x / t);
	else
		tmp = t * (x * ((1.0 / t) - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 1.5e-77], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(t * N[(x * N[(N[(1.0 / t), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.50000000000000008e-77

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf 63.3%

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

      1. mul-1-neg 63.3%

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

      2. distribute-lft-neg-out 63.3%

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

      3. *-commutative 63.3%

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

   5. Simplified 63.3%

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

   6. Taylor expanded in y around 0 28.0%

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

      1. mul-1-neg 28.0%

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

      2. unsub-neg 28.0%

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

      3. associate-*r* 28.9%

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

   8. Simplified 28.9%

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

   9. Taylor expanded in t around inf 33.1%

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

       1. *-commutative 33.1%

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

   11. Simplified 33.1%

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

   12. Taylor expanded in t around 0 31.7%

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]

   if 1.50000000000000008e-77 < x

   1. Initial program 97.4%

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

   3. Taylor expanded in t around inf 54.9%

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

      1. mul-1-neg 54.9%

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

      2. distribute-lft-neg-out 54.9%

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

      3. *-commutative 54.9%

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

   5. Simplified 54.9%

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

   6. Taylor expanded in y around 0 31.1%

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

      1. mul-1-neg 31.1%

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

      2. unsub-neg 31.1%

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

      3. associate-*r* 31.0%

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

   8. Simplified 31.0%

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

   9. Taylor expanded in t around inf 31.0%

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

       1. *-commutative 31.0%

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

   11. Simplified 31.0%

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

   12. Taylor expanded in x around 0 32.3%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-77}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot \left(\frac{1}{t} - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.7% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{1}{1 + y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot \left(\frac{1}{t} - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 2.1e+43) (* x (/ 1.0 (+ 1.0 (* y t)))) (* t (* x (- (/ 1.0 t) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 2.1e+43) {
		tmp = x * (1.0 / (1.0 + (y * t)));
	} else {
		tmp = t * (x * ((1.0 / t) - 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 (x <= 2.1d+43) then
        tmp = x * (1.0d0 / (1.0d0 + (y * t)))
    else
        tmp = t * (x * ((1.0d0 / t) - 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 (x <= 2.1e+43) {
		tmp = x * (1.0 / (1.0 + (y * t)));
	} else {
		tmp = t * (x * ((1.0 / t) - y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 2.1e+43:
		tmp = x * (1.0 / (1.0 + (y * t)))
	else:
		tmp = t * (x * ((1.0 / t) - y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 2.1e+43)
		tmp = Float64(x * Float64(1.0 / Float64(1.0 + Float64(y * t))));
	else
		tmp = Float64(t * Float64(x * Float64(Float64(1.0 / t) - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 2.1e+43)
		tmp = x * (1.0 / (1.0 + (y * t)));
	else
		tmp = t * (x * ((1.0 / t) - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 2.1e+43], N[(x * N[(1.0 / N[(1.0 + N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(x * N[(N[(1.0 / t), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000002e43

   1. Initial program 98.0%

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

   3. Taylor expanded in t around inf 62.4%

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

      1. mul-1-neg 62.4%

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

      2. distribute-lft-neg-out 62.4%

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

      3. *-commutative 62.4%

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

   5. Simplified 62.4%

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

      1. distribute-rgt-neg-out 62.4%

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

      2. exp-neg 62.4%

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

   7. Applied egg-rr 62.4%

      \[\leadsto x \cdot \color{blue}{\frac{1}{e^{y \cdot t}}} \]

   8. Taylor expanded in y around 0 30.1%

      \[\leadsto x \cdot \frac{1}{\color{blue}{1 + t \cdot y}} \]
   9. Step-by-step derivation

      1. *-commutative 30.1%

         \[\leadsto x \cdot \frac{1}{1 + \color{blue}{y \cdot t}} \]

