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

Percentage Accurate: 96.3% → 99.4%

Time: 15.9s

Alternatives: 17

Speedup: 1.0×

• 41.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.3% 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: 99.4% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (fma y (- (log z) t) (* a (- (log1p (- z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(fma(y, (log(z) - t), (a * (log1p(-z) - b))));
}

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(fma(y, Float64(log(z) - t), Float64(a * Float64(log1p(Float64(-z)) - b)))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.6%

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

   1. fma-def 95.6%

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

   2. sub-neg 95.6%

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

   3. log1p-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

2. Final simplification 95.6%

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

Alternative 3: 95.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{-202} \lor \neg \left(y \leq 8.5 \cdot 10^{-143}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.02e-202) (not (<= y 8.5e-143)))
   (* x (exp (- (* y (- (log z) t)) (* a b))))
   (* x (exp (- (* a (+ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.02e-202) || !(y <= 8.5e-143)) {
		tmp = x * exp(((y * (log(z) - t)) - (a * b)));
	} else {
		tmp = x * exp(-(a * (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 <= (-1.02d-202)) .or. (.not. (y <= 8.5d-143))) then
        tmp = x * exp(((y * (log(z) - t)) - (a * b)))
    else
        tmp = x * exp(-(a * (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 <= -1.02e-202) || !(y <= 8.5e-143)) {
		tmp = x * Math.exp(((y * (Math.log(z) - t)) - (a * b)));
	} else {
		tmp = x * Math.exp(-(a * (z + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.02e-202) or not (y <= 8.5e-143):
		tmp = x * math.exp(((y * (math.log(z) - t)) - (a * b)))
	else:
		tmp = x * math.exp(-(a * (z + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.02e-202) || !(y <= 8.5e-143))
		tmp = Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) - Float64(a * b))));
	else
		tmp = Float64(x * exp(Float64(-Float64(a * Float64(z + b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.02e-202) || ~((y <= 8.5e-143)))
		tmp = x * exp(((y * (log(z) - t)) - (a * b)));
	else
		tmp = x * exp(-(a * (z + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.02e-202], N[Not[LessEqual[y, 8.5e-143]], $MachinePrecision]], N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] - N[(a * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[(-N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.01999999999999997e-202 or 8.50000000000000072e-143 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in z around 0 99.0%

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

   if -1.01999999999999997e-202 < y < 8.50000000000000072e-143

   1. Initial program 84.5%

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

   2. Taylor expanded in y around 0 80.5%

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

      1. sub-neg 80.5%

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

      2. sub-neg 80.5%

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

      3. neg-mul-1 80.5%

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

      4. log1p-def 96.1%

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(-1 \cdot z\right)} + \left(-b\right)\right)} \]

      5. neg-mul-1 96.1%

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

      6. sub-neg 96.1%

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

   4. Simplified 96.1%

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

   5. Taylor expanded in z around 0 96.1%

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

      1. +-commutative 96.1%

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

      2. associate-*r* 96.1%

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

      3. associate-*r* 96.1%

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

      4. distribute-lft-out 96.1%

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

      5. neg-mul-1 96.1%

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

   7. Simplified 96.1%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{-202} \lor \neg \left(y \leq 8.5 \cdot 10^{-143}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \end{array} \]

Alternative 4: 86.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+15} \lor \neg \left(y \leq 0.027\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.8e+15) (not (<= y 0.027)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (- (* a (+ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.8e+15) || !(y <= 0.027)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp(-(a * (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 <= (-2.8d+15)) .or. (.not. (y <= 0.027d0))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp(-(a * (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 <= -2.8e+15) || !(y <= 0.027)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp(-(a * (z + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.8e+15) or not (y <= 0.027):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp(-(a * (z + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.8e+15) || !(y <= 0.027))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(-Float64(a * Float64(z + b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.8e+15) || ~((y <= 0.027)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp(-(a * (z + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.8e+15], N[Not[LessEqual[y, 0.027]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[(-N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e15 or 0.0269999999999999997 < y

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 96.9%

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

   if -2.8e15 < y < 0.0269999999999999997

   1. Initial program 92.0%

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

   2. Taylor expanded in y around 0 83.1%

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

      1. sub-neg 83.1%

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

      2. sub-neg 83.1%

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

      3. neg-mul-1 83.1%

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

      4. log1p-def 91.9%

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(-1 \cdot z\right)} + \left(-b\right)\right)} \]

      5. neg-mul-1 91.9%

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

      6. sub-neg 91.9%

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

   4. Simplified 91.9%

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

   5. Taylor expanded in z around 0 91.9%

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

      1. +-commutative 91.9%

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

      2. associate-*r* 91.9%

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

      3. associate-*r* 91.9%

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

      4. distribute-lft-out 91.9%

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

      5. neg-mul-1 91.9%

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

   7. Simplified 91.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+15} \lor \neg \left(y \leq 0.027\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \end{array} \]

