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

Percentage Accurate: 96.6% → 99.1%

Time: 17.1s

Alternatives: 13

Speedup: 1.5×

• 40.8% 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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in z around 0 100.0%

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

   1. associate-*r* 100.0%

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

   2. associate-*r* 100.0%

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

   3. distribute-lft-out 100.0%

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

   4. mul-1-neg 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 86.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{-21}:\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-72}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -5.3e-21)
   (* x (pow (/ z (exp t)) y))
   (if (<= y 4e-72)
     (* x (exp (* a (- (- b) z))))
     (* x (exp (* y (- (log z) t)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5.3e-21) {
		tmp = x * pow((z / exp(t)), y);
	} else if (y <= 4e-72) {
		tmp = x * exp((a * (-b - z)));
	} 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 (y <= (-5.3d-21)) then
        tmp = x * ((z / exp(t)) ** y)
    else if (y <= 4d-72) then
        tmp = x * exp((a * (-b - z)))
    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 (y <= -5.3e-21) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else if (y <= 4e-72) {
		tmp = x * Math.exp((a * (-b - z)));
	} else {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -5.3e-21:
		tmp = x * math.pow((z / math.exp(t)), y)
	elif y <= 4e-72:
		tmp = x * math.exp((a * (-b - z)))
	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 (y <= -5.3e-21)
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	elseif (y <= 4e-72)
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-b) - z))));
	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 (y <= -5.3e-21)
		tmp = x * ((z / exp(t)) ^ y);
	elseif (y <= 4e-72)
		tmp = x * exp((a * (-b - z)));
	else
		tmp = x * exp((y * (log(z) - t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -5.3e-21], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e-72], N[(x * N[Exp[N[(a * N[((-b) - z), $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 3 regimes

2. if y < -5.2999999999999999e-21

   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 a around 0 87.7%

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

      1. *-commutative 87.7%

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

      2. exp-prod 87.7%

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

      3. exp-diff 87.7%

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

      4. rem-exp-log 87.7%

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

   5. Simplified 87.7%

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

   if -5.2999999999999999e-21 < y < 3.9999999999999999e-72

   1. Initial program 96.1%

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

   3. Taylor expanded in y around 0 80.8%

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

   4. Taylor expanded in z around 0 84.6%

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

      1. associate-*r* 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-lft-out 99.9%

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

      4. mul-1-neg 99.9%

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

   6. Simplified 84.6%

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

   if 3.9999999999999999e-72 < y

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0 91.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{-21}:\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-72}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-18} \lor \neg \left(y \leq 3.9 \cdot 10^{-72}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.5e-18) (not (<= y 3.9e-72)))
   (* x (pow (/ z (exp t)) y))
   (* x (exp (* a (- (- b) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.5e-18) || !(y <= 3.9e-72)) {
		tmp = x * pow((z / exp(t)), y);
	} else {
		tmp = x * exp((a * (-b - z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-6.5d-18)) .or. (.not. (y <= 3.9d-72))) then
        tmp = x * ((z / exp(t)) ** y)
    else
        tmp = x * exp((a * (-b - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.5e-18) || !(y <= 3.9e-72)) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else {
		tmp = x * Math.exp((a * (-b - z)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6.5e-18) or not (y <= 3.9e-72):
		tmp = x * math.pow((z / math.exp(t)), y)
	else:
		tmp = x * math.exp((a * (-b - z)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.5e-18) || !(y <= 3.9e-72))
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-b) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6.5e-18) || ~((y <= 3.9e-72)))
		tmp = x * ((z / exp(t)) ^ y);
	else
		tmp = x * exp((a * (-b - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.5e-18], N[Not[LessEqual[y, 3.9e-72]], $MachinePrecision]], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.50000000000000008e-18 or 3.9e-72 < y

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0 89.7%

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

      1. *-commutative 89.7%

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

      2. exp-prod 89.7%

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

      3. exp-diff 89.7%

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

      4. rem-exp-log 89.7%

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

   5. Simplified 89.7%

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

   if -6.50000000000000008e-18 < y < 3.9e-72

   1. Initial program 96.1%

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

   3. Taylor expanded in y around 0 80.8%

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

   4. Taylor expanded in z around 0 84.6%

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

      1. associate-*r* 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-lft-out 99.9%

