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

Percentage Accurate: 96.5% → 99.6%

Time: 15.8s

Alternatives: 18

Speedup: 1.0×

• 40.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. fma-def 96.6%

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

   2. sub-neg 96.6%

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

   3. log1p-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

2. Final simplification 96.2%

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

Alternative 3: 83.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+70} \lor \neg \left(a \leq 2.7 \cdot 10^{+17}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-\left(z + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -2e+70) (not (<= a 2.7e+17)))
   (* x (exp (* a (- (+ z b)))))
   (* x (exp (* y (- (log z) t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -2e+70) || !(a <= 2.7e+17)) {
		tmp = x * exp((a * -(z + b)));
	} else {
		tmp = x * exp((y * (log(z) - t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-2d+70)) .or. (.not. (a <= 2.7d+17))) then
        tmp = x * exp((a * -(z + b)))
    else
        tmp = x * exp((y * (log(z) - t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -2e+70) || !(a <= 2.7e+17)) {
		tmp = x * Math.exp((a * -(z + b)));
	} else {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -2e+70) or not (a <= 2.7e+17):
		tmp = x * math.exp((a * -(z + b)))
	else:
		tmp = x * math.exp((y * (math.log(z) - t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -2e+70) || !(a <= 2.7e+17))
		tmp = Float64(x * exp(Float64(a * Float64(-Float64(z + b)))));
	else
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -2e+70) || ~((a <= 2.7e+17)))
		tmp = x * exp((a * -(z + b)));
	else
		tmp = x * exp((y * (log(z) - t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -2e+70], N[Not[LessEqual[a, 2.7e+17]], $MachinePrecision]], N[(x * N[Exp[N[(a * (-N[(z + b), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.00000000000000015e70 or 2.7e17 < a

   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 78.7%

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

      1. sub-neg 78.7%

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

      2. sub-neg 78.7%

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

      3. neg-mul-1 78.7%

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

      4. log1p-def 89.0%

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

      5. neg-mul-1 89.0%

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

      6. sub-neg 89.0%

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

   4. Simplified 89.0%

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

   5. Taylor expanded in z around 0 89.0%

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

      1. associate-*r* 89.0%

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

      2. associate-*r* 89.0%

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

      3. distribute-lft-out 89.0%

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

      4. neg-mul-1 89.0%

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

   7. Simplified 89.0%

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

   if -2.00000000000000015e70 < a < 2.7e17

   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 y around inf 90.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+70} \lor \neg \left(a \leq 2.7 \cdot 10^{+17}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-\left(z + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]

Alternative 4: 69.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{if}\;a \leq -5.4 \cdot 10^{+192}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.9 \cdot 10^{+179}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-z\right)}\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{+17} \lor \neg \left(a \leq 1.05 \cdot 10^{-32}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- b))))))
   (if (<= a -5.4e+192)
     t_1
     (if (<= a -2.9e+179)
       (* x (exp (* a (- z))))
       (if (or (<= a -4.8e+17) (not (<= a 1.05e-32)))
         t_1
         (* x (exp (* t (- y)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * -b));
	double tmp;
	if (a <= -5.4e+192) {
		tmp = t_1;
	} else if (a <= -2.9e+179) {
		tmp = x * exp((a * -z));
	} else if ((a <= -4.8e+17) || !(a <= 1.05e-32)) {
		tmp = t_1;
	} else {
		tmp = x * exp((t * -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) :: t_1
    real(8) :: tmp
    t_1 = x * exp((a * -b))
    if (a <= (-5.4d+192)) then
        tmp = t_1
    else if (a <= (-2.9d+179)) then
        tmp = x * exp((a * -z))
    else if ((a <= (-4.8d+17)) .or. (.not. (a <= 1.05d-32))) then
        tmp = t_1
    else
        tmp = x * exp((t * -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 t_1 = x * Math.exp((a * -b));
	double tmp;
	if (a <= -5.4e+192) {
		tmp = t_1;
	} else if (a <= -2.9e+179) {
		tmp = x * Math.exp((a * -z));
	} else if ((a <= -4.8e+17) || !(a <= 1.05e-32)) {
		tmp = t_1;
	} else {
		tmp = x * Math.exp((t * -y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * -b))
	tmp = 0
	if a <= -5.4e+192:
		tmp = t_1
	elif a <= -2.9e+179:
		tmp = x * math.exp((a * -z))
	elif (a <= -4.8e+17) or not (a <= 1.05e-32):
		tmp = t_1
	else:
		tmp = x * math.exp((t * -y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(-b))))
	tmp = 0.0
	if (a <= -5.4e+192)
		tmp = t_1;
	elseif (a <= -2.9e+179)
		tmp = Float64(x * exp(Float64(a * Float64(-z))));
	elseif ((a <= -4.8e+17) || !(a <= 1.05e-32))
		tmp = t_1;
	else
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * -b));
	tmp = 0.0;
	if (a <= -5.4e+192)
		tmp = t_1;
	elseif (a <= -2.9e+179)
		tmp = x * exp((a * -z));
	elseif ((a <= -4.8e+17) || ~((a <= 1.05e-32)))
		tmp = t_1;
	else
		tmp = x * exp((t * -y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5.4e+192], t$95$1, If[LessEqual[a, -2.9e+179], N[(x * N[Exp[N[(a * (-z)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a, -4.8e+17], N[Not[LessEqual[a, 1.05e-32]], $MachinePrecision]], t$95$1, N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.39999999999999979e192 or -2.90000000000000019e179 < a < -4.8e17 or 1.05e-32 < a