Alternative 5: 83.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+112}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-204} \lor \neg \left(y \leq 1.8 \cdot 10^{-140}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right) - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.4e+112)
   (* x (pow z y))
   (if (or (<= y -9.2e-204) (not (<= y 1.8e-140)))
     (* x (exp (- (* t (- y)) (* a b))))
     (* x (exp (- (* a (+ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.4e+112) {
		tmp = x * pow(z, y);
	} else if ((y <= -9.2e-204) || !(y <= 1.8e-140)) {
		tmp = x * exp(((t * -y) - (a * b)));
	} else {
		tmp = x * exp(-(a * (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 <= (-4.4d+112)) then
        tmp = x * (z ** y)
    else if ((y <= (-9.2d-204)) .or. (.not. (y <= 1.8d-140))) then
        tmp = x * exp(((t * -y) - (a * b)))
    else
        tmp = x * exp(-(a * (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 <= -4.4e+112) {
		tmp = x * Math.pow(z, y);
	} else if ((y <= -9.2e-204) || !(y <= 1.8e-140)) {
		tmp = x * Math.exp(((t * -y) - (a * b)));
	} else {
		tmp = x * Math.exp(-(a * (z + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4.4e+112:
		tmp = x * math.pow(z, y)
	elif (y <= -9.2e-204) or not (y <= 1.8e-140):
		tmp = x * math.exp(((t * -y) - (a * b)))
	else:
		tmp = x * math.exp(-(a * (z + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.4e+112)
		tmp = Float64(x * (z ^ y));
	elseif ((y <= -9.2e-204) || !(y <= 1.8e-140))
		tmp = Float64(x * exp(Float64(Float64(t * Float64(-y)) - Float64(a * b))));
	else
		tmp = Float64(x * exp(Float64(-Float64(a * Float64(z + b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4.4e+112)
		tmp = x * (z ^ y);
	elseif ((y <= -9.2e-204) || ~((y <= 1.8e-140)))
		tmp = x * exp(((t * -y) - (a * b)));
	else
		tmp = x * exp(-(a * (z + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.4e+112], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -9.2e-204], N[Not[LessEqual[y, 1.8e-140]], $MachinePrecision]], N[(x * N[Exp[N[(N[(t * (-y)), $MachinePrecision] - N[(a * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[(-N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.3999999999999999e112

   1. Initial program 97.9%

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

   2. Taylor expanded in y around inf 95.8%

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

   3. Taylor expanded in t around 0 83.2%

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

   if -4.3999999999999999e112 < y < -9.1999999999999997e-204 or 1.8e-140 < y

   1. Initial program 99.4%

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

   2. Taylor expanded in z around 0 99.4%

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

   3. Taylor expanded in t around inf 88.1%

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

      1. neg-mul-1 88.1%

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

   5. Simplified 88.1%

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

   if -9.1999999999999997e-204 < y < 1.8e-140

   1. Initial program 84.5%

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

   2. Taylor expanded in y around 0 80.5%

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

      1. sub-neg 80.5%

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

      2. sub-neg 80.5%

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

      3. neg-mul-1 80.5%

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

      4. log1p-def 96.1%

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(-1 \cdot z\right)} + \left(-b\right)\right)} \]

      5. neg-mul-1 96.1%

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

      6. sub-neg 96.1%

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

   4. Simplified 96.1%

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

   5. Taylor expanded in z around 0 96.1%

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

      1. +-commutative 96.1%

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

      2. associate-*r* 96.1%

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

      3. associate-*r* 96.1%

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

      4. distribute-lft-out 96.1%

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

      5. neg-mul-1 96.1%

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

   7. Simplified 96.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+112}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-204} \lor \neg \left(y \leq 1.8 \cdot 10^{-140}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right) - a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \end{array} \]

Alternative 6: 76.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+15} \lor \neg \left(y \leq 6900000000\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3.4e+15) (not (<= y 6900000000.0)))
   (* x (pow z y))
   (* x (exp (- (* a (+ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.4e+15) || !(y <= 6900000000.0)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp(-(a * (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 <= (-3.4d+15)) .or. (.not. (y <= 6900000000.0d0))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp(-(a * (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 <= -3.4e+15) || !(y <= 6900000000.0)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp(-(a * (z + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -3.4e+15) or not (y <= 6900000000.0):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp(-(a * (z + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3.4e+15) || !(y <= 6900000000.0))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(-Float64(a * Float64(z + b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -3.4e+15) || ~((y <= 6900000000.0)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp(-(a * (z + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3.4e+15], N[Not[LessEqual[y, 6900000000.0]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[(-N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e15 or 6.9e9 < y

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 96.9%

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

   3. Taylor expanded in t around 0 79.7%

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

   if -3.4e15 < y < 6.9e9

   1. Initial program 92.1%

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

   2. Taylor expanded in y around 0 82.0%

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

      1. sub-neg 82.0%

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

      2. sub-neg 82.0%

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

      3. neg-mul-1 82.0%

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

      4. log1p-def 90.5%

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(-1 \cdot z\right)} + \left(-b\right)\right)} \]

      5. neg-mul-1 90.5%

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

      6. sub-neg 90.5%

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

   4. Simplified 90.5%

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

   5. Taylor expanded in z around 0 90.5%

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

      1. +-commutative 90.5%

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

      2. associate-*r* 90.5%

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

      3. associate-*r* 90.5%

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

      4. distribute-lft-out 90.5%

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

      5. neg-mul-1 90.5%

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

   7. Simplified 90.5%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+15} \lor \neg \left(y \leq 6900000000\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \end{array} \]

Alternative 7: 71.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -280 \lor \neg \left(t \leq 4 \cdot 10^{+38}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -280.0) (not (<= t 4e+38)))
   (* x (exp (* t (- y))))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -280.0) || !(t <= 4e+38)) {
		tmp = x * exp((t * -y));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-280.0d0)) .or. (.not. (t <= 4d+38))) then
        tmp = x * exp((t * -y))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -280.0) || !(t <= 4e+38)) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -280.0) or not (t <= 4e+38):
		tmp = x * math.exp((t * -y))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -280.0) || !(t <= 4e+38))
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -280.0) || ~((t <= 4e+38)))
		tmp = x * exp((t * -y));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -280.0], N[Not[LessEqual[t, 4e+38]], $MachinePrecision]], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -280 or 3.99999999999999991e38 < t

