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

Percentage Accurate: 96.8% → 99.5%

Time: 21.4s

Alternatives: 13

Speedup: 1.5×

• 20.4% 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: 96.8% 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 95.7%

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

   1. fma-def 95.7%

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

   2. sub-neg 95.7%

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

   3. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 85.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-17} \lor \neg \left(y \leq 3.6 \cdot 10^{-10}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3e-17) (not (<= y 3.6e-10)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3e-17) || !(y <= 3.6e-10)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((a * (log((1.0 - z)) - b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-3d-17)) .or. (.not. (y <= 3.6d-10))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp((a * (log((1.0d0 - z)) - b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3e-17) || !(y <= 3.6e-10)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((a * (Math.log((1.0 - z)) - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -3e-17) or not (y <= 3.6e-10):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((a * (math.log((1.0 - z)) - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3e-17) || !(y <= 3.6e-10))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(a * Float64(log(Float64(1.0 - z)) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -3e-17) || ~((y <= 3.6e-10)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp((a * (log((1.0 - z)) - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3e-17], N[Not[LessEqual[y, 3.6e-10]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.00000000000000006e-17 or 3.6e-10 < y

   1. Initial program 98.5%

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

   2. Taylor expanded in y around inf 92.8%

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

   if -3.00000000000000006e-17 < y < 3.6e-10

   1. Initial program 92.6%

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

   2. Taylor expanded in y around 0 83.8%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-17} \lor \neg \left(y \leq 3.6 \cdot 10^{-10}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

2. Taylor expanded in z around 0 99.2%

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

   1. mul-1-neg 99.2%

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

4. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 4: 85.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-16} \lor \neg \left(y \leq 1.5 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.35e-16) (not (<= y 1.5e-9)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.35e-16) || !(y <= 1.5e-9)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.35d-16)) .or. (.not. (y <= 1.5d-9))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.35e-16) || !(y <= 1.5e-9)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.35e-16) or not (y <= 1.5e-9):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.35e-16) || !(y <= 1.5e-9))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.35e-16) || ~((y <= 1.5e-9)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.35e-16], N[Not[LessEqual[y, 1.5e-9]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e-16 or 1.49999999999999999e-9 < y

   1. Initial program 98.5%

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

   2. Taylor expanded in y around inf 92.8%

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

   if -1.35e-16 < y < 1.49999999999999999e-9

   1. Initial program 92.6%

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

   2. Taylor expanded in b around inf 83.7%

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

      1. mul-1-neg 83.7%

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

      2. distribute-rgt-neg-out 83.7%

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

   4. Simplified 83.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-16} \lor \neg \left(y \leq 1.5 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]

Alternative 5: 73.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{t \cdot \left(-y\right)}\\ t_2 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{+168}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -8 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{+54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.1:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* t (- y))))) (t_2 (* x (pow z y))))
   (if (<= y -1.32e+168)
     t_2
     (if (<= y -8e+123)
       t_1
       (if (<= y -1.45e+54)
         t_2
         (if (<= y -4.2e-17)
           t_1
           (if (<= y 3.1) (* x (exp (* a (- b)))) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((t * -y));
	double t_2 = x * pow(z, y);
	double tmp;
	if (y <= -1.32e+168) {
		tmp = t_2;
	} else if (y <= -8e+123) {
		tmp = t_1;
	} else if (y <= -1.45e+54) {
		tmp = t_2;
	} else if (y <= -4.2e-17) {
		tmp = t_1;
	} else if (y <= 3.1) {
		tmp = x * exp((a * -b));
	} 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((t * -y))
    t_2 = x * (z ** y)
    if (y <= (-1.32d+168)) then
        tmp = t_2
    else if (y <= (-8d+123)) then
        tmp = t_1
    else if (y <= (-1.45d+54)) then
        tmp = t_2
    else if (y <= (-4.2d-17)) then
        tmp = t_1
    else if (y <= 3.1d0) then
        tmp = x * exp((a * -b))
    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((t * -y));
	double t_2 = x * Math.pow(z, y);
	double tmp;
	if (y <= -1.32e+168) {
		tmp = t_2;
	} else if (y <= -8e+123) {
		tmp = t_1;
	} else if (y <= -1.45e+54) {
		tmp = t_2;
	} else if (y <= -4.2e-17) {
		tmp = t_1;
	} else if (y <= 3.1) {
		tmp = x * Math.exp((a * -b));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((t * -y))
	t_2 = x * math.pow(z, y)
	tmp = 0
	if y <= -1.32e+168:
		tmp = t_2
	elif y <= -8e+123:
		tmp = t_1
	elif y <= -1.45e+54:
		tmp = t_2
	elif y <= -4.2e-17:
		tmp = t_1
	elif y <= 3.1:
		tmp = x * math.exp((a * -b))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(t * Float64(-y))))
	t_2 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -1.32e+168)
		tmp = t_2;
	elseif (y <= -8e+123)
		tmp = t_1;
	elseif (y <= -1.45e+54)
		tmp = t_2;
	elseif (y <= -4.2e-17)
		tmp = t_1;
	elseif (y <= 3.1)
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((t * -y));
	t_2 = x * (z ^ y);
	tmp = 0.0;
	if (y <= -1.32e+168)
		tmp = t_2;
	elseif (y <= -8e+123)
		tmp = t_1;
	elseif (y <= -1.45e+54)
		tmp = t_2;
	elseif (y <= -4.2e-17)
		tmp = t_1;
	elseif (y <= 3.1)
		tmp = x * exp((a * -b));
	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[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.32e+168], t$95$2, If[LessEqual[y, -8e+123], t$95$1, If[LessEqual[y, -1.45e+54], t$95$2, If[LessEqual[y, -4.2e-17], t$95$1, If[LessEqual[y, 3.1], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.32000000000000007e168 or -7.99999999999999982e123 < y < -1.4499999999999999e54 or 3.10000000000000009 < y

