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

Percentage Accurate: 96.5% → 99.6%

Time: 19.0s

Alternatives: 15

Speedup: 0.8×

• 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: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% 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 98.0%

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

   1. fma-define 98.8%

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

   2. sub-neg 98.8%

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

   3. log1p-define 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 88.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -480 \lor \neg \left(t \leq 4.8 \cdot 10^{+130}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \log z - a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -480.0) (not (<= t 4.8e+130)))
   (* x (exp (* t (- y))))
   (* x (exp (- (* y (log z)) (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -480.0) || !(t <= 4.8e+130)) {
		tmp = x * exp((t * -y));
	} else {
		tmp = x * exp(((y * log(z)) - (a * b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-480.0d0)) .or. (.not. (t <= 4.8d+130))) then
        tmp = x * exp((t * -y))
    else
        tmp = x * exp(((y * log(z)) - (a * b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -480.0) || !(t <= 4.8e+130)) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = x * Math.exp(((y * Math.log(z)) - (a * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -480.0) or not (t <= 4.8e+130):
		tmp = x * math.exp((t * -y))
	else:
		tmp = x * math.exp(((y * math.log(z)) - (a * b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -480.0) || !(t <= 4.8e+130))
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	else
		tmp = Float64(x * exp(Float64(Float64(y * log(z)) - Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -480.0) || ~((t <= 4.8e+130)))
		tmp = x * exp((t * -y));
	else
		tmp = x * exp(((y * log(z)) - (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -480.0], N[Not[LessEqual[t, 4.8e+130]], $MachinePrecision]], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - N[(a * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -480 or 4.80000000000000048e130 < t

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf 83.5%

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

      1. mul-1-neg 83.5%

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

      2. distribute-lft-neg-out 83.5%

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

      3. *-commutative 83.5%

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

   5. Simplified 83.5%

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

   if -480 < t < 4.80000000000000048e130

   1. Initial program 97.8%

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

   3. Taylor expanded in z around 0 97.8%

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

   4. Taylor expanded in t around 0 97.2%

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

      1. +-commutative 97.2%

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

      2. mul-1-neg 97.2%

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

      3. unsub-neg 97.2%

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

   6. Simplified 97.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -480 \lor \neg \left(t \leq 4.8 \cdot 10^{+130}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \log z - a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+16}:\\ \;\;\;\;x \cdot {\left(e^{-a}\right)}^{b}\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{+112}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.55e+16)
   (* x (pow (exp (- a)) b))
   (if (<= a 1.85e+112)
     (* x (exp (* y (- (log z) t))))
     (* x (exp (* a (- (log1p (- z)) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.55e+16) {
		tmp = x * pow(exp(-a), b);
	} else if (a <= 1.85e+112) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((a * (log1p(-z) - b)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.55e+16) {
		tmp = x * Math.pow(Math.exp(-a), b);
	} else if (a <= 1.85e+112) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((a * (Math.log1p(-z) - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.55e+16:
		tmp = x * math.pow(math.exp(-a), b)
	elif a <= 1.85e+112:
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((a * (math.log1p(-z) - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.55e+16)
		tmp = Float64(x * (exp(Float64(-a)) ^ b));
	elseif (a <= 1.85e+112)
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(a * Float64(log1p(Float64(-z)) - b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.55e+16], N[(x * N[Power[N[Exp[(-a)], $MachinePrecision], b], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.85e+112], 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[1 + (-z)], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.55e16