   1. Initial program 95.0%

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

   2. Taylor expanded in b around inf 80.4%

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

      1. mul-1-neg 80.4%

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

      2. distribute-rgt-neg-out 80.4%

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

   4. Simplified 80.4%

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

   if -5.39999999999999979e192 < a < -2.90000000000000019e179

   1. Initial program 51.6%

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

   2. Taylor expanded in y around 0 18.9%

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

      1. sub-neg 18.9%

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

      2. sub-neg 18.9%

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

      3. neg-mul-1 18.9%

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

      4. log1p-def 100.0%

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

      5. neg-mul-1 100.0%

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

      6. sub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

   10. Simplified 100.0%

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

   if -4.8e17 < a < 1.05e-32

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. *-commutative 76.2%

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

   4. Simplified 76.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+192}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;a \leq -2.9 \cdot 10^{+179}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-z\right)}\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{+17} \lor \neg \left(a \leq 1.05 \cdot 10^{-32}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \]

Alternative 5: 68.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ t_2 := \frac{x}{e^{y \cdot t}}\\ \mathbf{if}\;t \leq -2.35 \cdot 10^{+32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-260}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-271}:\\ \;\;\;\;x - x \cdot \left(a \cdot \left(z + b\right)\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))) (t_2 (/ x (exp (* y t)))))
   (if (<= t -2.35e+32)
     t_2
     (if (<= t -4.6e-260)
       t_1
       (if (<= t 5.6e-271)
         (- x (* x (* a (+ z b))))
         (if (<= t 4e+99) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double t_2 = x / exp((y * t));
	double tmp;
	if (t <= -2.35e+32) {
		tmp = t_2;
	} else if (t <= -4.6e-260) {
		tmp = t_1;
	} else if (t <= 5.6e-271) {
		tmp = x - (x * (a * (z + b)));
	} else if (t <= 4e+99) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * (z ** y)
    t_2 = x / exp((y * t))
    if (t <= (-2.35d+32)) then
        tmp = t_2
    else if (t <= (-4.6d-260)) then
        tmp = t_1
    else if (t <= 5.6d-271) then
        tmp = x - (x * (a * (z + b)))
    else if (t <= 4d+99) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.pow(z, y);
	double t_2 = x / Math.exp((y * t));
	double tmp;
	if (t <= -2.35e+32) {
		tmp = t_2;
	} else if (t <= -4.6e-260) {
		tmp = t_1;
	} else if (t <= 5.6e-271) {
		tmp = x - (x * (a * (z + b)));
	} else if (t <= 4e+99) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	t_2 = x / math.exp((y * t))
	tmp = 0
	if t <= -2.35e+32:
		tmp = t_2
	elif t <= -4.6e-260:
		tmp = t_1
	elif t <= 5.6e-271:
		tmp = x - (x * (a * (z + b)))
	elif t <= 4e+99:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	t_2 = Float64(x / exp(Float64(y * t)))
	tmp = 0.0
	if (t <= -2.35e+32)
		tmp = t_2;
	elseif (t <= -4.6e-260)
		tmp = t_1;
	elseif (t <= 5.6e-271)
		tmp = Float64(x - Float64(x * Float64(a * Float64(z + b))));
	elseif (t <= 4e+99)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	t_2 = x / exp((y * t));
	tmp = 0.0;
	if (t <= -2.35e+32)
		tmp = t_2;
	elseif (t <= -4.6e-260)
		tmp = t_1;
	elseif (t <= 5.6e-271)
		tmp = x - (x * (a * (z + b)));
	elseif (t <= 4e+99)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Exp[N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.35e+32], t$95$2, If[LessEqual[t, -4.6e-260], t$95$1, If[LessEqual[t, 5.6e-271], N[(x - N[(x * N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e+99], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.35000000000000012e32 or 3.9999999999999999e99 < t

   1. Initial program 98.8%

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

   2. Taylor expanded in t around inf 80.1%

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

      1. mul-1-neg 80.1%

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

      2. *-commutative 80.1%

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

   4. Simplified 80.1%

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

   5. Taylor expanded in x around 0 80.1%

      \[\leadsto \color{blue}{x \cdot e^{-t \cdot y}} \]
   6. Step-by-step derivation

      1. exp-neg 80.1%

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{t \cdot y}}} \]

      2. *-commutative 80.1%

         \[\leadsto x \cdot \frac{1}{e^{\color{blue}{y \cdot t}}} \]

      3. exp-prod 59.3%

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

      4. associate-*r/ 59.3%

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

      5. *-rgt-identity 59.3%

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

      6. exp-prod 80.1%

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

   7. Simplified 80.1%

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

   if -2.35000000000000012e32 < t < -4.6e-260 or 5.5999999999999995e-271 < t < 3.9999999999999999e99

   1. Initial program 95.1%

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

   2. Taylor expanded in y around inf 62.7%

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

   3. Taylor expanded in t around 0 62.7%

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

   if -4.6e-260 < t < 5.5999999999999995e-271

   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 y around 0 89.3%

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

      1. sub-neg 89.3%

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

      2. sub-neg 89.3%

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

      3. neg-mul-1 89.3%

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

      4. log1p-def 100.0%

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

      5. neg-mul-1 100.0%

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

      6. sub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 57.1%

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

      1. mul-1-neg 57.1%

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

      2. unsub-neg 57.1%

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

      3. *-commutative 57.1%

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

      4. associate-*l* 62.3%

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

      5. *-commutative 62.3%

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

   10. Simplified 62.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-260}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-271}:\\ \;\;\;\;x - x \cdot \left(a \cdot \left(z + b\right)\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+99}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \end{array} \]