   1. Initial program 98.1%

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

   2. Taylor expanded in y around inf 93.5%

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

   3. Taylor expanded in t around 0 75.9%

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

   if -1.32000000000000007e168 < y < -7.99999999999999982e123 or -1.4499999999999999e54 < y < -4.19999999999999984e-17

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 84.7%

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

      1. mul-1-neg 84.7%

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

      2. distribute-lft-neg-out 84.7%

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

      3. *-commutative 84.7%

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

   4. Simplified 84.7%

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

   if -4.19999999999999984e-17 < y < 3.10000000000000009

   1. Initial program 92.9%

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

   2. Taylor expanded in b around inf 82.7%

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

      1. mul-1-neg 82.7%

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

      2. distribute-rgt-neg-out 82.7%

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

   4. Simplified 82.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+168}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq -8 \cdot 10^{+123}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{+54}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-17}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq 3.1:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]

Alternative 6: 73.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -320000000000 \lor \neg \left(y \leq 1.78\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -320000000000.0) (not (<= y 1.78)))
   (* x (pow z y))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -320000000000.0) || !(y <= 1.78)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-320000000000.0d0)) .or. (.not. (y <= 1.78d0))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -320000000000.0) || !(y <= 1.78)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -320000000000.0) or not (y <= 1.78):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -320000000000.0) || !(y <= 1.78))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -320000000000.0) || ~((y <= 1.78)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -320000000000.0], N[Not[LessEqual[y, 1.78]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2e11 or 1.78000000000000003 < y

   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. Taylor expanded in y around inf 94.5%

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

   3. Taylor expanded in t around 0 72.4%

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

   if -3.2e11 < y < 1.78000000000000003

   1. Initial program 93.2%

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

   2. Taylor expanded in b around inf 80.5%

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

      1. mul-1-neg 80.5%

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

      2. distribute-rgt-neg-out 80.5%

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

   4. Simplified 80.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -320000000000 \lor \neg \left(y \leq 1.78\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]

Alternative 7: 52.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.8e+131) (* a (* x (- b))) (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.8e+131) {
		tmp = a * (x * -b);
	} 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 <= (-2.8d+131)) then
        tmp = a * (x * -b)
    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 <= -2.8e+131) {
		tmp = a * (x * -b);
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -2.8e+131:
		tmp = a * (x * -b)
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.8e+131)
		tmp = Float64(a * Float64(x * Float64(-b)));
	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 <= -2.8e+131)
		tmp = a * (x * -b);
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.8e+131], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8000000000000001e131