   1. Initial program 97.3%

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

   2. Taylor expanded in t around inf 82.0%

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

      1. mul-1-neg 82.0%

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

      2. *-commutative 82.0%

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

   4. Simplified 82.0%

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

   if -280 < t < 3.99999999999999991e38

   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. Taylor expanded in y around inf 61.1%

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

   3. Taylor expanded in t around 0 61.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -280 \lor \neg \left(t \leq 4 \cdot 10^{+38}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]

Alternative 8: 73.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+15} \lor \neg \left(y \leq 17000000000\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.8e+15) (not (<= y 17000000000.0)))
   (* x (pow z y))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.8e+15) || !(y <= 17000000000.0)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-2.8d+15)) .or. (.not. (y <= 17000000000.0d0))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.8e+15) || !(y <= 17000000000.0)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.8e+15) or not (y <= 17000000000.0):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.8e+15) || !(y <= 17000000000.0))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.8e+15) || ~((y <= 17000000000.0)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.8e+15], N[Not[LessEqual[y, 17000000000.0]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e15 or 1.7e10 < y

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 96.9%

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

   3. Taylor expanded in t around 0 79.7%

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

   if -2.8e15 < y < 1.7e10

   1. Initial program 92.1%

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

   2. Taylor expanded in b around inf 80.5%

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

      1. mul-1-neg 80.5%

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

      2. distribute-rgt-neg-out 80.5%

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

   4. Simplified 80.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+15} \lor \neg \left(y \leq 17000000000\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]

Alternative 9: 55.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-49} \lor \neg \left(y \leq 6900000000\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.7e-49) (not (<= y 6900000000.0)))
   (* x (pow z y))
   (* x (- 1.0 (* a (+ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.7e-49) || !(y <= 6900000000.0)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * (1.0 - (a * (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 <= (-1.7d-49)) .or. (.not. (y <= 6900000000.0d0))) then
        tmp = x * (z ** y)
    else
        tmp = x * (1.0d0 - (a * (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 <= -1.7e-49) || !(y <= 6900000000.0)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * (1.0 - (a * (z + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.7e-49) or not (y <= 6900000000.0):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * (1.0 - (a * (z + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.7e-49) || !(y <= 6900000000.0))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * Float64(1.0 - Float64(a * Float64(z + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.7e-49) || ~((y <= 6900000000.0)))
		tmp = x * (z ^ y);
	else
		tmp = x * (1.0 - (a * (z + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.7e-49], N[Not[LessEqual[y, 6900000000.0]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.70000000000000002e-49 or 6.9e9 < y

   1. Initial program 98.7%

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

   2. Taylor expanded in y around inf 92.0%

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

   3. Taylor expanded in t around 0 74.1%

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

   if -1.70000000000000002e-49 < y < 6.9e9

   1. Initial program 91.8%

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

   2. Taylor expanded in y around 0 82.0%

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

      1. sub-neg 82.0%

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

      2. sub-neg 82.0%

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

      3. neg-mul-1 82.0%

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

      4. log1p-def 90.2%

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(-1 \cdot z\right)} + \left(-b\right)\right)} \]

      5. neg-mul-1 90.2%

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

      6. sub-neg 90.2%

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

   4. Simplified 90.2%

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

   5. Taylor expanded in z around 0 90.2%

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

      1. +-commutative 90.2%

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

      2. associate-*r* 90.2%

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

      3. associate-*r* 90.2%

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

      4. distribute-lft-out 90.2%

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

      5. neg-mul-1 90.2%

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

   7. Simplified 90.2%

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

   8. Taylor expanded in a around 0 44.8%

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

      1. neg-mul-1 44.8%

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

      2. unsub-neg 44.8%

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

   10. Simplified 44.8%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-49} \lor \neg \left(y \leq 6900000000\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \end{array} \]

Alternative 10: 21.6% accurate, 22.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-b\right) \cdot \left(x \cdot a\right)\\ \mathbf{if}\;b \leq -1.45 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-280}:\\ \;\;\;\;x\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-176}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{elif}\;b \leq 6.9 \cdot 10^{+159}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- b) (* x a))))
   (if (<= b -1.45e+123)
     t_1
     (if (<= b 6.4e-280)
       x
       (if (<= b 5.5e-176)
         (* x (* a (- b)))
         (if (<= b 6.9e+159) (* x (* t (- y))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -b * (x * a);
	double tmp;
	if (b <= -1.45e+123) {
		tmp = t_1;
	} else if (b <= 6.4e-280) {
		tmp = x;
	} else if (b <= 5.5e-176) {
		tmp = x * (a * -b);
	} else if (b <= 6.9e+159) {
		tmp = x * (t * -y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -b * (x * a)
    if (b <= (-1.45d+123)) then
        tmp = t_1
    else if (b <= 6.4d-280) then
        tmp = x
    else if (b <= 5.5d-176) then
        tmp = x * (a * -b)
    else if (b <= 6.9d+159) then
        tmp = x * (t * -y)
    else
        tmp = t_1
    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 = -b * (x * a);
	double tmp;
	if (b <= -1.45e+123) {
		tmp = t_1;
	} else if (b <= 6.4e-280) {
		tmp = x;
	} else if (b <= 5.5e-176) {
		tmp = x * (a * -b);
	} else if (b <= 6.9e+159) {
		tmp = x * (t * -y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = -b * (x * a)
	tmp = 0
	if b <= -1.45e+123:
		tmp = t_1
	elif b <= 6.4e-280:
		tmp = x
	elif b <= 5.5e-176:
		tmp = x * (a * -b)
	elif b <= 6.9e+159:
		tmp = x * (t * -y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(-b) * Float64(x * a))
	tmp = 0.0
	if (b <= -1.45e+123)
		tmp = t_1;
	elseif (b <= 6.4e-280)
		tmp = x;
	elseif (b <= 5.5e-176)
		tmp = Float64(x * Float64(a * Float64(-b)));
	elseif (b <= 6.9e+159)
		tmp = Float64(x * Float64(t * Float64(-y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -b * (x * a);
	tmp = 0.0;
	if (b <= -1.45e+123)
		tmp = t_1;
	elseif (b <= 6.4e-280)
		tmp = x;
	elseif (b <= 5.5e-176)
		tmp = x * (a * -b);
	elseif (b <= 6.9e+159)
		tmp = x * (t * -y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[((-b) * N[(x * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.45e+123], t$95$1, If[LessEqual[b, 6.4e-280], x, If[LessEqual[b, 5.5e-176], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.9e+159], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.45000000000000005e123 or 6.9000000000000002e159 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in b around inf 72.5%