Alternative 6: 68.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;t \leq -2.35 \cdot 10^{+32}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-260}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-271}:\\ \;\;\;\;x - x \cdot \left(a \cdot \left(z + b\right)\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= t -2.35e+32)
     (* x (exp (* t (- y))))
     (if (<= t -4.6e-260)
       t_1
       (if (<= t 5.6e-271)
         (- x (* x (* a (+ z b))))
         (if (<= t 5e+88) t_1 (/ x (exp (* y t)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double tmp;
	if (t <= -2.35e+32) {
		tmp = x * exp((t * -y));
	} else if (t <= -4.6e-260) {
		tmp = t_1;
	} else if (t <= 5.6e-271) {
		tmp = x - (x * (a * (z + b)));
	} else if (t <= 5e+88) {
		tmp = t_1;
	} else {
		tmp = x / exp((y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (z ** y)
    if (t <= (-2.35d+32)) then
        tmp = x * exp((t * -y))
    else if (t <= (-4.6d-260)) then
        tmp = t_1
    else if (t <= 5.6d-271) then
        tmp = x - (x * (a * (z + b)))
    else if (t <= 5d+88) then
        tmp = t_1
    else
        tmp = x / exp((y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.pow(z, y);
	double tmp;
	if (t <= -2.35e+32) {
		tmp = x * Math.exp((t * -y));
	} else if (t <= -4.6e-260) {
		tmp = t_1;
	} else if (t <= 5.6e-271) {
		tmp = x - (x * (a * (z + b)));
	} else if (t <= 5e+88) {
		tmp = t_1;
	} else {
		tmp = x / Math.exp((y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	tmp = 0
	if t <= -2.35e+32:
		tmp = x * math.exp((t * -y))
	elif t <= -4.6e-260:
		tmp = t_1
	elif t <= 5.6e-271:
		tmp = x - (x * (a * (z + b)))
	elif t <= 5e+88:
		tmp = t_1
	else:
		tmp = x / math.exp((y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	tmp = 0.0
	if (t <= -2.35e+32)
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	elseif (t <= -4.6e-260)
		tmp = t_1;
	elseif (t <= 5.6e-271)
		tmp = Float64(x - Float64(x * Float64(a * Float64(z + b))));
	elseif (t <= 5e+88)
		tmp = t_1;
	else
		tmp = Float64(x / exp(Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	tmp = 0.0;
	if (t <= -2.35e+32)
		tmp = x * exp((t * -y));
	elseif (t <= -4.6e-260)
		tmp = t_1;
	elseif (t <= 5.6e-271)
		tmp = x - (x * (a * (z + b)));
	elseif (t <= 5e+88)
		tmp = t_1;
	else
		tmp = x / exp((y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.35e+32], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.6e-260], t$95$1, If[LessEqual[t, 5.6e-271], N[(x - N[(x * N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e+88], t$95$1, N[(x / N[Exp[N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.35000000000000012e32

   1. Initial program 99.7%

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

   2. Taylor expanded in t around inf 74.0%

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

      1. mul-1-neg 74.0%

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

      2. *-commutative 74.0%

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

   4. Simplified 74.0%

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

   if -2.35000000000000012e32 < t < -4.6e-260 or 5.5999999999999995e-271 < t < 4.99999999999999997e88

   1. Initial program 95.1%

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

   2. Taylor expanded in y around inf 62.7%

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

   3. Taylor expanded in t around 0 62.7%

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

   if -4.6e-260 < t < 5.5999999999999995e-271

   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 y around 0 89.3%

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

      1. sub-neg 89.3%

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

      2. sub-neg 89.3%

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

      3. neg-mul-1 89.3%

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

      4. log1p-def 100.0%

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

      5. neg-mul-1 100.0%

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

      6. sub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 57.1%

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

      1. mul-1-neg 57.1%

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

      2. unsub-neg 57.1%

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

      3. *-commutative 57.1%

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

      4. associate-*l* 62.3%

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

      5. *-commutative 62.3%

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

   10. Simplified 62.3%

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

   if 4.99999999999999997e88 < t

   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 t around inf 87.0%

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

      1. mul-1-neg 87.0%

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

      2. *-commutative 87.0%

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

   4. Simplified 87.0%

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

   5. Taylor expanded in x around 0 87.0%

      \[\leadsto \color{blue}{x \cdot e^{-t \cdot y}} \]
   6. Step-by-step derivation

      1. exp-neg 87.0%

         \[\leadsto x \cdot \color{blue}{\frac{1}{e^{t \cdot y}}} \]

      2. *-commutative 87.0%

         \[\leadsto x \cdot \frac{1}{e^{\color{blue}{y \cdot t}}} \]

      3. exp-prod 60.7%

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

      4. associate-*r/ 60.7%

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

      5. *-rgt-identity 60.7%

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

      6. exp-prod 87.0%

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

   7. Simplified 87.0%

      \[\leadsto \color{blue}{\frac{x}{e^{y \cdot t}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{+32}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-260}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-271}:\\ \;\;\;\;x - x \cdot \left(a \cdot \left(z + b\right)\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+88}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{y \cdot t}}\\ \end{array} \]