   1. Initial program 95.0%

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

   2. Taylor expanded in b around inf 37.1%

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

      1. mul-1-neg 37.1%

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

      2. distribute-rgt-neg-out 37.1%

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

   4. Simplified 37.1%

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

   5. Taylor expanded in a around 0 8.1%

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

      1. mul-1-neg 8.1%

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

      2. unsub-neg 8.1%

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

   7. Simplified 8.1%

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

   8. Taylor expanded in a around inf 25.0%

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

      1. associate-*r* 24.9%

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

      2. *-commutative 24.9%

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

      3. mul-1-neg 24.9%

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

      4. distribute-lft-neg-in 24.9%

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

   10. Simplified 24.9%

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

       1. distribute-lft-neg-out 24.9%

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

       2. *-commutative 24.9%

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

       3. associate-*l* 25.0%

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

   12. Applied egg-rr 25.0%

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

   if -2.8000000000000001e131 < t

   1. Initial program 95.9%

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

   2. Taylor expanded in y around inf 71.5%

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

   3. Taylor expanded in t around 0 61.7%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]

Alternative 8: 27.1% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+201}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-111} \lor \neg \left(y \leq 1.1 \cdot 10^{-50}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.7e+201)
   (* x (* t (- y)))
   (if (or (<= y -4.2e-111) (not (<= y 1.1e-50))) (* a (* x (- b))) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.7e+201) {
		tmp = x * (t * -y);
	} else if ((y <= -4.2e-111) || !(y <= 1.1e-50)) {
		tmp = a * (x * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-3.7d+201)) then
        tmp = x * (t * -y)
    else if ((y <= (-4.2d-111)) .or. (.not. (y <= 1.1d-50))) then
        tmp = a * (x * -b)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.7e+201) {
		tmp = x * (t * -y);
	} else if ((y <= -4.2e-111) || !(y <= 1.1e-50)) {
		tmp = a * (x * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.7e+201:
		tmp = x * (t * -y)
	elif (y <= -4.2e-111) or not (y <= 1.1e-50):
		tmp = a * (x * -b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.7e+201)
		tmp = Float64(x * Float64(t * Float64(-y)));
	elseif ((y <= -4.2e-111) || !(y <= 1.1e-50))
		tmp = Float64(a * Float64(x * Float64(-b)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3.7e+201)
		tmp = x * (t * -y);
	elseif ((y <= -4.2e-111) || ~((y <= 1.1e-50)))
		tmp = a * (x * -b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.7e+201], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -4.2e-111], N[Not[LessEqual[y, 1.1e-50]], $MachinePrecision]], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if y < -3.6999999999999999e201

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 76.7%

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

      1. mul-1-neg 76.7%

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

      2. distribute-lft-neg-out 76.7%

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

      3. *-commutative 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in y around 0 34.8%

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

      1. *-commutative 34.8%

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

      2. mul-1-neg 34.8%

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

      3. unsub-neg 34.8%

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

   7. Simplified 34.8%

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

   8. Taylor expanded in y around inf 25.5%

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

      1. mul-1-neg 25.5%

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

      2. *-commutative 25.5%

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

      3. associate-*l* 34.6%

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

      4. distribute-rgt-neg-out 34.6%

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

      5. distribute-lft-neg-in 34.6%

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

      6. *-commutative 34.6%

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

   10. Simplified 34.6%

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

   if -3.6999999999999999e201 < y < -4.1999999999999997e-111 or 1.0999999999999999e-50 < y

   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. Taylor expanded in b around inf 37.4%