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

      4. mul-1-neg 99.9%

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

   6. Simplified 84.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-18} \lor \neg \left(y \leq 3.9 \cdot 10^{-72}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ t_2 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -3.25 \cdot 10^{+15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-123}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- (- b) z))))) (t_2 (* x (exp (* y (- t))))))
   (if (<= t -3.25e+15)
     t_2
     (if (<= t -4e-50)
       t_1
       (if (<= t -1.25e-123) (* x (pow z y)) (if (<= t 3.8e+37) 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 - z)));
	double t_2 = x * exp((y * -t));
	double tmp;
	if (t <= -3.25e+15) {
		tmp = t_2;
	} else if (t <= -4e-50) {
		tmp = t_1;
	} else if (t <= -1.25e-123) {
		tmp = x * pow(z, y);
	} else if (t <= 3.8e+37) {
		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 - z)))
    t_2 = x * exp((y * -t))
    if (t <= (-3.25d+15)) then
        tmp = t_2
    else if (t <= (-4d-50)) then
        tmp = t_1
    else if (t <= (-1.25d-123)) then
        tmp = x * (z ** y)
    else if (t <= 3.8d+37) 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 - z)));
	double t_2 = x * Math.exp((y * -t));
	double tmp;
	if (t <= -3.25e+15) {
		tmp = t_2;
	} else if (t <= -4e-50) {
		tmp = t_1;
	} else if (t <= -1.25e-123) {
		tmp = x * Math.pow(z, y);
	} else if (t <= 3.8e+37) {
		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 - z)))
	t_2 = x * math.exp((y * -t))
	tmp = 0
	if t <= -3.25e+15:
		tmp = t_2
	elif t <= -4e-50:
		tmp = t_1
	elif t <= -1.25e-123:
		tmp = x * math.pow(z, y)
	elif t <= 3.8e+37:
		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(Float64(-b) - z))))
	t_2 = Float64(x * exp(Float64(y * Float64(-t))))
	tmp = 0.0
	if (t <= -3.25e+15)
		tmp = t_2;
	elseif (t <= -4e-50)
		tmp = t_1;
	elseif (t <= -1.25e-123)
		tmp = Float64(x * (z ^ y));
	elseif (t <= 3.8e+37)
		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 - z)));
	t_2 = x * exp((y * -t));
	tmp = 0.0;
	if (t <= -3.25e+15)
		tmp = t_2;
	elseif (t <= -4e-50)
		tmp = t_1;
	elseif (t <= -1.25e-123)
		tmp = x * (z ^ y);
	elseif (t <= 3.8e+37)
		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 * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.25e+15], t$95$2, If[LessEqual[t, -4e-50], t$95$1, If[LessEqual[t, -1.25e-123], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e+37], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.25e15 or 3.7999999999999999e37 < t

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0 86.8%

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

   4. Taylor expanded in t around inf 86.8%

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

      1. mul-1-neg 86.8%

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

      2. distribute-lft-neg-out 86.8%

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

      3. *-commutative 86.8%

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

   6. Simplified 86.8%

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

   if -3.25e15 < t < -4.00000000000000003e-50 or -1.25000000000000007e-123 < t < 3.7999999999999999e37