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

      1. mul-1-neg 72.5%

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

      2. distribute-rgt-neg-out 72.5%

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

   4. Simplified 72.5%

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

   5. Taylor expanded in a around 0 27.9%

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

      1. mul-1-neg 27.9%

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

      2. unsub-neg 27.9%

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

      3. associate-*r* 26.7%

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

      4. *-commutative 26.7%

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

      5. associate-*l* 25.5%

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

   7. Simplified 25.5%

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

   8. Taylor expanded in b around inf 27.1%

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

      1. mul-1-neg 27.1%

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

      2. *-commutative 27.1%

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

      3. associate-*r* 30.8%

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

      4. distribute-rgt-neg-in 30.8%

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

      5. distribute-lft-neg-in 30.8%

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

   10. Simplified 30.8%

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

   if -1.45000000000000005e123 < b < 6.4000000000000001e-280

   1. Initial program 92.3%

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

   2. Taylor expanded in t around inf 67.0%

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

      1. mul-1-neg 67.0%

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

      2. *-commutative 67.0%

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

   4. Simplified 67.0%

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

   5. Taylor expanded in y around 0 26.9%

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

   if 6.4000000000000001e-280 < b < 5.5e-176

   1. Initial program 89.1%

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

   2. Taylor expanded in b around inf 16.3%

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

      1. mul-1-neg 16.3%

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

      2. distribute-rgt-neg-out 16.3%

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

   4. Simplified 16.3%

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

   5. Taylor expanded in a around 0 16.2%

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

      1. mul-1-neg 16.2%

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

      2. unsub-neg 16.2%

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

      3. associate-*r* 16.2%

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

      4. *-commutative 16.2%

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

      5. associate-*l* 15.7%

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

   7. Simplified 15.7%

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

   8. Taylor expanded in b around inf 32.4%

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

      1. mul-1-neg 32.4%

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

      2. *-commutative 32.4%

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

      3. associate-*r* 32.3%

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

      4. distribute-rgt-neg-in 32.3%

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

      5. distribute-lft-neg-in 32.3%

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

   10. Simplified 32.3%

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

   11. Taylor expanded in b around 0 32.4%

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

       1. mul-1-neg 32.4%

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

       2. associate-*r* 43.7%

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

       3. *-commutative 43.7%

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

       4. distribute-lft-neg-in 43.7%

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

   13. Simplified 43.7%

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

   if 5.5e-176 < b < 6.9000000000000002e159

   1. Initial program 97.1%

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

   2. Taylor expanded in t around inf 50.7%

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

      1. mul-1-neg 50.7%

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

      2. *-commutative 50.7%

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

   4. Simplified 50.7%

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

   5. Taylor expanded in y around 0 23.0%

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

      1. mul-1-neg 23.0%

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

      2. unsub-neg 23.0%

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

      3. *-commutative 23.0%

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

   7. Simplified 23.0%

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

   8. Taylor expanded in t around inf 21.3%

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

      1. mul-1-neg 21.3%

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

      2. associate-*r* 24.1%

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

      3. *-commutative 24.1%

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

      4. associate-*r* 23.8%

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

      5. distribute-rgt-neg-in 23.8%

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

      6. distribute-rgt-neg-in 23.8%

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

   10. Simplified 23.8%

       \[\leadsto \color{blue}{x \cdot \left(t \cdot \left(-y\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+123}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-280}:\\ \;\;\;\;x\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-176}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{elif}\;b \leq 6.9 \cdot 10^{+159}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \]