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

      1. mul-1-neg 37.4%

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

      2. distribute-rgt-neg-out 37.4%

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

   4. Simplified 37.4%

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

   5. Taylor expanded in a around 0 11.4%

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

      1. mul-1-neg 11.4%

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

      2. unsub-neg 11.4%

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

   7. Simplified 11.4%

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

   8. Taylor expanded in a around inf 23.8%

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

      1. associate-*r* 25.0%

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

      2. *-commutative 25.0%

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

      3. mul-1-neg 25.0%

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

      4. distribute-lft-neg-in 25.0%

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

   10. Simplified 25.0%

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

       1. distribute-lft-neg-out 25.0%

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

       2. *-commutative 25.0%

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

       3. associate-*l* 23.8%

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

   12. Applied egg-rr 23.8%

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

   if -4.1999999999999997e-111 < y < 1.0999999999999999e-50

   1. Initial program 94.0%

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

   2. Taylor expanded in b around inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. distribute-rgt-neg-out 87.1%

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

   4. Simplified 87.1%

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

   5. Taylor expanded in a around 0 40.8%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+201}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-111} \lor \neg \left(y \leq 1.1 \cdot 10^{-50}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 30.0% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-50}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -9.5e-107)
   (* x (* a (- b)))
   (if (<= y 1.15e-50) (* x (- 1.0 (* a b))) (* a (* x (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9.5e-107) {
		tmp = x * (a * -b);
	} else if (y <= 1.15e-50) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-9.5d-107)) then
        tmp = x * (a * -b)
    else if (y <= 1.15d-50) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = a * (x * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9.5e-107) {
		tmp = x * (a * -b);
	} else if (y <= 1.15e-50) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -9.5e-107:
		tmp = x * (a * -b)
	elif y <= 1.15e-50:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = a * (x * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -9.5e-107)
		tmp = Float64(x * Float64(a * Float64(-b)));
	elseif (y <= 1.15e-50)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(a * Float64(x * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -9.5e-107)
		tmp = x * (a * -b);
	elseif (y <= 1.15e-50)
		tmp = x * (1.0 - (a * b));
	else
		tmp = a * (x * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -9.5e-107], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e-50], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.4999999999999999e-107

   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. Taylor expanded in b around inf 31.1%

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

      1. mul-1-neg 31.1%

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

      2. distribute-rgt-neg-out 31.1%

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

   4. Simplified 31.1%

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

   5. Taylor expanded in a around 0 13.5%

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

      1. mul-1-neg 13.5%

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

      2. unsub-neg 13.5%

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

   7. Simplified 13.5%

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

   8. Taylor expanded in a around inf 17.8%

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

      1. associate-*r* 26.1%

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

      2. *-commutative 26.1%

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

      3. mul-1-neg 26.1%

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

      4. distribute-lft-neg-in 26.1%

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

   10. Simplified 26.1%

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

   if -9.4999999999999999e-107 < y < 1.1500000000000001e-50

   1. Initial program 94.0%

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

   2. Taylor expanded in b around inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. distribute-rgt-neg-out 87.1%

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

   4. Simplified 87.1%

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

   5. Taylor expanded in a around 0 49.8%

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

      1. mul-1-neg 49.8%

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

      2. unsub-neg 49.8%

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

   7. Simplified 49.8%

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

   if 1.1500000000000001e-50 < y

   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. Taylor expanded in b around inf 38.4%

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

      1. mul-1-neg 38.4%

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

      2. distribute-rgt-neg-out 38.4%

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

   4. Simplified 38.4%

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

   5. Taylor expanded in a around 0 9.5%

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

      1. mul-1-neg 9.5%

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

      2. unsub-neg 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in a around inf 25.7%

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

      1. associate-*r* 21.7%

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

      2. *-commutative 21.7%

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

      3. mul-1-neg 21.7%

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

      4. distribute-lft-neg-in 21.7%

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

   10. Simplified 21.7%

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

       1. distribute-lft-neg-out 21.7%

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

       2. *-commutative 21.7%

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

       3. associate-*l* 25.7%

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

   12. Applied egg-rr 25.7%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-50}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \]

Alternative 10: 30.1% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-106}:\\ \;\;\;\;\left(x \cdot \left(a \cdot b\right)\right) \cdot -3\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-50}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.3e-106)
   (* (* x (* a b)) -3.0)
   (if (<= y 1.15e-50) (* x (- 1.0 (* a b))) (* a (* x (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.3e-106) {
		tmp = (x * (a * b)) * -3.0;
	} else if (y <= 1.15e-50) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-1.3d-106)) then
        tmp = (x * (a * b)) * (-3.0d0)
    else if (y <= 1.15d-50) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = a * (x * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.3e-106) {
		tmp = (x * (a * b)) * -3.0;
	} else if (y <= 1.15e-50) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.3e-106:
		tmp = (x * (a * b)) * -3.0
	elif y <= 1.15e-50:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = a * (x * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.3e-106)
		tmp = Float64(Float64(x * Float64(a * b)) * -3.0);
	elseif (y <= 1.15e-50)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(a * Float64(x * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.3e-106)
		tmp = (x * (a * b)) * -3.0;
	elseif (y <= 1.15e-50)
		tmp = x * (1.0 - (a * b));
	else
		tmp = a * (x * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.3e-106], N[(N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 1.15e-50], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3e-106

   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. Taylor expanded in b around inf 31.1%

