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

Percentage Accurate: 96.2% → 99.0%

Time: 14.4s

Alternatives: 17

Speedup: 1.5×

• 42.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.2% 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: 99.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in z around 0 100.0%

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

   1. associate-+r+ 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-lft-out 100.0%

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

   4. mul-1-neg 100.0%

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

   5. unsub-neg 100.0%

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

   6. distribute-lft-out 100.0%

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

5. Simplified 100.0%

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

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: 82.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-49} \lor \neg \left(y \leq 2.5 \cdot 10^{-148}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.1e-49) (not (<= y 2.5e-148)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.1e-49) || !(y <= 2.5e-148)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((a * (log((1.0 - z)) - b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-2.1d-49)) .or. (.not. (y <= 2.5d-148))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp((a * (log((1.0d0 - z)) - b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.1e-49) || !(y <= 2.5e-148)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((a * (Math.log((1.0 - z)) - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.1e-49) or not (y <= 2.5e-148):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((a * (math.log((1.0 - z)) - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.1e-49) || !(y <= 2.5e-148))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(a * Float64(log(Float64(1.0 - z)) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.1e-49) || ~((y <= 2.5e-148)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp((a * (log((1.0 - z)) - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.1e-49], N[Not[LessEqual[y, 2.5e-148]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0999999999999999e-49 or 2.4999999999999999e-148 < y

   1. Initial program 96.6%

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

   3. Taylor expanded in z around 0 96.6%

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

      1. +-commutative 96.6%

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

      2. mul-1-neg 96.6%

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

      3. unsub-neg 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in y around inf 83.4%

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

   if -2.0999999999999999e-49 < y < 2.4999999999999999e-148

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0 87.9%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-49} \lor \neg \left(y \leq 2.5 \cdot 10^{-148}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-50} \lor \neg \left(y \leq 2.9 \cdot 10^{-148}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -9.5e-50) (not (<= y 2.9e-148)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -9.5e-50) || !(y <= 2.9e-148)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-9.5d-50)) .or. (.not. (y <= 2.9d-148))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -9.5e-50) || !(y <= 2.9e-148)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -9.5e-50) or not (y <= 2.9e-148):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -9.5e-50) || !(y <= 2.9e-148))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -9.5e-50) || ~((y <= 2.9e-148)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -9.5e-50], N[Not[LessEqual[y, 2.9e-148]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.4999999999999993e-50 or 2.8999999999999998e-148 < y

   1. Initial program 96.6%

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

   3. Taylor expanded in z around 0 96.6%

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

      1. +-commutative 96.6%

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

      2. mul-1-neg 96.6%

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

      3. unsub-neg 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in y around inf 83.4%

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

   if -9.4999999999999993e-50 < y < 2.8999999999999998e-148

   1. Initial program 93.1%

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

   3. Taylor expanded in z around 0 92.0%

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

      1. +-commutative 92.0%

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

      2. mul-1-neg 92.0%

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

      3. unsub-neg 92.0%

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

   5. Simplified 92.0%

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

   6. Taylor expanded in y around 0 86.8%

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

      1. associate-*r* 86.8%

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

      2. neg-mul-1 86.8%

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

   8. Simplified 86.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-50} \lor \neg \left(y \leq 2.9 \cdot 10^{-148}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.8% 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 95.4%