   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 z around 0 96.3%

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

   4. Taylor expanded in y around 0 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*r* 61.6%

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

      3. neg-mul-1 61.6%

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

      4. exp-prod 71.0%

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

   6. Simplified 71.0%

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

   if -1.55e16 < a < 1.85000000000000002e112

   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 y around inf 89.1%

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

   if 1.85000000000000002e112 < a

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0 86.5%

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

      1. sub-neg 86.5%

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

      2. log1p-define 94.5%

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

   5. Simplified 94.5%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+16}:\\ \;\;\;\;x \cdot {\left(e^{-a}\right)}^{b}\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{+112}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot {\left(e^{-a}\right)}^{b}\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+108}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -3.2e+16)
   (* x (pow (exp (- a)) b))
   (if (<= a 1.65e+108) (* 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 (a <= -3.2e+16) {
		tmp = x * pow(exp(-a), b);
	} else if (a <= 1.65e+108) {
		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 (a <= (-3.2d+16)) then
        tmp = x * (exp(-a) ** b)
    else if (a <= 1.65d+108) 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 (a <= -3.2e+16) {
		tmp = x * Math.pow(Math.exp(-a), b);
	} else if (a <= 1.65e+108) {
		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 a <= -3.2e+16:
		tmp = x * math.pow(math.exp(-a), b)
	elif a <= 1.65e+108:
		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 (a <= -3.2e+16)
		tmp = Float64(x * (exp(Float64(-a)) ^ b));
	elseif (a <= 1.65e+108)
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x / exp(Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -3.2e+16)
		tmp = x * (exp(-a) ^ b);
	elseif (a <= 1.65e+108)
		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[LessEqual[a, -3.2e+16], N[(x * N[Power[N[Exp[(-a)], $MachinePrecision], b], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.65e+108], 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 3 regimes

2. if a < -3.2e16

   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 z around 0 96.3%

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

   4. Taylor expanded in y around 0 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*r* 61.6%

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

      3. neg-mul-1 61.6%

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

      4. exp-prod 71.0%

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

   6. Simplified 71.0%

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

   if -3.2e16 < a < 1.6500000000000001e108

   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 y around inf 89.1%

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

   if 1.6500000000000001e108 < a

   1. Initial program 94.5%

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

   3. Taylor expanded in z around 0 94.5%

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

   4. Taylor expanded in y around inf 94.5%

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

      1. sub-neg 94.5%

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

      2. mul-1-neg 94.5%

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

      3. associate-+r+ 94.5%

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

      4. mul-1-neg 94.5%

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

      5. sub-neg 94.5%

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

      6. *-commutative 94.5%

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

      7. distribute-lft-out-- 94.5%

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

      8. +-commutative 94.5%

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

      9. mul-1-neg 94.5%

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

      10. sub-neg 94.5%

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

      11. exp-diff 75.0%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in y around 0 86.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot {\left(e^{-a}\right)}^{b}\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+108}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-70} \lor \neg \left(y \leq 5.2 \cdot 10^{-153}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.5e-70) (not (<= y 5.2e-153)))
   (* x (pow (/ z (exp t)) y))
   (/ x (exp (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.5e-70) || !(y <= 5.2e-153)) {
		tmp = x * pow((z / exp(t)), 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 <= (-2.5d-70)) .or. (.not. (y <= 5.2d-153))) then
        tmp = x * ((z / exp(t)) ** 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 <= -2.5e-70) || !(y <= 5.2e-153)) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else {
		tmp = x / Math.exp((a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.5e-70) or not (y <= 5.2e-153):
		tmp = x * math.pow((z / math.exp(t)), y)
	else:
		tmp = x / math.exp((a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.5e-70) || !(y <= 5.2e-153))
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	else
		tmp = Float64(x / exp(Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.5e-70) || ~((y <= 5.2e-153)))
		tmp = x * ((z / exp(t)) ^ y);
	else
		tmp = x / exp((a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.5e-70], N[Not[LessEqual[y, 5.2e-153]], $MachinePrecision]], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x / N[Exp[N[(a * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4999999999999999e-70 or 5.2000000000000003e-153 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in z around 0 97.8%