Alternative 7: 74.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+28} \lor \neg \left(a \leq 5.7 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-\left(z + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -8.5e+28) (not (<= a 5.7e-36)))
   (* x (exp (* a (- (+ z b)))))
   (* x (exp (* t (- y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -8.5e+28) || !(a <= 5.7e-36)) {
		tmp = x * exp((a * -(z + b)));
	} else {
		tmp = x * exp((t * -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 ((a <= (-8.5d+28)) .or. (.not. (a <= 5.7d-36))) then
        tmp = x * exp((a * -(z + b)))
    else
        tmp = x * exp((t * -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 ((a <= -8.5e+28) || !(a <= 5.7e-36)) {
		tmp = x * Math.exp((a * -(z + b)));
	} else {
		tmp = x * Math.exp((t * -y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -8.5e+28) or not (a <= 5.7e-36):
		tmp = x * math.exp((a * -(z + b)))
	else:
		tmp = x * math.exp((t * -y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -8.5e+28) || !(a <= 5.7e-36))
		tmp = Float64(x * exp(Float64(a * Float64(-Float64(z + b)))));
	else
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -8.5e+28) || ~((a <= 5.7e-36)))
		tmp = x * exp((a * -(z + b)));
	else
		tmp = x * exp((t * -y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -8.5e+28], N[Not[LessEqual[a, 5.7e-36]], $MachinePrecision]], N[(x * N[Exp[N[(a * (-N[(z + b), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.49999999999999954e28 or 5.6999999999999999e-36 < a

   1. Initial program 93.0%

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

   2. Taylor expanded in y around 0 77.9%

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

      1. sub-neg 77.9%

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

      2. sub-neg 77.9%

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

      3. neg-mul-1 77.9%

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

      4. log1p-def 86.9%

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

      5. neg-mul-1 86.9%

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

      6. sub-neg 86.9%

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

   4. Simplified 86.9%

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

   5. Taylor expanded in z around 0 86.9%

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

      1. associate-*r* 86.9%

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

      2. associate-*r* 86.9%

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

      3. distribute-lft-out 86.9%

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

      4. neg-mul-1 86.9%

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

   7. Simplified 86.9%

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

   if -8.49999999999999954e28 < a < 5.6999999999999999e-36

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. *-commutative 76.2%

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

   4. Simplified 76.2%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+28} \lor \neg \left(a \leq 5.7 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-\left(z + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \]

Alternative 8: 69.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{+17} \lor \neg \left(a \leq 1.56 \cdot 10^{-31}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -9.2e+17) (not (<= a 1.56e-31)))
   (* x (exp (* a (- b))))
   (* x (exp (* t (- y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -9.2e+17) || !(a <= 1.56e-31)) {
		tmp = x * exp((a * -b));
	} else {
		tmp = x * exp((t * -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 ((a <= (-9.2d+17)) .or. (.not. (a <= 1.56d-31))) then
        tmp = x * exp((a * -b))
    else
        tmp = x * exp((t * -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 ((a <= -9.2e+17) || !(a <= 1.56e-31)) {
		tmp = x * Math.exp((a * -b));
	} else {
		tmp = x * Math.exp((t * -y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -9.2e+17) or not (a <= 1.56e-31):
		tmp = x * math.exp((a * -b))
	else:
		tmp = x * math.exp((t * -y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -9.2e+17) || !(a <= 1.56e-31))
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	else
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -9.2e+17) || ~((a <= 1.56e-31)))
		tmp = x * exp((a * -b));
	else
		tmp = x * exp((t * -y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -9.2e+17], N[Not[LessEqual[a, 1.56e-31]], $MachinePrecision]], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.2e17 or 1.5600000000000001e-31 < a

   1. Initial program 93.2%

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

   2. Taylor expanded in b around inf 77.9%

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

      1. mul-1-neg 77.9%

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

      2. distribute-rgt-neg-out 77.9%

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

   4. Simplified 77.9%

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

   if -9.2e17 < a < 1.5600000000000001e-31

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. *-commutative 76.2%

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

   4. Simplified 76.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{+17} \lor \neg \left(a \leq 1.56 \cdot 10^{-31}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \]

Alternative 9: 53.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{-161} \lor \neg \left(y \leq 1.8 \cdot 10^{+20}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(a \cdot \left(z + b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.85e-161) (not (<= y 1.8e+20)))
   (* x (pow z y))
   (- x (* x (* a (+ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.85e-161) || !(y <= 1.8e+20)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x - (x * (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 <= (-2.85d-161)) .or. (.not. (y <= 1.8d+20))) then
        tmp = x * (z ** y)
    else
        tmp = x - (x * (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 <= -2.85e-161) || !(y <= 1.8e+20)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x - (x * (a * (z + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.85e-161) or not (y <= 1.8e+20):
		tmp = x * math.pow(z, y)
	else:
		tmp = x - (x * (a * (z + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.85e-161) || !(y <= 1.8e+20))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x - Float64(x * 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 <= -2.85e-161) || ~((y <= 1.8e+20)))
		tmp = x * (z ^ y);
	else
		tmp = x - (x * (a * (z + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.85e-161], N[Not[LessEqual[y, 1.8e+20]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x - N[(x * N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.85000000000000011e-161 or 1.8e20 < y