   10. Simplified 30.1%

       \[\leadsto x \cdot \frac{1}{\color{blue}{1 + y \cdot t}} \]

   if 2.10000000000000002e43 < x

   1. Initial program 96.2%

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

   3. Taylor expanded in t around inf 54.1%

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

      1. mul-1-neg 54.1%

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

      2. distribute-lft-neg-out 54.1%

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

      3. *-commutative 54.1%

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

   5. Simplified 54.1%

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

   6. Taylor expanded in y around 0 40.3%

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

      1. mul-1-neg 40.3%

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

      2. unsub-neg 40.3%

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

      3. associate-*r* 40.1%

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

   8. Simplified 40.1%

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

   9. Taylor expanded in t around inf 40.2%

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

       1. *-commutative 40.2%

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

   11. Simplified 40.2%

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

   12. Taylor expanded in x around 0 42.2%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{1}{1 + y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot \left(\frac{1}{t} - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 28.0% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{-77}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(x \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 2.85e-77) (* t (/ x t)) (- x (* y (* x t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 2.85e-77) {
		tmp = t * (x / t);
	} else {
		tmp = x - (y * (x * 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 (x <= 2.85d-77) then
        tmp = t * (x / t)
    else
        tmp = x - (y * (x * 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 (x <= 2.85e-77) {
		tmp = t * (x / t);
	} else {
		tmp = x - (y * (x * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 2.85e-77:
		tmp = t * (x / t)
	else:
		tmp = x - (y * (x * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 2.85e-77)
		tmp = Float64(t * Float64(x / t));
	else
		tmp = Float64(x - Float64(y * Float64(x * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 2.85e-77)
		tmp = t * (x / t);
	else
		tmp = x - (y * (x * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 2.85e-77], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.84999999999999991e-77

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf 63.3%

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

      1. mul-1-neg 63.3%

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

      2. distribute-lft-neg-out 63.3%

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

      3. *-commutative 63.3%

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

   5. Simplified 63.3%

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

   6. Taylor expanded in y around 0 28.0%

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

      1. mul-1-neg 28.0%

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

      2. unsub-neg 28.0%

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

      3. associate-*r* 28.9%

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

   8. Simplified 28.9%

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

   9. Taylor expanded in t around inf 33.1%

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

       1. *-commutative 33.1%

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

   11. Simplified 33.1%

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

   12. Taylor expanded in t around 0 31.7%

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]

   if 2.84999999999999991e-77 < x

   1. Initial program 97.4%

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

   3. Taylor expanded in t around inf 54.9%

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

      1. mul-1-neg 54.9%

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

      2. distribute-lft-neg-out 54.9%

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

      3. *-commutative 54.9%

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

   5. Simplified 54.9%

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

   6. Taylor expanded in y around 0 31.1%

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

      1. mul-1-neg 31.1%

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

      2. unsub-neg 31.1%

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

      3. associate-*r* 31.0%

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

   8. Simplified 31.0%

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

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{-77}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(x \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 25.4% accurate, 28.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{+78}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot t\right) \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 6.2e+78) (* t (/ x t)) (* (* x t) (- y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 6.2e+78) {
		tmp = t * (x / t);
	} else {
		tmp = (x * t) * -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 <= 6.2d+78) then
        tmp = t * (x / t)
    else
        tmp = (x * t) * -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 <= 6.2e+78) {
		tmp = t * (x / t);
	} else {
		tmp = (x * t) * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 6.2e+78:
		tmp = t * (x / t)
	else:
		tmp = (x * t) * -y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 6.2e+78)
		tmp = Float64(t * Float64(x / t));
	else
		tmp = Float64(Float64(x * t) * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 6.2e+78)
		tmp = t * (x / t);
	else
		tmp = (x * t) * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 6.2e+78], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(N[(x * t), $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.2e78

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf 55.5%

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

      1. mul-1-neg 55.5%

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

      2. distribute-lft-neg-out 55.5%

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

      3. *-commutative 55.5%

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

   5. Simplified 55.5%

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

   6. Taylor expanded in y around 0 28.2%

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

      1. mul-1-neg 28.2%

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

      2. unsub-neg 28.2%

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

      3. associate-*r* 26.7%

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

   8. Simplified 26.7%

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

   9. Taylor expanded in t around inf 30.8%

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

       1. *-commutative 30.8%

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

   11. Simplified 30.8%

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

   12. Taylor expanded in t around 0 31.0%

       \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]