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

   4. Taylor expanded in a around 0 82.7%

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

      1. *-commutative 82.7%

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

      2. *-commutative 82.7%

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

      3. exp-prod 82.7%

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

      4. exp-diff 82.7%

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

      5. rem-exp-log 82.7%

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

   6. Simplified 82.7%

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

   if -2.4999999999999999e-70 < y < 5.2000000000000003e-153

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0 98.6%

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

   4. Taylor expanded in y around inf 98.6%

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

      1. sub-neg 98.6%

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

      2. mul-1-neg 98.6%

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

      3. associate-+r+ 98.6%

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

      4. mul-1-neg 98.6%

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

      5. sub-neg 98.6%

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

      6. *-commutative 98.6%

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

      7. distribute-lft-out-- 98.6%

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

      8. +-commutative 98.6%

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

      9. mul-1-neg 98.6%

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

      10. sub-neg 98.6%

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

      11. exp-diff 95.7%

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

   6. Simplified 70.6%

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

   7. Taylor expanded in y around 0 84.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-70} \lor \neg \left(y \leq 5.2 \cdot 10^{-153}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in z around 0 98.0%

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

4. Final simplification 98.0%

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

Alternative 7: 71.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -420 \lor \neg \left(t \leq 2 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -420.0) (not (<= t 2e-19)))
   (* x (exp (* t (- y))))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -420.0) || !(t <= 2e-19)) {
		tmp = x * exp((t * -y));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-420.0d0)) .or. (.not. (t <= 2d-19))) then
        tmp = x * exp((t * -y))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -420.0) || !(t <= 2e-19)) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -420.0) or not (t <= 2e-19):
		tmp = x * math.exp((t * -y))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -420.0) || !(t <= 2e-19))
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -420.0) || ~((t <= 2e-19)))
		tmp = x * exp((t * -y));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -420.0], N[Not[LessEqual[t, 2e-19]], $MachinePrecision]], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -420 or 2e-19 < t

   1. Initial program 97.2%

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

   3. Taylor expanded in t around inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. distribute-lft-neg-out 81.8%

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

      3. *-commutative 81.8%

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

   5. Simplified 81.8%

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

   if -420 < t < 2e-19

   1. Initial program 99.0%

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

   3. Taylor expanded in y around inf 70.5%

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

   4. Taylor expanded in t around 0 70.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -420 \lor \neg \left(t \leq 2 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+97} \lor \neg \left(y \leq 3.2 \cdot 10^{+57}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.15e+97) (not (<= y 3.2e+57)))
   (* 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 <= -2.15e+97) || !(y <= 3.2e+57)) {
		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 <= (-2.15d+97)) .or. (.not. (y <= 3.2d+57))) 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 <= -2.15e+97) || !(y <= 3.2e+57)) {
		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 <= -2.15e+97) or not (y <= 3.2e+57):
		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 <= -2.15e+97) || !(y <= 3.2e+57))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x / exp(Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.15e+97) || ~((y <= 3.2e+57)))
		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, -2.15e+97], N[Not[LessEqual[y, 3.2e+57]], $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 < -2.1499999999999999e97 or 3.20000000000000029e57 < y