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

3. Taylor expanded in z around 0 95.0%

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

   1. +-commutative 95.0%

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

   2. mul-1-neg 95.0%

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

   3. unsub-neg 95.0%

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

5. Simplified 95.0%

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

6. Final simplification 95.0%

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

Alternative 5: 71.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(-b\right)}\\ t_2 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -2050000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{-298}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+16}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- b))))) (t_2 (* x (exp (* y (- t))))))
   (if (<= t -2050000.0)
     t_2
     (if (<= t 6.3e-298)
       t_1
       (if (<= t 2.6e+16) (* x (pow z y)) (if (<= t 1.26e+98) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * -b));
	double t_2 = x * exp((y * -t));
	double tmp;
	if (t <= -2050000.0) {
		tmp = t_2;
	} else if (t <= 6.3e-298) {
		tmp = t_1;
	} else if (t <= 2.6e+16) {
		tmp = x * pow(z, y);
	} else if (t <= 1.26e+98) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * exp((a * -b))
    t_2 = x * exp((y * -t))
    if (t <= (-2050000.0d0)) then
        tmp = t_2
    else if (t <= 6.3d-298) then
        tmp = t_1
    else if (t <= 2.6d+16) then
        tmp = x * (z ** y)
    else if (t <= 1.26d+98) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((a * -b));
	double t_2 = x * Math.exp((y * -t));
	double tmp;
	if (t <= -2050000.0) {
		tmp = t_2;
	} else if (t <= 6.3e-298) {
		tmp = t_1;
	} else if (t <= 2.6e+16) {
		tmp = x * Math.pow(z, y);
	} else if (t <= 1.26e+98) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * -b))
	t_2 = x * math.exp((y * -t))
	tmp = 0
	if t <= -2050000.0:
		tmp = t_2
	elif t <= 6.3e-298:
		tmp = t_1
	elif t <= 2.6e+16:
		tmp = x * math.pow(z, y)
	elif t <= 1.26e+98:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(-b))))
	t_2 = Float64(x * exp(Float64(y * Float64(-t))))
	tmp = 0.0
	if (t <= -2050000.0)
		tmp = t_2;
	elseif (t <= 6.3e-298)
		tmp = t_1;
	elseif (t <= 2.6e+16)
		tmp = Float64(x * (z ^ y));
	elseif (t <= 1.26e+98)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * -b));
	t_2 = x * exp((y * -t));
	tmp = 0.0;
	if (t <= -2050000.0)
		tmp = t_2;
	elseif (t <= 6.3e-298)
		tmp = t_1;
	elseif (t <= 2.6e+16)
		tmp = x * (z ^ y);
	elseif (t <= 1.26e+98)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2050000.0], t$95$2, If[LessEqual[t, 6.3e-298], t$95$1, If[LessEqual[t, 2.6e+16], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.26e+98], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.05e6 or 1.25999999999999999e98 < t

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0 96.4%

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

      1. +-commutative 96.4%

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

      2. mul-1-neg 96.4%

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

      3. unsub-neg 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in t around inf 79.4%

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

      1. mul-1-neg 79.4%

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

      2. distribute-rgt-neg-in 79.4%

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

   8. Simplified 79.4%

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

   if -2.05e6 < t < 6.2999999999999997e-298 or 2.6e16 < t < 1.25999999999999999e98

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0 96.4%

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

      1. +-commutative 96.4%

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

      2. mul-1-neg 96.4%

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

      3. unsub-neg 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in y around 0 79.4%

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

      1. associate-*r* 79.4%

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

      2. neg-mul-1 79.4%

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

   8. Simplified 79.4%

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

   if 6.2999999999999997e-298 < t < 2.6e16

   1. Initial program 92.8%

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

   3. Taylor expanded in z around 0 91.3%

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

      1. +-commutative 91.3%

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

      2. mul-1-neg 91.3%

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

      3. unsub-neg 91.3%

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

   5. Simplified 91.3%

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

   6. Taylor expanded in y around inf 78.5%

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

   7. Taylor expanded in t around 0 78.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2050000:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{-298}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+16}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{+98}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -600000 \lor \neg \left(t \leq 10^{-59}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -600000.0) (not (<= t 1e-59)))
   (* x (exp (* y (- t))))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -600000.0) || !(t <= 1e-59)) {
		tmp = x * exp((y * -t));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-600000.0d0)) .or. (.not. (t <= 1d-59))) then
        tmp = x * exp((y * -t))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -600000.0) || !(t <= 1e-59)) {
		tmp = x * Math.exp((y * -t));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -600000.0) or not (t <= 1e-59):
		tmp = x * math.exp((y * -t))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -600000.0) || !(t <= 1e-59))
		tmp = Float64(x * exp(Float64(y * Float64(-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 <= -600000.0) || ~((t <= 1e-59)))
		tmp = x * exp((y * -t));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -600000.0], N[Not[LessEqual[t, 1e-59]], $MachinePrecision]], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6e5 or 1e-59 < t

   1. Initial program 95.9%

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

   3. Taylor expanded in z around 0 95.9%

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

      1. +-commutative 95.9%

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

      2. mul-1-neg 95.9%

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

      3. unsub-neg 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in t around inf 78.5%

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

      1. mul-1-neg 78.5%

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

      2. distribute-rgt-neg-in 78.5%

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

   8. Simplified 78.5%

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

   if -6e5 < t < 1e-59

   1. Initial program 94.8%

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

   3. Taylor expanded in z around 0 93.9%

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

      1. +-commutative 93.9%

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

      2. mul-1-neg 93.9%

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

      3. unsub-neg 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in y around inf 63.6%

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

   7. Taylor expanded in t around 0 63.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -600000 \lor \neg \left(t \leq 10^{-59}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1700000:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1700000.0) (* a (- (/ x a) (* x z))) (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1700000.0) {
		tmp = a * ((x / a) - (x * z));
	} 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 <= (-1700000.0d0)) then
        tmp = a * ((x / a) - (x * z))
    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 <= -1700000.0) {
		tmp = a * ((x / a) - (x * z));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1700000.0:
		tmp = a * ((x / a) - (x * z))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1700000.0)
		tmp = Float64(a * Float64(Float64(x / a) - Float64(x * z)));
	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 <= -1700000.0)
		tmp = a * ((x / a) - (x * z));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1700000.0], N[(a * N[(N[(x / a), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7e6