   1. Initial program 97.1%

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

   3. Taylor expanded in y around 0 70.9%

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

   4. Taylor expanded in z around 0 76.3%

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

      1. associate-*r* 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

   6. Simplified 76.3%

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

   if -4.00000000000000003e-50 < t < -1.25000000000000007e-123

   1. Initial program 94.2%

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

   3. Taylor expanded in a around 0 89.1%

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

      1. *-commutative 89.1%

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

      2. exp-prod 89.1%

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

      3. exp-diff 89.1%

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

      4. rem-exp-log 89.1%

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

   5. Simplified 89.1%

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

   6. Taylor expanded in t around 0 89.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.25 \cdot 10^{+15}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-50}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-123}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+37}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-38} \lor \neg \left(b \leq 1.25 \cdot 10^{+51}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -2.7e-38) (not (<= b 1.25e+51)))
   (* x (exp (* a (- b))))
   (* x (exp (* y (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -2.7e-38) || !(b <= 1.25e+51)) {
		tmp = x * exp((a * -b));
	} else {
		tmp = x * exp((y * -t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-2.7d-38)) .or. (.not. (b <= 1.25d+51))) then
        tmp = x * exp((a * -b))
    else
        tmp = x * exp((y * -t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -2.7e-38) || !(b <= 1.25e+51)) {
		tmp = x * Math.exp((a * -b));
	} else {
		tmp = x * Math.exp((y * -t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -2.7e-38) or not (b <= 1.25e+51):
		tmp = x * math.exp((a * -b))
	else:
		tmp = x * math.exp((y * -t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -2.7e-38) || !(b <= 1.25e+51))
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	else
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -2.7e-38) || ~((b <= 1.25e+51)))
		tmp = x * exp((a * -b));
	else
		tmp = x * exp((y * -t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -2.7e-38], N[Not[LessEqual[b, 1.25e+51]], $MachinePrecision]], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.70000000000000005e-38 or 1.25e51 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 77.2%

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

   4. Taylor expanded in z around 0 77.2%

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

      1. mul-1-neg 77.2%

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

      2. distribute-rgt-neg-out 77.2%

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

   6. Simplified 77.2%

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

   if -2.70000000000000005e-38 < b < 1.25e51

   1. Initial program 95.5%

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

   3. Taylor expanded in a around 0 89.9%

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

   4. Taylor expanded in t around inf 74.4%

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

      1. mul-1-neg 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. *-commutative 74.4%

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

   6. Simplified 74.4%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-38} \lor \neg \left(b \leq 1.25 \cdot 10^{+51}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+49} \lor \neg \left(y \leq 5.8 \cdot 10^{+64}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.05e+49) (not (<= y 5.8e+64)))
   (* x (pow z y))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.05e+49) || !(y <= 5.8e+64)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.05d+49)) .or. (.not. (y <= 5.8d+64))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.05e+49) || !(y <= 5.8e+64)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.05e+49) or not (y <= 5.8e+64):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.05e+49) || !(y <= 5.8e+64))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.05e+49) || ~((y <= 5.8e+64)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.05e+49], N[Not[LessEqual[y, 5.8e+64]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05000000000000005e49 or 5.79999999999999986e64 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0 92.8%

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

      1. *-commutative 92.8%

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

      2. exp-prod 92.8%

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

      3. exp-diff 92.8%

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

      4. rem-exp-log 92.8%

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

   5. Simplified 92.8%

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

   6. Taylor expanded in t around 0 70.2%

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

   if -1.05000000000000005e49 < y < 5.79999999999999986e64

   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 y around 0 76.6%

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

   4. Taylor expanded in z around 0 75.9%

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

      1. mul-1-neg 75.9%

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

      2. distribute-rgt-neg-out 75.9%

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

   6. Simplified 75.9%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+49} \lor \neg \left(y \leq 5.8 \cdot 10^{+64}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{+181}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;t \leq -850:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.15e+181)
   (* a (* x (- z)))
   (if (<= t -850.0) (* x (- 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 <= -1.15e+181) {
		tmp = a * (x * -z);
	} else if (t <= -850.0) {
		tmp = x * (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 <= (-1.15d+181)) then
        tmp = a * (x * -z)
    else if (t <= (-850.0d0)) then
        tmp = x * (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 <= -1.15e+181) {
		tmp = a * (x * -z);
	} else if (t <= -850.0) {
		tmp = x * (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 <= -1.15e+181:
		tmp = a * (x * -z)
	elif t <= -850.0:
		tmp = x * (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 <= -1.15e+181)
		tmp = Float64(a * Float64(x * Float64(-z)));
	elseif (t <= -850.0)
		tmp = Float64(x * 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 <= -1.15e+181)
		tmp = a * (x * -z);
	elseif (t <= -850.0)
		tmp = x * (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, -1.15e+181], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -850.0], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1499999999999999e181