   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 inf 90.5%

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

   4. Taylor expanded in t around 0 65.0%

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

   if -2.1499999999999999e97 < y < 3.20000000000000029e57

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0 98.6%

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

   4. Taylor expanded in y around inf 98.6%

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

      1. sub-neg 98.6%

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

      2. mul-1-neg 98.6%

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

      3. associate-+r+ 98.6%

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

      4. mul-1-neg 98.6%

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

      5. sub-neg 98.6%

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

      6. *-commutative 98.6%

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

      7. distribute-lft-out-- 98.6%

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

      8. +-commutative 98.6%

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

      9. mul-1-neg 98.6%

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

      10. sub-neg 98.6%

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

      11. exp-diff 88.1%

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

   6. Simplified 75.3%

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

   7. Taylor expanded in y around 0 72.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+97} \lor \neg \left(y \leq 3.2 \cdot 10^{+57}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+265}:\\ \;\;\;\;x - x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t \leq -14500000:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -3.4e+265)
   (- x (* x (* y t)))
   (if (<= t -14500000.0) (* a (- (/ x 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 <= -3.4e+265) {
		tmp = x - (x * (y * t));
	} else if (t <= -14500000.0) {
		tmp = a * ((x / 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 <= (-3.4d+265)) then
        tmp = x - (x * (y * t))
    else if (t <= (-14500000.0d0)) then
        tmp = a * ((x / 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 <= -3.4e+265) {
		tmp = x - (x * (y * t));
	} else if (t <= -14500000.0) {
		tmp = a * ((x / 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 <= -3.4e+265:
		tmp = x - (x * (y * t))
	elif t <= -14500000.0:
		tmp = a * ((x / 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 <= -3.4e+265)
		tmp = Float64(x - Float64(x * Float64(y * t)));
	elseif (t <= -14500000.0)
		tmp = Float64(a * Float64(Float64(x / a) - Float64(x * 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 <= -3.4e+265)
		tmp = x - (x * (y * t));
	elseif (t <= -14500000.0)
		tmp = a * ((x / a) - (x * b));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -3.4e+265], N[(x - N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -14500000.0], N[(a * N[(N[(x / a), $MachinePrecision] - N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.3999999999999999e265

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

   4. Taylor expanded in y around 0 56.9%

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

      1. *-commutative 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in t around inf 56.9%

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

      1. neg-mul-1 56.9%

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

   9. Simplified 56.9%

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

   if -3.3999999999999999e265 < t < -1.45e7

   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 z around 0 100.0%

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

   4. Taylor expanded in y around 0 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*r* 50.0%

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

      3. neg-mul-1 50.0%

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

      4. exp-prod 41.7%

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

   6. Simplified 41.7%

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

   7. Taylor expanded in a around 0 25.5%

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

      1. neg-mul-1 25.5%

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

      2. distribute-rgt-neg-in 25.5%

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

   9. Simplified 25.5%

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

   10. Taylor expanded in a around inf 33.0%

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

       1. +-commutative 33.0%

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

       2. mul-1-neg 33.0%

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

       3. unsub-neg 33.0%

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

       4. *-commutative 33.0%

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

   12. Simplified 33.0%

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

   if -1.45e7 < t

   1. Initial program 97.2%

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

   3. Taylor expanded in y around inf 75.6%

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

   4. Taylor expanded in t around 0 65.6%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+265}:\\ \;\;\;\;x - x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t \leq -14500000:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-137}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-25}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 2.9e-137)
   (* x (- 1.0 (* a b)))
   (if (<= z 1.75e-25) (- x (* t (* x y))) (* a (* x (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 2.9e-137) {
		tmp = x * (1.0 - (a * b));
	} else if (z <= 1.75e-25) {
		tmp = x - (t * (x * y));
	} 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 (z <= 2.9d-137) then
        tmp = x * (1.0d0 - (a * b))
    else if (z <= 1.75d-25) then
        tmp = x - (t * (x * y))
    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 (z <= 2.9e-137) {
		tmp = x * (1.0 - (a * b));
	} else if (z <= 1.75e-25) {
		tmp = x - (t * (x * y));
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= 2.9e-137:
		tmp = x * (1.0 - (a * b))
	elif z <= 1.75e-25:
		tmp = x - (t * (x * y))
	else:
		tmp = a * (x * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 2.9e-137)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	elseif (z <= 1.75e-25)
		tmp = Float64(x - Float64(t * Float64(x * y)));
	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 (z <= 2.9e-137)
		tmp = x * (1.0 - (a * b));
	elseif (z <= 1.75e-25)
		tmp = x - (t * (x * y));
	else
		tmp = a * (x * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 2.9e-137], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.75e-25], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.89999999999999985e-137