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 21.7%

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

      1. associate-*r* 21.7%

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

      2. neg-mul-1 21.7%

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

   8. Simplified 21.7%

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

   9. Taylor expanded in a around 0 9.2%

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

       1. associate-*r* 9.2%

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

       2. mul-1-neg 9.2%

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

   11. Simplified 9.2%

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

   12. Taylor expanded in a around inf 30.0%

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

       1. +-commutative 30.0%

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

       2. mul-1-neg 30.0%

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

       3. unsub-neg 30.0%

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

   14. Simplified 30.0%

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

   if -1.7e6 < t

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0 94.7%

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

      1. +-commutative 94.7%

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

      2. mul-1-neg 94.7%

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

      3. unsub-neg 94.7%

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

   5. Simplified 94.7%

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

   6. Taylor expanded in y around inf 71.2%

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

   7. Taylor expanded in t around 0 60.5%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1700000:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.0% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-277}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.65e-277)
   (- x (* t (* x y)))
   (if (<= y 6.2e-12) (* a (- (/ x a) (* x z))) (* a (* x (- z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.65e-277) {
		tmp = x - (t * (x * y));
	} else if (y <= 6.2e-12) {
		tmp = a * ((x / a) - (x * z));
	} 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 <= (-1.65d-277)) then
        tmp = x - (t * (x * y))
    else if (y <= 6.2d-12) then
        tmp = a * ((x / a) - (x * z))
    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 <= -1.65e-277) {
		tmp = x - (t * (x * y));
	} else if (y <= 6.2e-12) {
		tmp = a * ((x / a) - (x * z));
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.65e-277:
		tmp = x - (t * (x * y))
	elif y <= 6.2e-12:
		tmp = a * ((x / a) - (x * z))
	else:
		tmp = a * (x * -z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.65e-277)
		tmp = Float64(x - Float64(t * Float64(x * y)));
	elseif (y <= 6.2e-12)
		tmp = Float64(a * Float64(Float64(x / a) - Float64(x * z)));
	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 <= -1.65e-277)
		tmp = x - (t * (x * y));
	elseif (y <= 6.2e-12)
		tmp = a * ((x / a) - (x * z));
	else
		tmp = a * (x * -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.65e-277], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e-12], N[(a * N[(N[(x / a), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.64999999999999991e-277

   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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 64.0%