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 51.6%

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

      1. *-commutative 51.6%

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

      2. exp-prod 51.6%

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

      3. exp-diff 51.6%

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

      4. rem-exp-log 51.6%

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

   5. Simplified 51.6%

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

   6. Taylor expanded in b around 0 6.6%

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

   7. Taylor expanded in z around 0 6.4%

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

      1. mul-1-neg 6.4%

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

      2. unsub-neg 6.4%

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

   9. Simplified 6.4%

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

   10. Taylor expanded in a around inf 34.6%

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

       1. mul-1-neg 34.6%

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

       2. distribute-rgt-neg-in 34.6%

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

       3. *-commutative 34.6%

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

       4. distribute-rgt-neg-in 34.6%

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

   12. Simplified 34.6%

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

   if -1.1499999999999999e181 < t < -850

   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 a around 0 85.6%

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

   4. Taylor expanded in t around inf 85.6%

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   5. 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%

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

   6. Simplified 85.6%

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

   7. Taylor expanded in y around 0 51.2%

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

      1. mul-1-neg 51.2%

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

      2. unsub-neg 51.2%

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

      3. *-commutative 51.2%

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

   9. Simplified 51.2%

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

   if -850 < t

   1. Initial program 97.1%

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

   3. Taylor expanded in a around 0 71.5%

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

      1. *-commutative 71.5%

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

      2. exp-prod 70.0%

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

      3. exp-diff 70.0%

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

      4. rem-exp-log 70.0%

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

   5. Simplified 70.0%

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

   6. Taylor expanded in t around 0 60.8%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{+181}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;t \leq -850:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 24.4% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-242} \lor \neg \left(x \leq 6.5 \cdot 10^{-202}\right) \land x \leq 3.6 \cdot 10^{-128}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= x 8.5e-242) (and (not (<= x 6.5e-202)) (<= x 3.6e-128)))
   (* a (* x (- z)))
   (* x (- 1.0 (* y t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x <= 8.5e-242) || (!(x <= 6.5e-202) && (x <= 3.6e-128))) {
		tmp = a * (x * -z);
	} 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 ((x <= 8.5d-242) .or. (.not. (x <= 6.5d-202)) .and. (x <= 3.6d-128)) then
        tmp = a * (x * -z)
    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 ((x <= 8.5e-242) || (!(x <= 6.5e-202) && (x <= 3.6e-128))) {
		tmp = a * (x * -z);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x <= 8.5e-242) or (not (x <= 6.5e-202) and (x <= 3.6e-128)):
		tmp = a * (x * -z)
	else:
		tmp = x * (1.0 - (y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((x <= 8.5e-242) || (!(x <= 6.5e-202) && (x <= 3.6e-128)))
		tmp = Float64(a * Float64(x * Float64(-z)));
	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 ((x <= 8.5e-242) || (~((x <= 6.5e-202)) && (x <= 3.6e-128)))
		tmp = a * (x * -z);
	else
		tmp = x * (1.0 - (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[x, 8.5e-242], And[N[Not[LessEqual[x, 6.5e-202]], $MachinePrecision], LessEqual[x, 3.6e-128]]], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.4999999999999997e-242 or 6.49999999999999956e-202 < x < 3.60000000000000025e-128

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0 63.4%

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

      1. *-commutative 63.4%

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

      2. exp-prod 62.0%

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

      3. exp-diff 62.0%

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

      4. rem-exp-log 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in b around 0 21.6%

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

   7. Taylor expanded in z around 0 21.0%

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

      1. mul-1-neg 21.0%

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

      2. unsub-neg 21.0%

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

   9. Simplified 21.0%

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

   10. Taylor expanded in a around inf 24.5%

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

       1. mul-1-neg 24.5%

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

       2. distribute-rgt-neg-in 24.5%

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

       3. *-commutative 24.5%

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

       4. distribute-rgt-neg-in 24.5%

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

   12. Simplified 24.5%

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

   if 8.4999999999999997e-242 < x < 6.49999999999999956e-202 or 3.60000000000000025e-128 < x