Alternative 11: 21.5% accurate, 22.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+120}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \mathbf{elif}\;b \leq 10^{-279}:\\ \;\;\;\;x\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{-176}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+159}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -4.4e+120)
   (* (- b) (* x a))
   (if (<= b 1e-279)
     x
     (if (<= b 2.85e-176)
       (* x (* a (- b)))
       (if (<= b 4.4e+159) (* x (* t (- y))) (* a (- (* x b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -4.4e+120) {
		tmp = -b * (x * a);
	} else if (b <= 1e-279) {
		tmp = x;
	} else if (b <= 2.85e-176) {
		tmp = x * (a * -b);
	} else if (b <= 4.4e+159) {
		tmp = x * (t * -y);
	} else {
		tmp = a * -(x * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-4.4d+120)) then
        tmp = -b * (x * a)
    else if (b <= 1d-279) then
        tmp = x
    else if (b <= 2.85d-176) then
        tmp = x * (a * -b)
    else if (b <= 4.4d+159) then
        tmp = x * (t * -y)
    else
        tmp = a * -(x * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -4.4e+120) {
		tmp = -b * (x * a);
	} else if (b <= 1e-279) {
		tmp = x;
	} else if (b <= 2.85e-176) {
		tmp = x * (a * -b);
	} else if (b <= 4.4e+159) {
		tmp = x * (t * -y);
	} else {
		tmp = a * -(x * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -4.4e+120:
		tmp = -b * (x * a)
	elif b <= 1e-279:
		tmp = x
	elif b <= 2.85e-176:
		tmp = x * (a * -b)
	elif b <= 4.4e+159:
		tmp = x * (t * -y)
	else:
		tmp = a * -(x * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -4.4e+120)
		tmp = Float64(Float64(-b) * Float64(x * a));
	elseif (b <= 1e-279)
		tmp = x;
	elseif (b <= 2.85e-176)
		tmp = Float64(x * Float64(a * Float64(-b)));
	elseif (b <= 4.4e+159)
		tmp = Float64(x * Float64(t * Float64(-y)));
	else
		tmp = Float64(a * Float64(-Float64(x * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -4.4e+120)
		tmp = -b * (x * a);
	elseif (b <= 1e-279)
		tmp = x;
	elseif (b <= 2.85e-176)
		tmp = x * (a * -b);
	elseif (b <= 4.4e+159)
		tmp = x * (t * -y);
	else
		tmp = a * -(x * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -4.4e+120], N[((-b) * N[(x * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-279], x, If[LessEqual[b, 2.85e-176], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e+159], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], N[(a * (-N[(x * b), $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -4.4000000000000003e120

   1. Initial program 100.0%

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

   2. Taylor expanded in b around inf 63.9%

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

      1. mul-1-neg 63.9%

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

      2. distribute-rgt-neg-out 63.9%

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

   4. Simplified 63.9%

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

   5. Taylor expanded in a around 0 24.8%

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

      1. mul-1-neg 24.8%

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

      2. unsub-neg 24.8%

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

      3. associate-*r* 22.9%

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

      4. *-commutative 22.9%

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

      5. associate-*l* 22.9%

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

   7. Simplified 22.9%

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

   8. Taylor expanded in b around inf 25.5%

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

      1. mul-1-neg 25.5%

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

      2. *-commutative 25.5%

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

      3. associate-*r* 31.5%

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

      4. distribute-rgt-neg-in 31.5%

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

      5. distribute-lft-neg-in 31.5%

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

   10. Simplified 31.5%

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

   if -4.4000000000000003e120 < b < 1.00000000000000006e-279

   1. Initial program 92.3%

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

   2. Taylor expanded in t around inf 67.0%

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

      1. mul-1-neg 67.0%

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

      2. *-commutative 67.0%

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

   4. Simplified 67.0%

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

   5. Taylor expanded in y around 0 26.9%

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

   if 1.00000000000000006e-279 < b < 2.84999999999999992e-176

   1. Initial program 89.1%

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

   2. Taylor expanded in b around inf 16.3%

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

      1. mul-1-neg 16.3%

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

      2. distribute-rgt-neg-out 16.3%

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

   4. Simplified 16.3%

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

   5. Taylor expanded in a around 0 16.2%

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

      1. mul-1-neg 16.2%

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

      2. unsub-neg 16.2%

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

      3. associate-*r* 16.2%

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

      4. *-commutative 16.2%

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

      5. associate-*l* 15.7%

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

   7. Simplified 15.7%

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

   8. Taylor expanded in b around inf 32.4%

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

      1. mul-1-neg 32.4%

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

      2. *-commutative 32.4%

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

      3. associate-*r* 32.3%

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

      4. distribute-rgt-neg-in 32.3%

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

      5. distribute-lft-neg-in 32.3%

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

   10. Simplified 32.3%

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

   11. Taylor expanded in b around 0 32.4%

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

       1. mul-1-neg 32.4%

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

       2. associate-*r* 43.7%

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

       3. *-commutative 43.7%

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

       4. distribute-lft-neg-in 43.7%

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

   13. Simplified 43.7%

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

   if 2.84999999999999992e-176 < b < 4.3999999999999998e159

   1. Initial program 97.1%

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

   2. Taylor expanded in t around inf 50.7%

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

      1. mul-1-neg 50.7%

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

      2. *-commutative 50.7%

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

   4. Simplified 50.7%

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

   5. Taylor expanded in y around 0 23.0%

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

      1. mul-1-neg 23.0%

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

      2. unsub-neg 23.0%

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

      3. *-commutative 23.0%

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

   7. Simplified 23.0%

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

   8. Taylor expanded in t around inf 21.3%

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

      1. mul-1-neg 21.3%

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

      2. associate-*r* 24.1%

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

      3. *-commutative 24.1%

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

      4. associate-*r* 23.8%

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

      5. distribute-rgt-neg-in 23.8%

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

      6. distribute-rgt-neg-in 23.8%

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

   10. Simplified 23.8%

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

   if 4.3999999999999998e159 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in b around inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. distribute-rgt-neg-out 86.5%