   1. Initial program 98.6%

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

   2. Taylor expanded in y around inf 78.6%

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

   3. Taylor expanded in t around 0 63.9%

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

   if -2.85000000000000011e-161 < y < 1.8e20

   1. Initial program 92.9%

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

   2. Taylor expanded in y around 0 78.2%

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

      1. sub-neg 78.2%

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

      2. sub-neg 78.2%

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

      3. neg-mul-1 78.2%

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

      4. log1p-def 85.2%

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

      5. neg-mul-1 85.2%

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

      6. sub-neg 85.2%

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

   4. Simplified 85.2%

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

   5. Taylor expanded in z around 0 85.2%

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

      1. associate-*r* 85.2%

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

      2. associate-*r* 85.2%

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

      3. distribute-lft-out 85.2%

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

      4. neg-mul-1 85.2%

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

   7. Simplified 85.2%

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

   8. Taylor expanded in a around 0 47.5%

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

      1. mul-1-neg 47.5%

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

      2. unsub-neg 47.5%

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

      3. *-commutative 47.5%

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

      4. associate-*l* 50.2%

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

      5. *-commutative 50.2%

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

   10. Simplified 50.2%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{-161} \lor \neg \left(y \leq 1.8 \cdot 10^{+20}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(a \cdot \left(z + b\right)\right)\\ \end{array} \]

Alternative 10: 31.2% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-13} \lor \neg \left(y \leq 10^{+19}\right):\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.2e-13) (not (<= y 1e+19)))
   (* x (* a (- b)))
   (- x (* a (* x b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.2e-13) || !(y <= 1e+19)) {
		tmp = x * (a * -b);
	} else {
		tmp = x - (a * (x * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-6.2d-13)) .or. (.not. (y <= 1d+19))) then
        tmp = x * (a * -b)
    else
        tmp = x - (a * (x * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.2e-13) || !(y <= 1e+19)) {
		tmp = x * (a * -b);
	} else {
		tmp = x - (a * (x * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6.2e-13) or not (y <= 1e+19):
		tmp = x * (a * -b)
	else:
		tmp = x - (a * (x * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.2e-13) || !(y <= 1e+19))
		tmp = Float64(x * Float64(a * Float64(-b)));
	else
		tmp = Float64(x - Float64(a * Float64(x * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6.2e-13) || ~((y <= 1e+19)))
		tmp = x * (a * -b);
	else
		tmp = x - (a * (x * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.2e-13], N[Not[LessEqual[y, 1e+19]], $MachinePrecision]], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision], N[(x - N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1999999999999998e-13 or 1e19 < 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 b around inf 43.4%

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

      1. mul-1-neg 43.4%

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

      2. distribute-rgt-neg-out 43.4%

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

   4. Simplified 43.4%

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

   5. Taylor expanded in a around 0 10.3%

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

      1. mul-1-neg 10.3%

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

      2. unsub-neg 10.3%

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

   7. Simplified 10.3%

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

   8. Taylor expanded in a around inf 17.5%

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

      1. mul-1-neg 17.5%

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

      2. associate-*r* 23.8%

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

      3. distribute-rgt-neg-in 23.8%

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

   10. Simplified 23.8%

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

   if -6.1999999999999998e-13 < y < 1e19

   1. Initial program 93.4%

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

   2. Taylor expanded in b around inf 78.3%

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

      1. mul-1-neg 78.3%

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

      2. distribute-rgt-neg-out 78.3%

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

   4. Simplified 78.3%

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

   5. Taylor expanded in a around 0 41.1%

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

      1. mul-1-neg 41.1%

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

      2. unsub-neg 41.1%

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

   7. Simplified 41.1%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-13} \lor \neg \left(y \leq 10^{+19}\right):\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \end{array} \]

Alternative 11: 31.8% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+19}:\\ \;\;\;\;x - x \cdot \left(a \cdot \left(z + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 2.25e+19) (- x (* x (* a (+ z b)))) (* x (* a (- b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 2.25e+19) {
		tmp = x - (x * (a * (z + b)));
	} 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 <= 2.25d+19) then
        tmp = x - (x * (a * (z + b)))
    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 <= 2.25e+19) {
		tmp = x - (x * (a * (z + b)));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 2.25e+19:
		tmp = x - (x * (a * (z + b)))
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 2.25e+19)
		tmp = Float64(x - Float64(x * Float64(a * Float64(z + b))));
	else
		tmp = Float64(x * Float64(a * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 2.25e+19)
		tmp = x - (x * (a * (z + b)));
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 2.25e+19], N[(x - N[(x * N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.25e19

   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 y around 0 65.9%

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

      1. sub-neg 65.9%

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

      2. sub-neg 65.9%

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

      3. neg-mul-1 65.9%

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

      4. log1p-def 70.8%

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

      5. neg-mul-1 70.8%

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

      6. sub-neg 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in z around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. associate-*r* 70.8%