   1. Initial program 96.6%

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

   3. Taylor expanded in a around 0 74.9%

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

   4. Taylor expanded in t around inf 61.7%

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

      1. mul-1-neg 61.7%

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

      2. distribute-lft-neg-out 61.7%

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

      3. *-commutative 61.7%

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

   6. Simplified 61.7%

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

   7. Taylor expanded in y around 0 21.3%

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

      1. mul-1-neg 21.3%

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

      2. unsub-neg 21.3%

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

      3. *-commutative 21.3%

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

   9. Simplified 21.3%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-242} \lor \neg \left(x \leq 6.5 \cdot 10^{-202}\right) \land x \leq 3.6 \cdot 10^{-128}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 24.5% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{if}\;x \leq 8.5 \cdot 10^{-242}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-128}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (* x (- z)))))
   (if (<= x 8.5e-242)
     t_1
     (if (<= x 6.4e-202)
       (* x (- 1.0 (* y t)))
       (if (<= x 6.6e-128) t_1 (- x (* a (* x b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (x * -z);
	double tmp;
	if (x <= 8.5e-242) {
		tmp = t_1;
	} else if (x <= 6.4e-202) {
		tmp = x * (1.0 - (y * t));
	} else if (x <= 6.6e-128) {
		tmp = t_1;
	} else {
		tmp = x - (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) :: t_1
    real(8) :: tmp
    t_1 = a * (x * -z)
    if (x <= 8.5d-242) then
        tmp = t_1
    else if (x <= 6.4d-202) then
        tmp = x * (1.0d0 - (y * t))
    else if (x <= 6.6d-128) then
        tmp = t_1
    else
        tmp = x - (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 t_1 = a * (x * -z);
	double tmp;
	if (x <= 8.5e-242) {
		tmp = t_1;
	} else if (x <= 6.4e-202) {
		tmp = x * (1.0 - (y * t));
	} else if (x <= 6.6e-128) {
		tmp = t_1;
	} else {
		tmp = x - (a * (x * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (x * -z)
	tmp = 0
	if x <= 8.5e-242:
		tmp = t_1
	elif x <= 6.4e-202:
		tmp = x * (1.0 - (y * t))
	elif x <= 6.6e-128:
		tmp = t_1
	else:
		tmp = x - (a * (x * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(x * Float64(-z)))
	tmp = 0.0
	if (x <= 8.5e-242)
		tmp = t_1;
	elseif (x <= 6.4e-202)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	elseif (x <= 6.6e-128)
		tmp = t_1;
	else
		tmp = Float64(x - Float64(a * Float64(x * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (x * -z);
	tmp = 0.0;
	if (x <= 8.5e-242)
		tmp = t_1;
	elseif (x <= 6.4e-202)
		tmp = x * (1.0 - (y * t));
	elseif (x <= 6.6e-128)
		tmp = t_1;
	else
		tmp = x - (a * (x * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 8.5e-242], t$95$1, If[LessEqual[x, 6.4e-202], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e-128], t$95$1, N[(x - N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.4999999999999997e-242 or 6.4000000000000002e-202 < x < 6.6e-128

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0 63.4%

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

      1. *-commutative 63.4%

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

      2. exp-prod 62.0%

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

      3. exp-diff 62.0%

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

      4. rem-exp-log 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in b around 0 21.6%

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

   7. Taylor expanded in z around 0 21.0%

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

      1. mul-1-neg 21.0%

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

      2. unsub-neg 21.0%

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

   9. Simplified 21.0%

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

   10. Taylor expanded in a around inf 24.5%

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

       1. mul-1-neg 24.5%

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

       2. distribute-rgt-neg-in 24.5%

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

       3. *-commutative 24.5%

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

       4. distribute-rgt-neg-in 24.5%

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

   12. Simplified 24.5%

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

   if 8.4999999999999997e-242 < x < 6.4000000000000002e-202

   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 a around 0 100.0%

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

   4. Taylor expanded in t around inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. distribute-lft-neg-out 97.0%

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

      3. *-commutative 97.0%

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

   6. Simplified 97.0%

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

   7. Taylor expanded in y around 0 48.4%

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

      1. mul-1-neg 48.4%

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

      2. unsub-neg 48.4%

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

      3. *-commutative 48.4%

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

   9. Simplified 48.4%

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

   if 6.6e-128 < x

   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 y around 0 53.3%

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

   4. Taylor expanded in z around 0 53.3%

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

      1. mul-1-neg 53.3%

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

      2. distribute-rgt-neg-out 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in a around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. unsub-neg 21.8%