   1. Initial program 98.4%

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

   3. Taylor expanded in z around 0 98.4%

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

   4. Taylor expanded in y around 0 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-*r* 57.6%

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

      3. neg-mul-1 57.6%

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

      4. exp-prod 49.6%

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

   6. Simplified 49.6%

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

   7. Taylor expanded in a around 0 36.8%

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

      1. neg-mul-1 36.8%

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

      2. distribute-rgt-neg-in 36.8%

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

   9. Simplified 36.8%

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

   10. Taylor expanded in a around 0 31.9%

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

       1. mul-1-neg 31.9%

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

       2. *-lft-identity 31.9%

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

       3. associate-*r* 36.8%

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

       4. distribute-lft-neg-in 36.8%

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

       5. mul-1-neg 36.8%

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

       6. distribute-rgt-in 36.8%

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

       7. mul-1-neg 36.8%

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

       8. unsub-neg 36.8%

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

   12. Simplified 36.8%

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

   if 2.89999999999999985e-137 < z < 1.7500000000000001e-25

   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. Add Preprocessing

   3. Taylor expanded in y around inf 73.8%

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

   4. Taylor expanded in y around 0 30.0%

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

      1. *-commutative 30.0%

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

   6. Simplified 30.0%

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

   7. Taylor expanded in t around inf 26.8%

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

      1. associate-*r* 26.8%

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

      2. neg-mul-1 26.8%

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

   9. Simplified 26.8%

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

   if 1.7500000000000001e-25 < z

   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 z around 0 100.0%

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

   4. Taylor expanded in y around 0 39.0%

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

      1. *-commutative 39.0%

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

      2. associate-*r* 39.0%

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

      3. neg-mul-1 39.0%

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

      4. exp-prod 28.6%

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

   6. Simplified 28.6%

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

   7. Taylor expanded in a around 0 17.6%

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

      1. neg-mul-1 17.6%

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

      2. distribute-rgt-neg-in 17.6%

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

   9. Simplified 17.6%

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

   10. Taylor expanded in a around inf 49.4%

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

       1. mul-1-neg 49.4%

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

       2. *-commutative 49.4%

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

   12. Simplified 49.4%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-137}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-25}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.0% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{-137}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-25}:\\ \;\;\;\;x - x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 8.5e-137)
   (* x (- 1.0 (* a b)))
   (if (<= z 1.75e-25) (- x (* x (* y t))) (* a (* x (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 8.5e-137) {
		tmp = x * (1.0 - (a * b));
	} else if (z <= 1.75e-25) {
		tmp = x - (x * (y * t));
	} 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 (z <= 8.5d-137) then
        tmp = x * (1.0d0 - (a * b))
    else if (z <= 1.75d-25) then
        tmp = x - (x * (y * t))
    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 (z <= 8.5e-137) {
		tmp = x * (1.0 - (a * b));
	} else if (z <= 1.75e-25) {
		tmp = x - (x * (y * t));
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= 8.5e-137:
		tmp = x * (1.0 - (a * b))
	elif z <= 1.75e-25:
		tmp = x - (x * (y * t))
	else:
		tmp = a * (x * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 8.5e-137)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	elseif (z <= 1.75e-25)
		tmp = Float64(x - Float64(x * Float64(y * t)));
	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 (z <= 8.5e-137)
		tmp = x * (1.0 - (a * b));
	elseif (z <= 1.75e-25)
		tmp = x - (x * (y * t));
	else
		tmp = a * (x * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 8.5e-137], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.75e-25], N[(x - N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 8.5000000000000001e-137