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

      1. mul-1-neg 31.1%

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

      2. distribute-rgt-neg-out 31.1%

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

   4. Simplified 31.1%

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

   5. Taylor expanded in a around 0 13.5%

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

      1. mul-1-neg 13.5%

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

      2. unsub-neg 13.5%

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

   7. Simplified 13.5%

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

   8. Taylor expanded in a around inf 17.8%

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

      1. associate-*r* 26.1%

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

      2. *-commutative 26.1%

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

      3. mul-1-neg 26.1%

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

      4. distribute-lft-neg-in 26.1%

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

   10. Simplified 26.1%

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

       1. *-rgt-identity 26.1%

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

       2. associate-*r* 26.1%

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

       3. distribute-lft-neg-out 26.1%

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

       4. distribute-lft-neg-in 26.1%

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

       5. distribute-rgt-neg-in 26.1%

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

       6. metadata-eval 26.1%

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

       7. metadata-eval 26.1%

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

       8. distribute-rgt-out-- 16.0%

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

       9. *-un-lft-identity 16.0%

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

       10. *-un-lft-identity 16.0%

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

       11. cancel-sign-sub-inv 16.0%

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

   12. Applied egg-rr 15.1%

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

       1. *-rgt-identity 15.1%

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

       2. distribute-rgt-out 15.1%

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

       3. associate-*r* 15.1%

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

       4. associate-*r* 12.7%

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

       5. metadata-eval 12.7%

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

       6. associate-*l* 12.7%

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

       7. metadata-eval 12.7%

          \[\leadsto \left(a \cdot \left(b \cdot x\right)\right) \cdot 1 + \left(a \cdot \left(b \cdot x\right)\right) \cdot \color{blue}{-4} \]

       8. distribute-lft-out 17.8%

          \[\leadsto \color{blue}{\left(a \cdot \left(b \cdot x\right)\right) \cdot \left(1 + -4\right)} \]

       9. associate-*r* 26.1%

          \[\leadsto \color{blue}{\left(\left(a \cdot b\right) \cdot x\right)} \cdot \left(1 + -4\right) \]

       10. *-commutative 26.1%

           \[\leadsto \color{blue}{\left(x \cdot \left(a \cdot b\right)\right)} \cdot \left(1 + -4\right) \]

       11. metadata-eval 26.1%

           \[\leadsto \left(x \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{-3} \]

   14. Simplified 26.1%

       \[\leadsto \color{blue}{\left(x \cdot \left(a \cdot b\right)\right) \cdot -3} \]

   if -1.3e-106 < y < 1.1500000000000001e-50

   1. Initial program 94.0%

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

   2. Taylor expanded in b around inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. distribute-rgt-neg-out 87.1%

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

   4. Simplified 87.1%

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

   5. Taylor expanded in a around 0 49.8%

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

      1. mul-1-neg 49.8%

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

      2. unsub-neg 49.8%

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

   7. Simplified 49.8%

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

   if 1.1500000000000001e-50 < y

   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. Taylor expanded in b around inf 38.4%

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

      1. mul-1-neg 38.4%

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

      2. distribute-rgt-neg-out 38.4%

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

   4. Simplified 38.4%

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

   5. Taylor expanded in a around 0 9.5%

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

      1. mul-1-neg 9.5%

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

      2. unsub-neg 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in a around inf 25.7%

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

      1. associate-*r* 21.7%

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

      2. *-commutative 21.7%

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

      3. mul-1-neg 21.7%

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

      4. distribute-lft-neg-in 21.7%

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

   10. Simplified 21.7%

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

       1. distribute-lft-neg-out 21.7%

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

       2. *-commutative 21.7%

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

       3. associate-*l* 25.7%

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

   12. Applied egg-rr 25.7%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-106}:\\ \;\;\;\;\left(x \cdot \left(a \cdot b\right)\right) \cdot -3\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-50}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \]

Alternative 11: 26.0% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-109} \lor \neg \left(y \leq 1.05 \cdot 10^{-50}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3.7e-109) (not (<= y 1.05e-50))) (* a (* x (- b))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.7e-109) || !(y <= 1.05e-50)) {
		tmp = a * (x * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-3.7d-109)) .or. (.not. (y <= 1.05d-50))) then
        tmp = a * (x * -b)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.7e-109) || !(y <= 1.05e-50)) {
		tmp = a * (x * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -3.7e-109) or not (y <= 1.05e-50):
		tmp = a * (x * -b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3.7e-109) || !(y <= 1.05e-50))
		tmp = Float64(a * Float64(x * Float64(-b)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -3.7e-109) || ~((y <= 1.05e-50)))
		tmp = a * (x * -b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3.7e-109], N[Not[LessEqual[y, 1.05e-50]], $MachinePrecision]], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -3.69999999999999981e-109 or 1.05e-50 < y