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

      1. mul-1-neg 64.0%

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

      2. distribute-rgt-neg-in 64.0%

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

   8. Simplified 64.0%

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

   9. Taylor expanded in t around 0 31.3%

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

       1. mul-1-neg 31.3%

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

       2. unsub-neg 31.3%

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

       3. *-commutative 31.3%

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

   11. Simplified 31.3%

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

   if -1.64999999999999991e-277 < y < 6.2000000000000002e-12

   1. Initial program 89.3%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 56.7%

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

      1. associate-*r* 56.7%

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

      2. neg-mul-1 56.7%

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

   8. Simplified 56.7%

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

   9. Taylor expanded in a around 0 39.6%

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

       1. associate-*r* 39.6%

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

       2. mul-1-neg 39.6%

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

   11. Simplified 39.6%

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

   12. Taylor expanded in a around inf 47.2%

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

       1. +-commutative 47.2%

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

       2. mul-1-neg 47.2%

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

       3. unsub-neg 47.2%

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

   14. Simplified 47.2%

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

   if 6.2000000000000002e-12 < y

   1. Initial program 94.6%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 23.2%

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

      1. associate-*r* 23.2%

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

      2. neg-mul-1 23.2%

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

   8. Simplified 23.2%

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

   9. Taylor expanded in a around 0 7.2%

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

       1. associate-*r* 7.2%

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

       2. mul-1-neg 7.2%

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

   11. Simplified 7.2%

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

   12. Taylor expanded in a around inf 26.7%

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

       1. associate-*r* 26.7%

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

       2. mul-1-neg 26.7%

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

   14. Simplified 26.7%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-277}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 27.8% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-252}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{+59}:\\ \;\;\;\;z \cdot \left(\frac{x}{z} - x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -8.5e-252)
   (- x (* t (* x y)))
   (if (<= y 1.08e+59) (* z (- (/ x z) (* x a))) (* a (* x (- z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8.5e-252) {
		tmp = x - (t * (x * y));
	} else if (y <= 1.08e+59) {
		tmp = z * ((x / z) - (x * a));
	} 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 <= (-8.5d-252)) then
        tmp = x - (t * (x * y))
    else if (y <= 1.08d+59) then
        tmp = z * ((x / z) - (x * a))
    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 <= -8.5e-252) {
		tmp = x - (t * (x * y));
	} else if (y <= 1.08e+59) {
		tmp = z * ((x / z) - (x * a));
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -8.5e-252:
		tmp = x - (t * (x * y))
	elif y <= 1.08e+59:
		tmp = z * ((x / z) - (x * a))
	else:
		tmp = a * (x * -z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -8.5e-252)
		tmp = Float64(x - Float64(t * Float64(x * y)));
	elseif (y <= 1.08e+59)
		tmp = Float64(z * Float64(Float64(x / z) - Float64(x * a)));
	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 <= -8.5e-252)
		tmp = x - (t * (x * y));
	elseif (y <= 1.08e+59)
		tmp = z * ((x / z) - (x * a));
	else
		tmp = a * (x * -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -8.5e-252], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.08e+59], N[(z * N[(N[(x / z), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.50000000000000042e-252

   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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 64.4%

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

      1. mul-1-neg 64.4%

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

      2. distribute-rgt-neg-in 64.4%

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

   8. Simplified 64.4%

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

   9. Taylor expanded in t around 0 30.5%

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

       1. mul-1-neg 30.5%

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

       2. unsub-neg 30.5%

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

       3. *-commutative 30.5%

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

   11. Simplified 30.5%

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

   if -8.50000000000000042e-252 < y < 1.08e59

   1. Initial program 88.5%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 53.4%

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

      1. associate-*r* 53.4%

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

      2. neg-mul-1 53.4%

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

   8. Simplified 53.4%

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

   9. Taylor expanded in a around 0 35.9%

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

       1. associate-*r* 35.9%

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

       2. mul-1-neg 35.9%

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

   11. Simplified 35.9%

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

   12. Taylor expanded in z around inf 43.0%

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

       1. +-commutative 43.0%

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

       2. mul-1-neg 43.0%

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

       3. unsub-neg 43.0%

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

       4. *-commutative 43.0%

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

   14. Simplified 43.0%

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

   if 1.08e59 < y

   1. Initial program 98.3%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 18.7%

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

      1. associate-*r* 18.7%

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

      2. neg-mul-1 18.7%

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

   8. Simplified 18.7%

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

   9. Taylor expanded in a around 0 5.1%

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

       1. associate-*r* 5.1%

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

       2. mul-1-neg 5.1%

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

   11. Simplified 5.1%

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

   12. Taylor expanded in a around inf 28.4%

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

       1. associate-*r* 28.4%

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

       2. mul-1-neg 28.4%

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

   14. Simplified 28.4%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-252}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{+59}:\\ \;\;\;\;z \cdot \left(\frac{x}{z} - x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.0% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-39}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(1 - z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.8e-39)
   (* t (* x (- y)))
   (if (<= y 4.8e-28) (* x (- 1.0 (* z a))) (* a (* x (- z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.8e-39) {
		tmp = t * (x * -y);
	} else if (y <= 4.8e-28) {
		tmp = x * (1.0 - (z * a));
	} 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 <= (-3.8d-39)) then
        tmp = t * (x * -y)
    else if (y <= 4.8d-28) then
        tmp = x * (1.0d0 - (z * a))
    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 <= -3.8e-39) {
		tmp = t * (x * -y);
	} else if (y <= 4.8e-28) {
		tmp = x * (1.0 - (z * a));
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.8e-39:
		tmp = t * (x * -y)
	elif y <= 4.8e-28:
		tmp = x * (1.0 - (z * a))
	else:
		tmp = a * (x * -z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.8e-39)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 4.8e-28)
		tmp = Float64(x * Float64(1.0 - Float64(z * a)));
	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 <= -3.8e-39)
		tmp = t * (x * -y);
	elseif (y <= 4.8e-28)
		tmp = x * (1.0 - (z * a));
	else
		tmp = a * (x * -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.8e-39], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e-28], N[(x * N[(1.0 - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8000000000000002e-39

   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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 68.9%