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

      3. *-commutative 21.8%

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

   9. Simplified 21.8%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-242}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-128}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.6% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-127} \lor \neg \left(y \leq 3.75 \cdot 10^{-34}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3.1e-127) (not (<= y 3.75e-34)))
   (* a (* x (- z)))
   (* x (- 1.0 (* z a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.1e-127) || !(y <= 3.75e-34)) {
		tmp = a * (x * -z);
	} else {
		tmp = x * (1.0 - (z * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-3.1d-127)) .or. (.not. (y <= 3.75d-34))) then
        tmp = a * (x * -z)
    else
        tmp = x * (1.0d0 - (z * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.1e-127) || !(y <= 3.75e-34)) {
		tmp = a * (x * -z);
	} else {
		tmp = x * (1.0 - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -3.1e-127) or not (y <= 3.75e-34):
		tmp = a * (x * -z)
	else:
		tmp = x * (1.0 - (z * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3.1e-127) || !(y <= 3.75e-34))
		tmp = Float64(a * Float64(x * Float64(-z)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -3.1e-127) || ~((y <= 3.75e-34)))
		tmp = a * (x * -z);
	else
		tmp = x * (1.0 - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3.1e-127], N[Not[LessEqual[y, 3.75e-34]], $MachinePrecision]], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1e-127 or 3.7500000000000002e-34 < y

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 46.8%

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

      1. *-commutative 46.8%

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

      2. exp-prod 45.4%

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

      3. exp-diff 45.4%

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

      4. rem-exp-log 45.4%

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

   5. Simplified 45.4%

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

   6. Taylor expanded in b around 0 7.3%

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

   7. Taylor expanded in z around 0 6.9%

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

      1. mul-1-neg 6.9%

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

      2. unsub-neg 6.9%

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

   9. Simplified 6.9%

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

   10. Taylor expanded in a around inf 23.0%

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

       1. mul-1-neg 23.0%

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

       2. distribute-rgt-neg-in 23.0%

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

       3. *-commutative 23.0%

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

       4. distribute-rgt-neg-in 23.0%

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

   12. Simplified 23.0%

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

   if -3.1e-127 < y < 3.7500000000000002e-34

   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 y around 0 82.0%

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

      1. *-commutative 82.0%

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

      2. exp-prod 71.9%

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

      3. exp-diff 71.9%

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

      4. rem-exp-log 71.9%

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

   5. Simplified 71.9%

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

   6. Taylor expanded in b around 0 43.4%

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

   7. Taylor expanded in z around 0 41.5%

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

      1. mul-1-neg 41.5%

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

      2. unsub-neg 41.5%

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

   9. Simplified 41.5%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-127} \lor \neg \left(y \leq 3.75 \cdot 10^{-34}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.2% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-101}:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot z\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -5e-101)
   (* a (- (/ x a) (* x z)))
   (if (<= y 5e-11) (- x (* a (* x b))) (* a (* x (- z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5e-101) {
		tmp = a * ((x / a) - (x * z));
	} else if (y <= 5e-11) {
		tmp = x - (a * (x * b));
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-5d-101)) then
        tmp = a * ((x / a) - (x * z))
    else if (y <= 5d-11) then
        tmp = x - (a * (x * b))
    else
        tmp = a * (x * -z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5e-101) {
		tmp = a * ((x / a) - (x * z));
	} else if (y <= 5e-11) {
		tmp = x - (a * (x * b));
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -5e-101:
		tmp = a * ((x / a) - (x * z))
	elif y <= 5e-11:
		tmp = x - (a * (x * b))
	else:
		tmp = a * (x * -z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -5e-101)
		tmp = Float64(a * Float64(Float64(x / a) - Float64(x * z)));
	elseif (y <= 5e-11)
		tmp = Float64(x - Float64(a * Float64(x * b)));
	else
		tmp = Float64(a * Float64(x * Float64(-z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -5e-101)
		tmp = a * ((x / a) - (x * z));
	elseif (y <= 5e-11)
		tmp = x - (a * (x * b));
	else
		tmp = a * (x * -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -5e-101], N[(a * N[(N[(x / a), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-11], N[(x - N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.0000000000000001e-101