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

      3. distribute-lft-out 70.8%

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

      4. neg-mul-1 70.8%

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

   7. Simplified 70.8%

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

   8. Taylor expanded in a around 0 31.7%

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

      1. mul-1-neg 31.7%

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

      2. unsub-neg 31.7%

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

      3. *-commutative 31.7%

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

      4. associate-*l* 35.1%

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

      5. *-commutative 35.1%

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

   10. Simplified 35.1%

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

   if 2.25e19 < y

   1. Initial program 98.3%

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

   2. Taylor expanded in b around inf 46.5%

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

      1. mul-1-neg 46.5%

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

      2. distribute-rgt-neg-out 46.5%

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

   4. Simplified 46.5%

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

   5. Taylor expanded in a around 0 9.8%

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

      1. mul-1-neg 9.8%

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

      2. unsub-neg 9.8%

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

   7. Simplified 9.8%

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

   8. Taylor expanded in a around inf 21.6%

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

      1. mul-1-neg 21.6%

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

      2. associate-*r* 26.4%

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

      3. distribute-rgt-neg-in 26.4%

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

   10. Simplified 26.4%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+19}:\\ \;\;\;\;x - x \cdot \left(a \cdot \left(z + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]

Alternative 12: 26.2% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{-8} \lor \neg \left(y \leq 2.5 \cdot 10^{-137}\right):\\ \;\;\;\;\left(-a\right) \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.42e-8) (not (<= y 2.5e-137))) (* (- a) (* x b)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.42e-8) || !(y <= 2.5e-137)) {
		tmp = -a * (x * b);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.42d-8)) .or. (.not. (y <= 2.5d-137))) then
        tmp = -a * (x * b)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.42e-8) || !(y <= 2.5e-137)) {
		tmp = -a * (x * b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.42e-8) or not (y <= 2.5e-137):
		tmp = -a * (x * b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.42e-8) || !(y <= 2.5e-137))
		tmp = Float64(Float64(-a) * Float64(x * b));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.42e-8) || ~((y <= 2.5e-137)))
		tmp = -a * (x * b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.42e-8], N[Not[LessEqual[y, 2.5e-137]], $MachinePrecision]], N[((-a) * N[(x * b), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.41999999999999998e-8 or 2.5e-137 < y

   1. Initial program 98.1%

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

   2. Taylor expanded in b around inf 48.5%

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

      1. mul-1-neg 48.5%

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

      2. distribute-rgt-neg-out 48.5%

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

   4. Simplified 48.5%

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

   5. Taylor expanded in a around 0 14.9%

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

      1. mul-1-neg 14.9%

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

      2. unsub-neg 14.9%

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

   7. Simplified 14.9%

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

      1. expm1-log1p-u 9.1%

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

      2. expm1-udef 9.1%

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

      3. associate-*r* 10.9%

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

      4. *-commutative 10.9%

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

   9. Applied egg-rr 10.9%

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

       1. expm1-def 10.9%

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

       2. expm1-log1p 18.5%

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

   11. Simplified 18.5%

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

   12. Taylor expanded in a around inf 19.3%

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

       1. mul-1-neg 19.3%

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

       2. *-commutative 19.3%

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

   14. Simplified 19.3%

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

   if -1.41999999999999998e-8 < y < 2.5e-137

   1. Initial program 93.0%

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

   2. Taylor expanded in t around inf 49.1%

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

      1. mul-1-neg 49.1%

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

      2. *-commutative 49.1%

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

   4. Simplified 49.1%

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

   5. Taylor expanded in y around 0 37.8%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{-8} \lor \neg \left(y \leq 2.5 \cdot 10^{-137}\right):\\ \;\;\;\;\left(-a\right) \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13: 26.4% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-14} \lor \neg \left(y \leq 2.5 \cdot 10^{-137}\right):\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -5.5e-14) (not (<= y 2.5e-137))) (* x (* a (- b))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.5e-14) || !(y <= 2.5e-137)) {
		tmp = x * (a * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-5.5d-14)) .or. (.not. (y <= 2.5d-137))) then
        tmp = x * (a * -b)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.5e-14) || !(y <= 2.5e-137)) {
		tmp = x * (a * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -5.5e-14) or not (y <= 2.5e-137):
		tmp = x * (a * -b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -5.5e-14) || !(y <= 2.5e-137))
		tmp = Float64(x * Float64(a * Float64(-b)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -5.5e-14) || ~((y <= 2.5e-137)))
		tmp = x * (a * -b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -5.5e-14], N[Not[LessEqual[y, 2.5e-137]], $MachinePrecision]], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -5.49999999999999991e-14 or 2.5e-137 < y