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

      1. mul-1-neg 68.9%

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

      2. distribute-rgt-neg-in 68.9%

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

   8. Simplified 68.9%

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

   9. Taylor expanded in t around 0 27.7%

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

       1. associate-*r* 27.7%

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

       2. neg-mul-1 27.7%

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

       3. *-commutative 27.7%

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

       4. associate-*r* 21.7%

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

       5. distribute-rgt1-in 21.7%

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

       6. +-commutative 21.7%

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

       7. cancel-sign-sub-inv 21.7%

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

   11. Simplified 21.7%

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

   12. Taylor expanded in t around inf 29.1%

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

       1. mul-1-neg 29.1%

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

       2. distribute-rgt-neg-out 29.1%

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

       3. distribute-rgt-neg-in 29.1%

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

   14. Simplified 29.1%

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

   if -3.8000000000000002e-39 < y < 4.8000000000000004e-28

   1. Initial program 94.4%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 54.8%

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

      1. associate-*r* 54.8%

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

      2. neg-mul-1 54.8%

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

   8. Simplified 54.8%

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

   9. Taylor expanded in a around 0 38.1%

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

       1. associate-*r* 38.1%

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

       2. mul-1-neg 38.1%

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

   11. Simplified 38.1%

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

   12. Taylor expanded in x around 0 38.1%

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

       1. mul-1-neg 38.1%

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

       2. *-commutative 38.1%

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

       3. sub-neg 38.1%

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

   14. Simplified 38.1%

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

   if 4.8000000000000004e-28 < y

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 24.5%

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

      1. associate-*r* 24.5%

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

      2. neg-mul-1 24.5%

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

   8. Simplified 24.5%

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

   9. Taylor expanded in a around 0 9.2%

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

       1. associate-*r* 9.2%

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

       2. mul-1-neg 9.2%

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

   11. Simplified 9.2%

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

   12. Taylor expanded in a around inf 28.4%

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

       1. associate-*r* 28.4%

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

       2. mul-1-neg 28.4%

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

   14. Simplified 28.4%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-39}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(1 - z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 24.5% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-39} \lor \neg \left(y \leq 6 \cdot 10^{+168}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3.6e-39) (not (<= y 6e+168))) (* y (* x (- t))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.6e-39) || !(y <= 6e+168)) {
		tmp = y * (x * -t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-3.6d-39)) .or. (.not. (y <= 6d+168))) then
        tmp = y * (x * -t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.6e-39) || !(y <= 6e+168)) {
		tmp = y * (x * -t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -3.6e-39) or not (y <= 6e+168):
		tmp = y * (x * -t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3.6e-39) || !(y <= 6e+168))
		tmp = Float64(y * Float64(x * Float64(-t)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -3.6e-39) || ~((y <= 6e+168)))
		tmp = y * (x * -t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3.6e-39], N[Not[LessEqual[y, 6e+168]], $MachinePrecision]], N[(y * N[(x * (-t)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -3.6000000000000001e-39 or 5.9999999999999996e168 < y

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. mul-1-neg 98.9%

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

      3. unsub-neg 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in t around inf 64.8%

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

      1. mul-1-neg 64.8%

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

      2. distribute-rgt-neg-in 64.8%

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

   8. Simplified 64.8%

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

   9. Taylor expanded in t around 0 22.2%

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

       1. associate-*r* 22.2%

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

       2. neg-mul-1 22.2%

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

       3. *-commutative 22.2%

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

       4. associate-*r* 21.4%

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

       5. distribute-rgt1-in 21.4%

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

       6. +-commutative 21.4%

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

       7. cancel-sign-sub-inv 21.4%

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

   11. Simplified 21.4%

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

   12. Taylor expanded in t around inf 23.2%

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

       1. mul-1-neg 23.2%

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

       2. *-commutative 23.2%

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

       3. *-commutative 23.2%

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

       4. associate-*r* 24.3%

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

   14. Simplified 24.3%

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

   if -3.6000000000000001e-39 < y < 5.9999999999999996e168

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0 92.9%

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

      1. +-commutative 92.9%

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

      2. mul-1-neg 92.9%

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

      3. unsub-neg 92.9%

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

   5. Simplified 92.9%

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

   6. Taylor expanded in t around inf 58.9%

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

      1. mul-1-neg 58.9%

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

      2. distribute-rgt-neg-in 58.9%

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

   8. Simplified 58.9%

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

   9. Taylor expanded in t around 0 27.2%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-39} \lor \neg \left(y \leq 6 \cdot 10^{+168}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.1% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-39}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.5e-39)
   (* t (* x (- y)))
   (if (<= y 3.1e+168) x (* y (* x (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.5e-39) {
		tmp = t * (x * -y);
	} else if (y <= 3.1e+168) {
		tmp = x;
	} else {
		tmp = y * (x * -t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.5d-39)) then
        tmp = t * (x * -y)
    else if (y <= 3.1d+168) then
        tmp = x
    else
        tmp = y * (x * -t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.5e-39) {
		tmp = t * (x * -y);
	} else if (y <= 3.1e+168) {
		tmp = x;
	} else {
		tmp = y * (x * -t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.5e-39:
		tmp = t * (x * -y)
	elif y <= 3.1e+168:
		tmp = x
	else:
		tmp = y * (x * -t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.5e-39)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 3.1e+168)
		tmp = x;
	else
		tmp = Float64(y * Float64(x * Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.5e-39)
		tmp = t * (x * -y);
	elseif (y <= 3.1e+168)
		tmp = x;
	else
		tmp = y * (x * -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.5e-39], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+168], x, N[(y * N[(x * (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4999999999999999e-39