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 52.1%

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

      1. *-commutative 52.1%

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

      2. exp-prod 48.4%

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

      3. exp-diff 48.4%

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

      4. rem-exp-log 48.4%

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

   5. Simplified 48.4%

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

   6. Taylor expanded in b around 0 9.7%

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

   7. Taylor expanded in z around 0 10.5%

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

      1. mul-1-neg 10.5%

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

      2. unsub-neg 10.5%

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

   9. Simplified 10.5%

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

   10. Taylor expanded in a around inf 25.6%

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

       1. +-commutative 25.6%

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

       2. mul-1-neg 25.6%

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

       3. sub-neg 25.6%

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

   12. Simplified 25.6%

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

   if -5.0000000000000001e-101 < y < 5.00000000000000018e-11

   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. Add Preprocessing

   3. Taylor expanded in y around 0 79.0%

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

   4. Taylor expanded in z around 0 78.1%

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

      1. mul-1-neg 78.1%

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

      2. distribute-rgt-neg-out 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in a around 0 46.8%

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

      1. mul-1-neg 46.8%

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

      2. unsub-neg 46.8%

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

      3. *-commutative 46.8%

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

   9. Simplified 46.8%

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

   if 5.00000000000000018e-11 < 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. Add Preprocessing

   3. Taylor expanded in y around 0 37.9%

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

      1. *-commutative 37.9%

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

      2. exp-prod 39.0%

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

      3. exp-diff 39.0%

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

      4. rem-exp-log 39.0%

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

   5. Simplified 39.0%

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

   6. Taylor expanded in b around 0 5.1%

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

   7. Taylor expanded in z around 0 3.4%

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

      1. mul-1-neg 3.4%

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

      2. unsub-neg 3.4%

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

   9. Simplified 3.4%

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

   10. Taylor expanded in a around inf 27.4%

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

       1. mul-1-neg 27.4%

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

       2. distribute-rgt-neg-in 27.4%

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

       3. *-commutative 27.4%

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

       4. distribute-rgt-neg-in 27.4%

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

   12. Simplified 27.4%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-101}:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot z\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.2% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-127} \lor \neg \left(y \leq 8 \cdot 10^{-38}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4.1e-127) (not (<= y 8e-38))) (* a (* x (- z))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.1e-127) || !(y <= 8e-38)) {
		tmp = a * (x * -z);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-4.1d-127)) .or. (.not. (y <= 8d-38))) then
        tmp = a * (x * -z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.1e-127) || !(y <= 8e-38)) {
		tmp = a * (x * -z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -4.1e-127) or not (y <= 8e-38):
		tmp = a * (x * -z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -4.1e-127) || !(y <= 8e-38))
		tmp = Float64(a * Float64(x * Float64(-z)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -4.1e-127) || ~((y <= 8e-38)))
		tmp = a * (x * -z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -4.1e-127], N[Not[LessEqual[y, 8e-38]], $MachinePrecision]], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -4.1e-127 or 7.9999999999999997e-38 < y

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 46.8%

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

      1. *-commutative 46.8%

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

      2. exp-prod 45.4%

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

      3. exp-diff 45.4%

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

      4. rem-exp-log 45.4%

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

   5. Simplified 45.4%

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

   6. Taylor expanded in b around 0 7.3%

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

   7. Taylor expanded in z around 0 6.9%

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

      1. mul-1-neg 6.9%

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

      2. unsub-neg 6.9%

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

   9. Simplified 6.9%

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

   10. Taylor expanded in a around inf 23.0%

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

       1. mul-1-neg 23.0%

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

       2. distribute-rgt-neg-in 23.0%

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

       3. *-commutative 23.0%

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

       4. distribute-rgt-neg-in 23.0%

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

   12. Simplified 23.0%

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

   if -4.1e-127 < y < 7.9999999999999997e-38

   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 y around 0 82.0%

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

      1. *-commutative 82.0%

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

      2. exp-prod 71.9%

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

      3. exp-diff 71.9%

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

      4. rem-exp-log 71.9%

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

   5. Simplified 71.9%

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

   6. Taylor expanded in a around 0 40.3%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-127} \lor \neg \left(y \leq 8 \cdot 10^{-38}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 19.6% accurate, 315.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in y around 0 59.7%

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

   1. *-commutative 59.7%

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

   2. exp-prod 55.1%

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

   3. exp-diff 55.1%

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

   4. rem-exp-log 55.1%

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

5. Simplified 55.1%

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

6. Taylor expanded in a around 0 19.1%

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

Reproduce

herbie shell --seed 2024087 
(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))))))