   1. Initial program 98.1%

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

   2. Taylor expanded in b around inf 48.5%

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

      1. mul-1-neg 48.5%

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

      2. distribute-rgt-neg-out 48.5%

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

   4. Simplified 48.5%

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

   5. Taylor expanded in a around 0 14.9%

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

      1. mul-1-neg 14.9%

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

      2. unsub-neg 14.9%

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

   7. Simplified 14.9%

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

   8. Taylor expanded in a 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. associate-*r* 24.9%

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

      3. distribute-rgt-neg-in 24.9%

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

   10. Simplified 24.9%

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

   if -5.49999999999999991e-14 < y < 2.5e-137

   1. Initial program 93.0%

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

   2. Taylor expanded in t around inf 49.1%

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

      1. mul-1-neg 49.1%

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

      2. *-commutative 49.1%

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

   4. Simplified 49.1%

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

   5. Taylor expanded in y around 0 37.8%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-14} \lor \neg \left(y \leq 2.5 \cdot 10^{-137}\right):\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14: 26.3% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-89}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-137}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-a\right) \cdot \left(x \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.4e-89)
   (* t (* x (- y)))
   (if (<= y 2.4e-137) x (* (- a) (* x b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.4e-89) {
		tmp = t * (x * -y);
	} else if (y <= 2.4e-137) {
		tmp = x;
	} else {
		tmp = -a * (x * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-6.4d-89)) then
        tmp = t * (x * -y)
    else if (y <= 2.4d-137) then
        tmp = x
    else
        tmp = -a * (x * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.4e-89) {
		tmp = t * (x * -y);
	} else if (y <= 2.4e-137) {
		tmp = x;
	} else {
		tmp = -a * (x * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.4e-89:
		tmp = t * (x * -y)
	elif y <= 2.4e-137:
		tmp = x
	else:
		tmp = -a * (x * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.4e-89)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 2.4e-137)
		tmp = x;
	else
		tmp = Float64(Float64(-a) * Float64(x * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6.4e-89)
		tmp = t * (x * -y);
	elseif (y <= 2.4e-137)
		tmp = x;
	else
		tmp = -a * (x * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.4e-89], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e-137], x, N[((-a) * N[(x * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.39999999999999997e-89

   1. Initial program 98.8%

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

   2. Taylor expanded in t around inf 56.8%

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

      1. mul-1-neg 56.8%

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

      2. *-commutative 56.8%

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

   4. Simplified 56.8%

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

   5. Taylor expanded in y around 0 18.2%

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

      1. mul-1-neg 18.2%

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

      2. unsub-neg 18.2%

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

      3. associate-*r* 13.9%

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

      4. *-commutative 13.9%

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

   7. Simplified 13.9%

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

   8. Taylor expanded in y around inf 19.0%

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

      1. mul-1-neg 19.0%

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

      2. *-commutative 19.0%

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

      3. associate-*r* 18.1%

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

      4. distribute-rgt-neg-in 18.1%

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

      5. *-commutative 18.1%

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

      6. distribute-rgt-neg-in 18.1%

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

   10. Simplified 18.1%

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

   11. Taylor expanded in x around 0 19.0%

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

       1. mul-1-neg 19.0%

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

   13. Simplified 19.0%

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

   if -6.39999999999999997e-89 < y < 2.4e-137

   1. Initial program 92.8%

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

   2. Taylor expanded in t around inf 52.2%

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

      1. mul-1-neg 52.2%

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

      2. *-commutative 52.2%

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

   4. Simplified 52.2%

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

   5. Taylor expanded in y around 0 42.2%

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

   if 2.4e-137 < y

   1. Initial program 96.9%

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

   2. Taylor expanded in b around inf 54.1%

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

      1. mul-1-neg 54.1%

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

      2. distribute-rgt-neg-out 54.1%

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

   4. Simplified 54.1%

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

   5. Taylor expanded in a around 0 17.7%

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

      1. mul-1-neg 17.7%

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

      2. unsub-neg 17.7%

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

   7. Simplified 17.7%

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

      1. expm1-log1p-u 11.7%

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

      2. expm1-udef 11.7%

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

      3. associate-*r* 13.6%

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

      4. *-commutative 13.6%

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

   9. Applied egg-rr 13.6%

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

       1. expm1-def 13.6%

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

       2. expm1-log1p 19.8%

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

   11. Simplified 19.8%

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

   12. Taylor expanded in a around inf 23.2%

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

       1. mul-1-neg 23.2%

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

       2. *-commutative 23.2%

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

   14. Simplified 23.2%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-89}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-137}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-a\right) \cdot \left(x \cdot b\right)\\ \end{array} \]

Alternative 15: 31.4% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+19}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 1e+19) (- x (* x (* a b))) (* x (* a (- b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 1e+19) {
		tmp = x - (x * (a * b));
	} 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 <= 1d+19) then
        tmp = x - (x * (a * b))
    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 <= 1e+19) {
		tmp = x - (x * (a * b));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 1e+19:
		tmp = x - (x * (a * b))
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 1e+19)
		tmp = Float64(x - Float64(x * Float64(a * b)));
	else
		tmp = Float64(x * Float64(a * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 1e+19)
		tmp = x - (x * (a * b));
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 1e+19], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e19

   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 b around inf 65.9%

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

      1. mul-1-neg 65.9%

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

      2. distribute-rgt-neg-out 65.9%

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

   4. Simplified 65.9%

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

   5. Taylor expanded in a around 0 31.2%

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

      1. mul-1-neg 31.2%

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

      2. unsub-neg 31.2%

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

   7. Simplified 31.2%

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

      1. expm1-log1p-u 25.5%

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

      2. expm1-udef 25.4%

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

      3. associate-*r* 26.8%

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

      4. *-commutative 26.8%

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

   9. Applied egg-rr 26.8%

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

       1. expm1-def 26.9%

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

       2. expm1-log1p 34.5%

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

   11. Simplified 34.5%

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

   if 1e19 < y

   1. Initial program 98.3%

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

   2. Taylor expanded in b around inf 46.5%

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

      1. mul-1-neg 46.5%

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

      2. distribute-rgt-neg-out 46.5%

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

   4. Simplified 46.5%

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

   5. Taylor expanded in a around 0 9.8%

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

      1. mul-1-neg 9.8%

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

      2. unsub-neg 9.8%

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

   7. Simplified 9.8%

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

   8. Taylor expanded in a around inf 21.6%

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

      1. mul-1-neg 21.6%

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

      2. associate-*r* 26.4%

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

      3. distribute-rgt-neg-in 26.4%

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

   10. Simplified 26.4%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+19}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]