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

   4. Simplified 86.5%

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

   5. Taylor expanded in a around 0 33.0%

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

      1. mul-1-neg 33.0%

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

      2. unsub-neg 33.0%

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

      3. associate-*r* 32.9%

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

      4. *-commutative 32.9%

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

      5. associate-*l* 29.8%

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

   7. Simplified 29.8%

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

   8. Taylor expanded in b around inf 29.6%

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+120}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \mathbf{elif}\;b \leq 10^{-279}:\\ \;\;\;\;x\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{-176}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+159}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \end{array} \]

Alternative 12: 32.9% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -8e-32)
   (* x (* t (- y)))
   (if (<= y 3.6e+34) (* x (- 1.0 (* a (+ z b)))) (* (- b) (* x a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8e-32) {
		tmp = x * (t * -y);
	} else if (y <= 3.6e+34) {
		tmp = x * (1.0 - (a * (z + b)));
	} else {
		tmp = -b * (x * a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-8d-32)) then
        tmp = x * (t * -y)
    else if (y <= 3.6d+34) then
        tmp = x * (1.0d0 - (a * (z + b)))
    else
        tmp = -b * (x * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8e-32) {
		tmp = x * (t * -y);
	} else if (y <= 3.6e+34) {
		tmp = x * (1.0 - (a * (z + b)));
	} else {
		tmp = -b * (x * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -8e-32:
		tmp = x * (t * -y)
	elif y <= 3.6e+34:
		tmp = x * (1.0 - (a * (z + b)))
	else:
		tmp = -b * (x * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -8e-32)
		tmp = Float64(x * Float64(t * Float64(-y)));
	elseif (y <= 3.6e+34)
		tmp = Float64(x * Float64(1.0 - Float64(a * Float64(z + b))));
	else
		tmp = Float64(Float64(-b) * Float64(x * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -8e-32)
		tmp = x * (t * -y);
	elseif (y <= 3.6e+34)
		tmp = x * (1.0 - (a * (z + b)));
	else
		tmp = -b * (x * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -8e-32], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e+34], N[(x * N[(1.0 - N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) * N[(x * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000045e-32

   1. Initial program 97.6%

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

   2. Taylor expanded in t around inf 61.3%

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

      1. mul-1-neg 61.3%

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

      2. *-commutative 61.3%

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

   4. Simplified 61.3%

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

   5. Taylor expanded in y around 0 24.0%

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

      1. mul-1-neg 24.0%

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

      2. unsub-neg 24.0%

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

      3. *-commutative 24.0%

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

   7. Simplified 24.0%

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

   8. Taylor expanded in t around inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. associate-*r* 22.7%

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

      3. *-commutative 22.7%

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

      4. associate-*r* 26.3%

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

      5. distribute-rgt-neg-in 26.3%

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

      6. distribute-rgt-neg-in 26.3%

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

   10. Simplified 26.3%

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

   if -8.00000000000000045e-32 < y < 3.6e34

   1. Initial program 92.3%

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

   2. Taylor expanded in y around 0 80.0%

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

      1. sub-neg 80.0%

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

      2. sub-neg 80.0%

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

      3. neg-mul-1 80.0%

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

      4. log1p-def 87.7%

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(-1 \cdot z\right)} + \left(-b\right)\right)} \]

      5. neg-mul-1 87.7%

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

      6. sub-neg 87.7%

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

   4. Simplified 87.7%

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

   5. Taylor expanded in z around 0 87.7%

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

      1. +-commutative 87.7%

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

      2. associate-*r* 87.7%

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

      3. associate-*r* 87.7%

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

      4. distribute-lft-out 87.7%

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

      5. neg-mul-1 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in a around 0 43.6%

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

      1. neg-mul-1 43.6%

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

      2. unsub-neg 43.6%

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

   10. Simplified 43.6%

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

   if 3.6e34 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in b around inf 32.1%

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

      1. mul-1-neg 32.1%

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

      2. distribute-rgt-neg-out 32.1%

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

   4. Simplified 32.1%

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

   5. Taylor expanded in a around 0 5.0%

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

      1. mul-1-neg 5.0%

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

      2. unsub-neg 5.0%

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

      3. associate-*r* 5.0%

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

      4. *-commutative 5.0%

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

      5. associate-*l* 5.1%

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

   7. Simplified 5.1%

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

   8. Taylor expanded in b around inf 15.0%

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

      1. mul-1-neg 15.0%

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

      2. *-commutative 15.0%

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

      3. associate-*r* 18.4%

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

      4. distribute-rgt-neg-in 18.4%

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

      5. distribute-lft-neg-in 18.4%

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

   10. Simplified 18.4%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \]

Alternative 13: 32.5% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.8e-31)
   (* x (* t (- y)))
   (if (<= y 2.7e+34) (* x (- 1.0 (* a b))) (* (- b) (* x a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.8e-31) {
		tmp = x * (t * -y);
	} else if (y <= 2.7e+34) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -b * (x * a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.8d-31)) then
        tmp = x * (t * -y)
    else if (y <= 2.7d+34) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = -b * (x * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.8e-31) {
		tmp = x * (t * -y);
	} else if (y <= 2.7e+34) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -b * (x * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.8e-31:
		tmp = x * (t * -y)
	elif y <= 2.7e+34:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = -b * (x * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.8e-31)
		tmp = Float64(x * Float64(t * Float64(-y)));
	elseif (y <= 2.7e+34)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(Float64(-b) * Float64(x * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.8e-31)
		tmp = x * (t * -y);
	elseif (y <= 2.7e+34)
		tmp = x * (1.0 - (a * b));
	else
		tmp = -b * (x * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.8e-31], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+34], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) * N[(x * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7999999999999999e-31