   1. Initial program 96.8%

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

   2. Taylor expanded in b around inf 34.7%

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

      1. mul-1-neg 34.7%

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

      2. distribute-rgt-neg-out 34.7%

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

   4. Simplified 34.7%

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

   5. Taylor expanded in a around 0 11.5%

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

      1. mul-1-neg 11.5%

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

      2. unsub-neg 11.5%

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

   7. Simplified 11.5%

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

   8. Taylor expanded in a around inf 21.7%

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

      1. associate-*r* 24.0%

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

      2. *-commutative 24.0%

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

      3. mul-1-neg 24.0%

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

      4. distribute-lft-neg-in 24.0%

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

   10. Simplified 24.0%

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

       1. distribute-lft-neg-out 24.0%

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

       2. *-commutative 24.0%

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

       3. associate-*l* 21.7%

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

   12. Applied egg-rr 21.7%

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

   if -3.69999999999999981e-109 < y < 1.05e-50

   1. Initial program 94.0%

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

   2. Taylor expanded in b around inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. distribute-rgt-neg-out 87.1%

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

   4. Simplified 87.1%

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

   5. Taylor expanded in a around 0 40.8%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-109} \lor \neg \left(y \leq 1.05 \cdot 10^{-50}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 26.3% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.7e-112)
   (* x (* a (- b)))
   (if (<= y 1.3e-52) x (* a (* x (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.7e-112) {
		tmp = x * (a * -b);
	} else if (y <= 1.3e-52) {
		tmp = x;
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.7d-112)) then
        tmp = x * (a * -b)
    else if (y <= 1.3d-52) then
        tmp = x
    else
        tmp = a * (x * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.7e-112) {
		tmp = x * (a * -b);
	} else if (y <= 1.3e-52) {
		tmp = x;
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.7e-112:
		tmp = x * (a * -b)
	elif y <= 1.3e-52:
		tmp = x
	else:
		tmp = a * (x * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.7e-112)
		tmp = Float64(x * Float64(a * Float64(-b)));
	elseif (y <= 1.3e-52)
		tmp = x;
	else
		tmp = Float64(a * Float64(x * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.7e-112)
		tmp = x * (a * -b);
	elseif (y <= 1.3e-52)
		tmp = x;
	else
		tmp = a * (x * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.7e-112], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-52], x, N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7000000000000001e-112

   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. Taylor expanded in b around inf 31.1%

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

      1. mul-1-neg 31.1%

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

      2. distribute-rgt-neg-out 31.1%

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

   4. Simplified 31.1%

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

   5. Taylor expanded in a around 0 13.5%

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

      1. mul-1-neg 13.5%

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

      2. unsub-neg 13.5%

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

   7. Simplified 13.5%

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

   8. Taylor expanded in a around inf 17.8%

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

      1. associate-*r* 26.1%

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

      2. *-commutative 26.1%

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

      3. mul-1-neg 26.1%

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

      4. distribute-lft-neg-in 26.1%

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

   10. Simplified 26.1%

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

   if -2.7000000000000001e-112 < y < 1.2999999999999999e-52

   1. Initial program 94.0%

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

   2. Taylor expanded in b around inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. distribute-rgt-neg-out 87.1%

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

   4. Simplified 87.1%

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

   5. Taylor expanded in a around 0 40.8%

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

   if 1.2999999999999999e-52 < y

   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. Taylor expanded in b around inf 38.4%

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

      1. mul-1-neg 38.4%

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

      2. distribute-rgt-neg-out 38.4%

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

   4. Simplified 38.4%

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

   5. Taylor expanded in a around 0 9.5%

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

      1. mul-1-neg 9.5%

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

      2. unsub-neg 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in a around inf 25.7%

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

      1. associate-*r* 21.7%

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

      2. *-commutative 21.7%

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

      3. mul-1-neg 21.7%

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

      4. distribute-lft-neg-in 21.7%

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

   10. Simplified 21.7%

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

       1. distribute-lft-neg-out 21.7%

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

       2. *-commutative 21.7%

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

       3. associate-*l* 25.7%

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

   12. Applied egg-rr 25.7%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \]

Alternative 13: 19.2% 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 95.7%

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

2. Taylor expanded in b around inf 55.0%

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

   1. mul-1-neg 55.0%

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

   2. distribute-rgt-neg-out 55.0%

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

4. Simplified 55.0%

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

5. Taylor expanded in a around 0 18.5%

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

6. Final simplification 18.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023301 
(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))))))