   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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 68.9%

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

      1. mul-1-neg 68.9%

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

      2. distribute-rgt-neg-in 68.9%

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

   8. Simplified 68.9%

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

   9. Taylor expanded in t around 0 27.7%

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

       1. associate-*r* 27.7%

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

       2. neg-mul-1 27.7%

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

       3. *-commutative 27.7%

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

       4. associate-*r* 21.7%

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

       5. distribute-rgt1-in 21.7%

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

       6. +-commutative 21.7%

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

       7. cancel-sign-sub-inv 21.7%

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

   11. Simplified 21.7%

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

   12. Taylor expanded in t around inf 29.1%

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

       1. mul-1-neg 29.1%

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

       2. distribute-rgt-neg-out 29.1%

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

       3. distribute-rgt-neg-in 29.1%

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

   14. Simplified 29.1%

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

   if -2.4999999999999999e-39 < y < 3.09999999999999996e168

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0 92.9%

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

      1. +-commutative 92.9%

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

      2. mul-1-neg 92.9%

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

      3. unsub-neg 92.9%

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

   5. Simplified 92.9%

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

   6. Taylor expanded in t around inf 58.9%

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

      1. mul-1-neg 58.9%

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

      2. distribute-rgt-neg-in 58.9%

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

   8. Simplified 58.9%

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

   9. Taylor expanded in t around 0 27.2%

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

   if 3.09999999999999996e168 < y

   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. Step-by-step derivation

      1. +-commutative 96.9%

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

      2. mul-1-neg 96.9%

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

      3. unsub-neg 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in t around inf 57.3%

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

      1. mul-1-neg 57.3%

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

      2. distribute-rgt-neg-in 57.3%

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

   8. Simplified 57.3%

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

   9. Taylor expanded in t around 0 12.1%

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

       1. associate-*r* 12.1%

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

       2. neg-mul-1 12.1%

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

       3. *-commutative 12.1%

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

       4. associate-*r* 21.0%

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

       5. distribute-rgt1-in 21.0%

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

       6. +-commutative 21.0%

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

       7. cancel-sign-sub-inv 21.0%

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

   11. Simplified 21.0%

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

   12. Taylor expanded in t around inf 12.3%

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

       1. mul-1-neg 12.3%

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

       2. *-commutative 12.3%

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

       3. *-commutative 12.3%

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

       4. associate-*r* 27.0%

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

   14. Simplified 27.0%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-39}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.0% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-39}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot a\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.8e-39)
   (* t (* x (- y)))
   (if (<= y 1.45e-24) x (* (* z a) (- x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.8e-39) {
		tmp = t * (x * -y);
	} else if (y <= 1.45e-24) {
		tmp = x;
	} else {
		tmp = (z * a) * -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-3.8d-39)) then
        tmp = t * (x * -y)
    else if (y <= 1.45d-24) then
        tmp = x
    else
        tmp = (z * a) * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.8e-39) {
		tmp = t * (x * -y);
	} else if (y <= 1.45e-24) {
		tmp = x;
	} else {
		tmp = (z * a) * -x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.8e-39:
		tmp = t * (x * -y)
	elif y <= 1.45e-24:
		tmp = x
	else:
		tmp = (z * a) * -x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.8e-39)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 1.45e-24)
		tmp = x;
	else
		tmp = Float64(Float64(z * a) * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3.8e-39)
		tmp = t * (x * -y);
	elseif (y <= 1.45e-24)
		tmp = x;
	else
		tmp = (z * a) * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.8e-39], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e-24], x, N[(N[(z * a), $MachinePrecision] * (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8000000000000002e-39

   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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 68.9%