   1. Initial program 98.4%

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

   3. Taylor expanded in z around 0 98.4%

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

   4. Taylor expanded in y around 0 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-*r* 57.6%

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

      3. neg-mul-1 57.6%

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

      4. exp-prod 49.6%

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

   6. Simplified 49.6%

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

   7. Taylor expanded in a around 0 36.8%

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

      1. neg-mul-1 36.8%

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

      2. distribute-rgt-neg-in 36.8%

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

   9. Simplified 36.8%

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

   10. Taylor expanded in a around 0 31.9%

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

       1. mul-1-neg 31.9%

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

       2. *-lft-identity 31.9%

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

       3. associate-*r* 36.8%

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

       4. distribute-lft-neg-in 36.8%

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

       5. mul-1-neg 36.8%

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

       6. distribute-rgt-in 36.8%

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

       7. mul-1-neg 36.8%

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

       8. unsub-neg 36.8%

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

   12. Simplified 36.8%

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

   if 8.5000000000000001e-137 < z < 1.7500000000000001e-25

   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. Add Preprocessing

   3. Taylor expanded in y around inf 73.8%

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

   4. Taylor expanded in y around 0 30.0%

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

      1. *-commutative 30.0%

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

   6. Simplified 30.0%

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

   7. Taylor expanded in t around inf 28.8%

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

      1. neg-mul-1 28.8%

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

   9. Simplified 28.8%

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

   if 1.7500000000000001e-25 < z

   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 z around 0 100.0%

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

   4. Taylor expanded in y around 0 39.0%

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

      1. *-commutative 39.0%

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

      2. associate-*r* 39.0%

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

      3. neg-mul-1 39.0%

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

      4. exp-prod 28.6%

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

   6. Simplified 28.6%

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

   7. Taylor expanded in a around 0 17.6%

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

      1. neg-mul-1 17.6%

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

      2. distribute-rgt-neg-in 17.6%

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

   9. Simplified 17.6%

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

   10. Taylor expanded in a around inf 49.4%

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

       1. mul-1-neg 49.4%

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

       2. *-commutative 49.4%

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

   12. Simplified 49.4%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{-137}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-25}:\\ \;\;\;\;x - x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.1% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-92} \lor \neg \left(y \leq 2.05 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -8.5e-92) (not (<= y 2.05e-7))) (* a (* x (- b))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -8.5e-92) || !(y <= 2.05e-7)) {
		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 <= (-8.5d-92)) .or. (.not. (y <= 2.05d-7))) 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 <= -8.5e-92) || !(y <= 2.05e-7)) {
		tmp = a * (x * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -8.5e-92) or not (y <= 2.05e-7):
		tmp = a * (x * -b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -8.5e-92) || !(y <= 2.05e-7))
		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 <= -8.5e-92) || ~((y <= 2.05e-7)))
		tmp = a * (x * -b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -8.5e-92], N[Not[LessEqual[y, 2.05e-7]], $MachinePrecision]], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -8.50000000000000067e-92 or 2.05e-7 < y

   1. Initial program 98.1%

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

   3. Taylor expanded in z around 0 98.1%

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

   4. Taylor expanded in y around 0 42.5%

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

      1. *-commutative 42.5%

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

      2. associate-*r* 42.5%

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

      3. neg-mul-1 42.5%

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

      4. exp-prod 38.4%

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

   6. Simplified 38.4%

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

   7. Taylor expanded in a around 0 16.7%

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

      1. neg-mul-1 16.7%

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

      2. distribute-rgt-neg-in 16.7%

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

   9. Simplified 16.7%

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

   10. Taylor expanded in a around inf 21.3%

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

       1. mul-1-neg 21.3%

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

       2. *-commutative 21.3%

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

   12. Simplified 21.3%

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

   if -8.50000000000000067e-92 < y < 2.05e-7

   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 inf 66.2%

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

   4. Taylor expanded in y around 0 40.2%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-92} \lor \neg \left(y \leq 2.05 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.4% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-92}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.2e-92) (* a (* x (- b))) (if (<= y 2.7e-6) x (* (* a b) (- x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.2e-92) {
		tmp = a * (x * -b);
	} else if (y <= 2.7e-6) {
		tmp = x;
	} else {
		tmp = (a * b) * -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.2d-92)) then
        tmp = a * (x * -b)
    else if (y <= 2.7d-6) then
        tmp = x
    else
        tmp = (a * b) * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.2e-92) {
		tmp = a * (x * -b);
	} else if (y <= 2.7e-6) {
		tmp = x;
	} else {
		tmp = (a * b) * -x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.2e-92:
		tmp = a * (x * -b)
	elif y <= 2.7e-6:
		tmp = x
	else:
		tmp = (a * b) * -x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.2e-92)
		tmp = Float64(a * Float64(x * Float64(-b)));
	elseif (y <= 2.7e-6)
		tmp = x;
	else
		tmp = Float64(Float64(a * b) * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.2e-92)
		tmp = a * (x * -b);
	elseif (y <= 2.7e-6)
		tmp = x;
	else
		tmp = (a * b) * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.2e-92], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e-6], x, N[(N[(a * b), $MachinePrecision] * (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.19999999999999987e-92