   if 6.2e78 < t

   1. Initial program 96.2%

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

   3. Taylor expanded in t around inf 81.5%

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

      1. mul-1-neg 81.5%

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

      2. distribute-lft-neg-out 81.5%

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

      3. *-commutative 81.5%

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

   5. Simplified 81.5%

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

   6. Taylor expanded in y around 0 31.9%

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

      1. mul-1-neg 31.9%

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

      2. unsub-neg 31.9%

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

      3. associate-*r* 40.6%

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

   8. Simplified 40.6%

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

   9. Taylor expanded in y around inf 40.6%

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

   10. Taylor expanded in y around inf 37.2%

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

       1. associate-*r* 37.2%

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

       2. mul-1-neg 37.2%

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

   12. Simplified 37.2%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{+78}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot t\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 14.7% accurate, 31.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-181}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 1.65e-181) (* t (* x y)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 1.65e-181) {
		tmp = t * (x * y);
	} 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 (x <= 1.65d-181) then
        tmp = t * (x * y)
    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 (x <= 1.65e-181) {
		tmp = t * (x * y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 1.65e-181:
		tmp = t * (x * y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 1.65e-181)
		tmp = Float64(t * Float64(x * y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 1.65e-181)
		tmp = t * (x * y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 1.65e-181], N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < 1.65000000000000004e-181

   1. Initial program 97.4%

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

   3. Taylor expanded in t around inf 61.0%

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

      1. mul-1-neg 61.0%

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

      2. distribute-lft-neg-out 61.0%

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

      3. *-commutative 61.0%

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

   5. Simplified 61.0%

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

   6. Taylor expanded in y around 0 27.2%

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

      1. mul-1-neg 27.2%

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

      2. unsub-neg 27.2%

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

      3. associate-*r* 28.2%

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

   8. Simplified 28.2%

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

   9. Taylor expanded in t around inf 23.1%

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

       1. mul-1-neg 23.1%

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

       2. *-commutative 23.1%

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

       3. distribute-rgt-neg-in 23.1%

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

       4. associate-*r* 21.2%

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

   11. Simplified 21.2%

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

       1. pow1 21.2%

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

       2. *-commutative 21.2%

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

       3. *-commutative 21.2%

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

       4. associate-*l* 23.1%

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

       5. add-sqr-sqrt 9.2%

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

       6. sqrt-unprod 16.8%

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

       7. sqr-neg 16.8%

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

       8. sqrt-unprod 6.8%

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

       9. add-sqr-sqrt 14.0%

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

   13. Applied egg-rr 14.0%

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

       1. unpow1 14.0%

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

   15. Simplified 14.0%

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

   if 1.65000000000000004e-181 < x

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf 60.4%

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

      1. mul-1-neg 60.4%

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

      2. distribute-lft-neg-out 60.4%

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

      3. *-commutative 60.4%

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

   5. Simplified 60.4%

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

   6. Taylor expanded in y around 0 21.8%

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

4. Final simplification 16.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-181}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 25.8% accurate, 63.0× speedup

Math

\[\begin{array}{l} \\ t \cdot \frac{x}{t} \end{array} \]

FPCore

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

C

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

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 = t * (x / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(t * N[(x / t), $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. Add Preprocessing

3. Taylor expanded in t around inf 60.8%

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

   1. mul-1-neg 60.8%

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

   2. distribute-lft-neg-out 60.8%

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

   3. *-commutative 60.8%

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

5. Simplified 60.8%

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

6. Taylor expanded in y around 0 29.0%

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

   1. mul-1-neg 29.0%

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

   2. unsub-neg 29.0%

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

   3. associate-*r* 29.5%

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

8. Simplified 29.5%

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

9. Taylor expanded in t around inf 32.5%

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

    1. *-commutative 32.5%

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

11. Simplified 32.5%

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

12. Taylor expanded in t around 0 29.6%

    \[\leadsto t \cdot \color{blue}{\frac{x}{t}} \]

13. Final simplification 29.6%

    \[\leadsto t \cdot \frac{x}{t} \]
14. Add Preprocessing

Alternative 20: 19.5% 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. Add Preprocessing

3. Taylor expanded in t around inf 60.8%

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

   1. mul-1-neg 60.8%

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

   2. distribute-lft-neg-out 60.8%

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

   3. *-commutative 60.8%

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

5. Simplified 60.8%

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

6. Taylor expanded in y around 0 19.1%

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

7. Final simplification 19.1%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(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))))))