   1. Initial program 97.6%

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

   2. Taylor expanded in t around inf 61.3%

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

      1. mul-1-neg 61.3%

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

      2. *-commutative 61.3%

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

   4. Simplified 61.3%

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

   5. Taylor expanded in y around 0 24.0%

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

      1. mul-1-neg 24.0%

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

      2. unsub-neg 24.0%

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

      3. *-commutative 24.0%

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

   7. Simplified 24.0%

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

   8. Taylor expanded in t around inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. associate-*r* 22.7%

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

      3. *-commutative 22.7%

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

      4. associate-*r* 26.3%

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

      5. distribute-rgt-neg-in 26.3%

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

      6. distribute-rgt-neg-in 26.3%

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

   10. Simplified 26.3%

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

   if -2.7999999999999999e-31 < y < 2.7e34

   1. Initial program 92.3%

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

   2. Taylor expanded in b around inf 78.4%

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

      1. mul-1-neg 78.4%

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

      2. distribute-rgt-neg-out 78.4%

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

   4. Simplified 78.4%

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

   5. Taylor expanded in a around 0 40.1%

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

      1. mul-1-neg 40.1%

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

      2. unsub-neg 40.1%

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

      3. associate-*r* 41.6%

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

      4. *-commutative 41.6%

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

      5. associate-*l* 39.6%

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

   7. Simplified 39.6%

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

   8. Taylor expanded in x around 0 41.6%

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

   if 2.7e34 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in b around inf 32.1%

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

      1. mul-1-neg 32.1%

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

      2. distribute-rgt-neg-out 32.1%

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

   4. Simplified 32.1%

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

   5. Taylor expanded in a around 0 5.0%

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

      1. mul-1-neg 5.0%

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

      2. unsub-neg 5.0%

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

      3. associate-*r* 5.0%

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

      4. *-commutative 5.0%

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

      5. associate-*l* 5.1%

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

   7. Simplified 5.1%

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

   8. Taylor expanded in b around inf 15.0%

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

      1. mul-1-neg 15.0%

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

      2. *-commutative 15.0%

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

      3. associate-*r* 18.4%

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

      4. distribute-rgt-neg-in 18.4%

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

      5. distribute-lft-neg-in 18.4%

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

   10. Simplified 18.4%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \]

Alternative 14: 24.2% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-31} \lor \neg \left(y \leq 3 \cdot 10^{-248}\right):\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.85e-31) (not (<= y 3e-248))) (* (- b) (* x a)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.85e-31) || !(y <= 3e-248)) {
		tmp = -b * (x * a);
	} 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 <= (-1.85d-31)) .or. (.not. (y <= 3d-248))) then
        tmp = -b * (x * a)
    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 <= -1.85e-31) || !(y <= 3e-248)) {
		tmp = -b * (x * a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.85e-31) or not (y <= 3e-248):
		tmp = -b * (x * a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.85e-31) || !(y <= 3e-248))
		tmp = Float64(Float64(-b) * Float64(x * a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.85e-31) || ~((y <= 3e-248)))
		tmp = -b * (x * a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.85e-31], N[Not[LessEqual[y, 3e-248]], $MachinePrecision]], N[((-b) * N[(x * a), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.8499999999999999e-31 or 3.00000000000000014e-248 < y

   1. Initial program 97.7%

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

   2. Taylor expanded in b around inf 45.6%

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

      1. mul-1-neg 45.6%

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

      2. distribute-rgt-neg-out 45.6%

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

   4. Simplified 45.6%

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

   5. Taylor expanded in a around 0 17.0%

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

      1. mul-1-neg 17.0%

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

      2. unsub-neg 17.0%

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

      3. associate-*r* 16.5%

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

      4. *-commutative 16.5%

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

      5. associate-*l* 18.2%

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

   7. Simplified 18.2%

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

   8. Taylor expanded in b around inf 17.0%

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

      1. mul-1-neg 17.0%

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

      2. *-commutative 17.0%

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

      3. associate-*r* 18.5%

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

      4. distribute-rgt-neg-in 18.5%

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

      5. distribute-lft-neg-in 18.5%

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

   10. Simplified 18.5%

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

   if -1.8499999999999999e-31 < y < 3.00000000000000014e-248

   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. Taylor expanded in t around inf 49.6%

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

      1. mul-1-neg 49.6%

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

      2. *-commutative 49.6%

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

   4. Simplified 49.6%

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

   5. Taylor expanded in y around 0 38.9%

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

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-31} \lor \neg \left(y \leq 3 \cdot 10^{-248}\right):\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15: 24.9% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-248}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.8e-31)
   (* x (* t (- y)))
   (if (<= y 1.8e-248) x (* (- b) (* x a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.8e-31) {
		tmp = x * (t * -y);
	} else if (y <= 1.8e-248) {
		tmp = x;
	} else {
		tmp = -b * (x * a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.8d-31)) then
        tmp = x * (t * -y)
    else if (y <= 1.8d-248) then
        tmp = x
    else
        tmp = -b * (x * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.8e-31) {
		tmp = x * (t * -y);
	} else if (y <= 1.8e-248) {
		tmp = x;
	} else {
		tmp = -b * (x * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.8e-31:
		tmp = x * (t * -y)
	elif y <= 1.8e-248:
		tmp = x
	else:
		tmp = -b * (x * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.8e-31)
		tmp = Float64(x * Float64(t * Float64(-y)));
	elseif (y <= 1.8e-248)
		tmp = x;
	else
		tmp = Float64(Float64(-b) * Float64(x * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.8e-31)
		tmp = x * (t * -y);
	elseif (y <= 1.8e-248)
		tmp = x;
	else
		tmp = -b * (x * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.8e-31], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e-248], x, N[((-b) * N[(x * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7999999999999999e-31