   1. Initial program 96.9%

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

   3. Taylor expanded in z around 0 96.9%

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

   4. Taylor expanded in y around 0 48.4%

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

      1. *-commutative 48.4%

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

      2. associate-*r* 48.4%

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

      3. neg-mul-1 48.4%

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

      4. exp-prod 43.6%

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

   6. Simplified 43.6%

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

   7. Taylor expanded in a around 0 19.7%

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

      1. neg-mul-1 19.7%

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

      2. distribute-rgt-neg-in 19.7%

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

   9. Simplified 19.7%

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

   10. Taylor expanded in a around inf 20.4%

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

       1. mul-1-neg 20.4%

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

       2. *-commutative 20.4%

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

   12. Simplified 20.4%

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

   if -2.19999999999999987e-92 < y < 2.69999999999999998e-6

   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 inf 66.2%

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

   4. Taylor expanded in y around 0 40.2%

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

   if 2.69999999999999998e-6 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 99.8%

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

   4. Taylor expanded in y around 0 33.3%

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

      1. *-commutative 33.3%

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

      2. associate-*r* 33.3%

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

      3. neg-mul-1 33.3%

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

      4. exp-prod 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in a around 0 12.2%

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

      1. neg-mul-1 12.2%

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

      2. distribute-rgt-neg-in 12.2%

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

   9. Simplified 12.2%

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

   10. Taylor expanded in a around inf 22.6%

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

       1. mul-1-neg 22.6%

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

       2. associate-*r* 24.0%

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

       3. *-commutative 24.0%

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

       4. distribute-rgt-neg-in 24.0%

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

       5. *-commutative 24.0%

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

       6. distribute-rgt-neg-in 24.0%

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

   12. Simplified 24.0%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-92}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.8% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 3.8e-27) (* 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 (z <= 3.8e-27) {
		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 (z <= 3.8d-27) 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 (z <= 3.8e-27) {
		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 z <= 3.8e-27:
		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 (z <= 3.8e-27)
		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 (z <= 3.8e-27)
		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[z, 3.8e-27], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.8e-27

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0 97.7%

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

   4. Taylor expanded in y around 0 57.9%

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

      1. *-commutative 57.9%

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

      2. associate-*r* 57.9%

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

      3. neg-mul-1 57.9%

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

      4. exp-prod 51.9%

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

   6. Simplified 51.9%

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

   7. Taylor expanded in a around 0 30.3%

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

      1. neg-mul-1 30.3%

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

      2. distribute-rgt-neg-in 30.3%

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

   9. Simplified 30.3%

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

   10. Taylor expanded in a around 0 27.8%

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

       1. mul-1-neg 27.8%

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

       2. *-lft-identity 27.8%

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

       3. associate-*r* 30.3%

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

       4. distribute-lft-neg-in 30.3%

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

       5. mul-1-neg 30.3%

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

       6. distribute-rgt-in 30.3%

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

       7. mul-1-neg 30.3%

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

       8. unsub-neg 30.3%

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

   12. Simplified 30.3%

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

   if 3.8e-27 < z

   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 z around 0 100.0%

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

   4. Taylor expanded in y around 0 34.1%

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

      1. *-commutative 34.1%

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

      2. associate-*r* 34.1%

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

      3. neg-mul-1 34.1%

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

      4. exp-prod 25.1%

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

   6. Simplified 25.1%

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

   7. Taylor expanded in a around 0 15.6%

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

      1. neg-mul-1 15.6%

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

      2. distribute-rgt-neg-in 15.6%

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

   9. Simplified 15.6%

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

   10. Taylor expanded in a around inf 43.3%

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

       1. mul-1-neg 43.3%

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

       2. *-commutative 43.3%

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

   12. Simplified 43.3%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 19.8% 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 98.0%

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

3. Taylor expanded in y around inf 76.8%

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

4. Taylor expanded in y around 0 18.9%

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

5. Final simplification 18.9%

   \[\leadsto x \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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))))))