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

      1. mul-1-neg 68.9%

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

      2. distribute-rgt-neg-in 68.9%

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

   8. Simplified 68.9%

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

   9. Taylor expanded in t around 0 27.7%

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

       1. associate-*r* 27.7%

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

       2. neg-mul-1 27.7%

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

       3. *-commutative 27.7%

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

       4. associate-*r* 21.7%

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

       5. distribute-rgt1-in 21.7%

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

       6. +-commutative 21.7%

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

       7. cancel-sign-sub-inv 21.7%

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

   11. Simplified 21.7%

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

   12. Taylor expanded in t around inf 29.1%

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

       1. mul-1-neg 29.1%

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

       2. distribute-rgt-neg-out 29.1%

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

       3. distribute-rgt-neg-in 29.1%

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

   14. Simplified 29.1%

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

   if -3.8000000000000002e-39 < y < 1.4499999999999999e-24

   1. Initial program 94.4%

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

   3. Taylor expanded in z around 0 93.6%

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

      1. +-commutative 93.6%

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

      2. mul-1-neg 93.6%

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

      3. unsub-neg 93.6%

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

   5. Simplified 93.6%

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

   6. Taylor expanded in t around inf 57.0%

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

      1. mul-1-neg 57.0%

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

      2. distribute-rgt-neg-in 57.0%

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

   8. Simplified 57.0%

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

   9. Taylor expanded in t around 0 35.2%

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

   if 1.4499999999999999e-24 < y

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 24.5%

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

      1. associate-*r* 24.5%

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

      2. neg-mul-1 24.5%

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

   8. Simplified 24.5%

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

   9. Taylor expanded in a around 0 9.2%

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

       1. associate-*r* 9.2%

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

       2. mul-1-neg 9.2%

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

   11. Simplified 9.2%

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

   12. Taylor expanded in a around inf 26.1%

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

       1. associate-*r* 26.1%

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

       2. mul-1-neg 26.1%

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

   14. Simplified 26.1%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-39}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot a\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.8% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.3 \cdot 10^{-40}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -7.3e-40)
   (* t (* x (- y)))
   (if (<= y 2.05e-83) x (* a (* x (- z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7.3e-40) {
		tmp = t * (x * -y);
	} else if (y <= 2.05e-83) {
		tmp = x;
	} 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 <= (-7.3d-40)) then
        tmp = t * (x * -y)
    else if (y <= 2.05d-83) then
        tmp = x
    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 <= -7.3e-40) {
		tmp = t * (x * -y);
	} else if (y <= 2.05e-83) {
		tmp = x;
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -7.3e-40:
		tmp = t * (x * -y)
	elif y <= 2.05e-83:
		tmp = x
	else:
		tmp = a * (x * -z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -7.3e-40)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 2.05e-83)
		tmp = x;
	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 <= -7.3e-40)
		tmp = t * (x * -y);
	elseif (y <= 2.05e-83)
		tmp = x;
	else
		tmp = a * (x * -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -7.3e-40], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e-83], x, N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.30000000000000005e-40

   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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 68.9%

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

      1. mul-1-neg 68.9%

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

      2. distribute-rgt-neg-in 68.9%

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

   8. Simplified 68.9%

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

   9. Taylor expanded in t around 0 27.7%

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

       1. associate-*r* 27.7%

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

       2. neg-mul-1 27.7%

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

       3. *-commutative 27.7%

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

       4. associate-*r* 21.7%

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

       5. distribute-rgt1-in 21.7%

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

       6. +-commutative 21.7%

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

       7. cancel-sign-sub-inv 21.7%

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

   11. Simplified 21.7%

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

   12. Taylor expanded in t around inf 29.1%

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

       1. mul-1-neg 29.1%

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

       2. distribute-rgt-neg-out 29.1%

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

       3. distribute-rgt-neg-in 29.1%

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

   14. Simplified 29.1%

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

   if -7.30000000000000005e-40 < y < 2.05e-83

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 0 93.3%

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

      1. +-commutative 93.3%

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

      2. mul-1-neg 93.3%

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

      3. unsub-neg 93.3%

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

   5. Simplified 93.3%

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

   6. Taylor expanded in t around inf 57.0%

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

      1. mul-1-neg 57.0%

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

      2. distribute-rgt-neg-in 57.0%

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

   8. Simplified 57.0%

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

   9. Taylor expanded in t around 0 37.9%

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

   if 2.05e-83 < y

   1. Initial program 93.8%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 26.6%

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

      1. associate-*r* 26.6%

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

      2. neg-mul-1 26.6%

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

   8. Simplified 26.6%

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

   9. Taylor expanded in a around 0 12.3%

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

       1. associate-*r* 12.3%

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

       2. mul-1-neg 12.3%

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

   11. Simplified 12.3%

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

   12. Taylor expanded in a around inf 26.8%

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

       1. associate-*r* 26.8%

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

       2. mul-1-neg 26.8%

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

   14. Simplified 26.8%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.3 \cdot 10^{-40}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 28.7% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{+112}:\\ \;\;\;\;z \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -7.5e+112) (* z (* x (- a))) (* x (- 1.0 (* y t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -7.5e+112) {
		tmp = z * (x * -a);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-7.5d+112)) then
        tmp = z * (x * -a)
    else
        tmp = x * (1.0d0 - (y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -7.5e+112) {
		tmp = z * (x * -a);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -7.5e+112:
		tmp = z * (x * -a)
	else:
		tmp = x * (1.0 - (y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -7.5e+112)
		tmp = Float64(z * Float64(x * Float64(-a)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -7.5e+112)
		tmp = z * (x * -a);
	else
		tmp = x * (1.0 - (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -7.5e+112], N[(z * N[(x * (-a)), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.5e112