   1. Initial program 97.6%

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

   2. Taylor expanded in t around inf 61.3%

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

      1. mul-1-neg 61.3%

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

      2. *-commutative 61.3%

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

   4. Simplified 61.3%

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

   5. Taylor expanded in y around 0 24.0%

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

      1. mul-1-neg 24.0%

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

      2. unsub-neg 24.0%

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

      3. *-commutative 24.0%

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

   7. Simplified 24.0%

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

   8. Taylor expanded in t around inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. associate-*r* 22.7%

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

      3. *-commutative 22.7%

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

      4. associate-*r* 26.3%

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

      5. distribute-rgt-neg-in 26.3%

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

      6. distribute-rgt-neg-in 26.3%

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

   10. Simplified 26.3%

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

   if -2.7999999999999999e-31 < y < 1.79999999999999992e-248

   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. Taylor expanded in t around inf 49.6%

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

      1. mul-1-neg 49.6%

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

      2. *-commutative 49.6%

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

   4. Simplified 49.6%

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

   5. Taylor expanded in y around 0 38.9%

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

   if 1.79999999999999992e-248 < y

   1. Initial program 97.8%

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

   2. Taylor expanded in b around inf 52.9%

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

      1. mul-1-neg 52.9%

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

      2. distribute-rgt-neg-out 52.9%

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

   4. Simplified 52.9%

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

   5. Taylor expanded in a around 0 19.1%

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

      1. mul-1-neg 19.1%

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

      2. unsub-neg 19.1%

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

      3. associate-*r* 19.0%

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

      4. *-commutative 19.0%

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

      5. associate-*l* 20.4%

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

   7. Simplified 20.4%

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

   8. Taylor expanded in b around inf 19.3%

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

      1. mul-1-neg 19.3%

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

      2. *-commutative 19.3%

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

      3. associate-*r* 21.0%

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

      4. distribute-rgt-neg-in 21.0%

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

      5. distribute-lft-neg-in 21.0%

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

   10. Simplified 21.0%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-248}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \]

Alternative 16: 22.5% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= y 1.5e+39) x (* x (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 1.5e+39) {
		tmp = x;
	} else {
		tmp = x * (a * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= 1.5d+39) then
        tmp = x
    else
        tmp = x * (a * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 1.5e+39) {
		tmp = x;
	} else {
		tmp = x * (a * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 1.5e+39:
		tmp = x
	else:
		tmp = x * (a * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 1.5e+39)
		tmp = x;
	else
		tmp = Float64(x * Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 1.5e+39)
		tmp = x;
	else
		tmp = x * (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 1.5e+39], x, N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5e39

   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. Taylor expanded in t around inf 50.8%

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

      1. mul-1-neg 50.8%

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

      2. *-commutative 50.8%

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

   4. Simplified 50.8%

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

   5. Taylor expanded in y around 0 19.5%

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

   if 1.5e39 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in b around inf 30.9%

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

      1. mul-1-neg 30.9%

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

      2. distribute-rgt-neg-out 30.9%

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

   4. Simplified 30.9%

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

   5. Taylor expanded in a around 0 5.1%

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

      1. mul-1-neg 5.1%

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

      2. unsub-neg 5.1%

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

      3. associate-*r* 5.1%

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

      4. *-commutative 5.1%

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

      5. associate-*l* 5.2%

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

   7. Simplified 5.2%

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

   8. Taylor expanded in b around inf 15.2%

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

      1. mul-1-neg 15.2%

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

      2. *-commutative 15.2%

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

      3. associate-*r* 18.7%

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

      4. distribute-rgt-neg-in 18.7%

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

      5. distribute-lft-neg-in 18.7%

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

   10. Simplified 18.7%

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

       1. expm1-log1p-u 18.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left(\left(-x\right) \cdot a\right)\right)\right)} \]

       2. expm1-udef 38.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(b \cdot \left(\left(-x\right) \cdot a\right)\right)} - 1} \]

       3. *-commutative 38.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(-x\right) \cdot a\right) \cdot b}\right)} - 1 \]

       4. associate-*l* 38.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-x\right) \cdot \left(a \cdot b\right)}\right)} - 1 \]

       5. add-sqr-sqrt 11.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \left(a \cdot b\right)\right)} - 1 \]

       6. sqrt-unprod 31.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \cdot \left(a \cdot b\right)\right)} - 1 \]

       7. sqr-neg 31.1%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{x \cdot x}} \cdot \left(a \cdot b\right)\right)} - 1 \]

       8. sqrt-unprod 25.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(a \cdot b\right)\right)} - 1 \]

       9. add-sqr-sqrt 36.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x} \cdot \left(a \cdot b\right)\right)} - 1 \]

   12. Applied egg-rr 36.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(a \cdot b\right)\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 14.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(a \cdot b\right)\right)\right)} \]

       2. expm1-log1p 16.8%

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

   14. Simplified 16.8%

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

4. Final simplification 18.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot b\right)\\ \end{array} \]

Alternative 17: 19.5% accurate, 315.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

2. Taylor expanded in t around inf 53.9%

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

   1. mul-1-neg 53.9%

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

   2. *-commutative 53.9%

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

4. Simplified 53.9%

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

5. Taylor expanded in y around 0 16.1%

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

6. Final simplification 16.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023335 
(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))))))