   1. Initial program 87.4%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 56.9%

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

      1. associate-*r* 56.9%

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

      2. neg-mul-1 56.9%

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

   8. Simplified 56.9%

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

   9. Taylor expanded in a around 0 35.1%

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

       1. associate-*r* 35.1%

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

       2. mul-1-neg 35.1%

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

   11. Simplified 35.1%

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

   12. Taylor expanded in a around inf 43.1%

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

       1. associate-*r* 45.7%

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

       2. associate-*r* 45.7%

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

       3. *-commutative 45.7%

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

       4. mul-1-neg 45.7%

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

       5. *-commutative 45.7%

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

       6. distribute-rgt-neg-in 45.7%

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

   14. Simplified 45.7%

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

   if -7.5e112 < a

   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 96.4%

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

      1. +-commutative 96.4%

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

      2. mul-1-neg 96.4%

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

      3. unsub-neg 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in t around inf 65.6%

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

      1. mul-1-neg 65.6%

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

      2. distribute-rgt-neg-in 65.6%

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

   8. Simplified 65.6%

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

   9. Taylor expanded in t around 0 28.4%

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

       1. associate-*r* 28.4%

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

       2. neg-mul-1 28.4%

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

       3. *-commutative 28.4%

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

       4. associate-*r* 28.5%

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

       5. distribute-rgt1-in 28.5%

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

       6. +-commutative 28.5%

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

       7. cancel-sign-sub-inv 28.5%

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

   11. Simplified 28.5%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{+112}:\\ \;\;\;\;z \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 24.1% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-145}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 1.95e-145) (* a (* x (- z))) (- x (* t (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 1.95e-145) {
		tmp = a * (x * -z);
	} else {
		tmp = x - (t * (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= 1.95d-145) then
        tmp = a * (x * -z)
    else
        tmp = x - (t * (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 1.95e-145) {
		tmp = a * (x * -z);
	} else {
		tmp = x - (t * (x * y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 1.95e-145:
		tmp = a * (x * -z)
	else:
		tmp = x - (t * (x * y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 1.95e-145)
		tmp = Float64(a * Float64(x * Float64(-z)));
	else
		tmp = Float64(x - Float64(t * Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 1.95e-145)
		tmp = a * (x * -z);
	else
		tmp = x - (t * (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 1.95e-145], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.95000000000000015e-145

   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 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 33.1%

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

      1. associate-*r* 33.1%

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

      2. neg-mul-1 33.1%

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

   8. Simplified 33.1%

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

   9. Taylor expanded in a around 0 17.5%

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

       1. associate-*r* 17.5%

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

       2. mul-1-neg 17.5%

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

   11. Simplified 17.5%

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

   12. Taylor expanded in a around inf 23.5%

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

       1. associate-*r* 23.5%

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

       2. mul-1-neg 23.5%

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

   14. Simplified 23.5%

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

   if 1.95000000000000015e-145 < x

   1. Initial program 93.9%

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

   3. Taylor expanded in z around 0 93.9%

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

      1. +-commutative 93.9%

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

      2. mul-1-neg 93.9%

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

      3. unsub-neg 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in t around inf 62.2%

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

      1. mul-1-neg 62.2%

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

      2. distribute-rgt-neg-in 62.2%

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

   8. Simplified 62.2%

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

   9. Taylor expanded in t around 0 36.5%

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

       1. mul-1-neg 36.5%

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

       2. unsub-neg 36.5%

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

       3. *-commutative 36.5%

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

   11. Simplified 36.5%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-145}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 18.6% 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 95.4%

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

3. Taylor expanded in z around 0 95.0%

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

   1. +-commutative 95.0%

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

   2. mul-1-neg 95.0%

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

   3. unsub-neg 95.0%

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

5. Simplified 95.0%

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

6. Taylor expanded in t around inf 61.0%

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

   1. mul-1-neg 61.0%

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

   2. distribute-rgt-neg-in 61.0%

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

8. Simplified 61.0%

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

9. Taylor expanded in t around 0 18.9%

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

10. Final simplification 18.9%

    \[\leadsto x \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(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))))))