Alternative 16: 21.1% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.85 \cdot 10^{+145}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a 2.85e+145) x (* t (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 2.85e+145) {
		tmp = x;
	} else {
		tmp = t * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 2.85d+145) then
        tmp = x
    else
        tmp = t * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 2.85e+145) {
		tmp = x;
	} else {
		tmp = t * (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= 2.85e+145:
		tmp = x
	else:
		tmp = t * (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= 2.85e+145)
		tmp = x;
	else
		tmp = Float64(t * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= 2.85e+145)
		tmp = x;
	else
		tmp = t * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, 2.85e+145], x, N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.85e145

   1. Initial program 96.3%

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

   2. Taylor expanded in t around inf 59.5%

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

      1. mul-1-neg 59.5%

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

      2. *-commutative 59.5%

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

   4. Simplified 59.5%

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

   5. Taylor expanded in y around 0 20.7%

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

   if 2.85e145 < a

   1. Initial program 95.4%

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

   2. Taylor expanded in t around inf 26.0%

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

      1. mul-1-neg 26.0%

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

      2. *-commutative 26.0%

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

   4. Simplified 26.0%

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

   5. Taylor expanded in y around 0 3.8%

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

      1. mul-1-neg 3.8%

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

      2. unsub-neg 3.8%

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

      3. associate-*r* 3.7%

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

      4. *-commutative 3.7%

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

   7. Simplified 3.7%

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

   8. Taylor expanded in y around inf 19.6%

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

      1. mul-1-neg 19.6%

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

      2. *-commutative 19.6%

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

      3. associate-*r* 24.5%

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

      4. distribute-rgt-neg-in 24.5%

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

      5. *-commutative 24.5%

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

      6. distribute-rgt-neg-in 24.5%

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

   10. Simplified 24.5%

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

       1. expm1-log1p-u 21.4%

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

       2. expm1-udef 31.9%

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

       3. associate-*r* 31.8%

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

       4. *-commutative 31.8%

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

       5. add-sqr-sqrt 18.5%

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

       6. sqrt-unprod 41.5%

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

       7. sqr-neg 41.5%

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

       8. sqrt-unprod 13.3%

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

       9. add-sqr-sqrt 31.6%

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

   12. Applied egg-rr 31.6%

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

       1. expm1-def 16.4%

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

       2. expm1-log1p 19.3%

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

       3. *-commutative 19.3%

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

       4. *-commutative 19.3%

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

       5. associate-*r* 19.2%

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

   14. Simplified 19.2%

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

4. Final simplification 20.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.85 \cdot 10^{+145}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 17: 20.9% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{+145}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= a 2.8e+145) x (* x (* y t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 2.8e+145) {
		tmp = x;
	} else {
		tmp = x * (y * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 2.8d+145) then
        tmp = x
    else
        tmp = x * (y * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 2.8e+145) {
		tmp = x;
	} else {
		tmp = x * (y * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= 2.8e+145:
		tmp = x
	else:
		tmp = x * (y * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= 2.8e+145)
		tmp = x;
	else
		tmp = Float64(x * Float64(y * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= 2.8e+145)
		tmp = x;
	else
		tmp = x * (y * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, 2.8e+145], x, N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.7999999999999999e145

   1. Initial program 96.3%

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

   2. Taylor expanded in t around inf 59.5%

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

      1. mul-1-neg 59.5%

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

      2. *-commutative 59.5%

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

   4. Simplified 59.5%

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

   5. Taylor expanded in y around 0 20.7%

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

   if 2.7999999999999999e145 < a

   1. Initial program 95.4%

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

   2. Taylor expanded in t around inf 26.0%

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

      1. mul-1-neg 26.0%

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

      2. *-commutative 26.0%

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

   4. Simplified 26.0%

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

   5. Taylor expanded in y around 0 3.8%

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

      1. mul-1-neg 3.8%

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

      2. unsub-neg 3.8%

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

      3. associate-*r* 3.7%

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

      4. *-commutative 3.7%

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

   7. Simplified 3.7%

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

   8. Taylor expanded in y around inf 19.6%

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

      1. mul-1-neg 19.6%

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

      2. *-commutative 19.6%

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

      3. associate-*r* 24.5%

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

      4. distribute-rgt-neg-in 24.5%

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

      5. *-commutative 24.5%

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

      6. distribute-rgt-neg-in 24.5%

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

   10. Simplified 24.5%

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

       1. expm1-log1p-u 21.4%

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

       2. expm1-udef 31.9%

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

       3. associate-*r* 31.8%

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

       4. *-commutative 31.8%

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

       5. add-sqr-sqrt 18.5%

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

       6. sqrt-unprod 41.5%

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

       7. sqr-neg 41.5%

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

       8. sqrt-unprod 13.3%

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

       9. add-sqr-sqrt 31.6%

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

   12. Applied egg-rr 31.6%

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

       1. expm1-def 16.4%

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

       2. expm1-log1p 19.3%

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

       3. *-commutative 19.3%

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

       4. associate-*l* 21.6%

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

   14. Simplified 21.6%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{+145}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 18: 19.6% accurate, 315.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

2. Taylor expanded in t around inf 54.4%

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

   1. mul-1-neg 54.4%

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

   2. *-commutative 54.4%

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

4. Simplified 54.4%

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

5. Taylor expanded in y around 0 18.1%

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

6. Final simplification 18.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023333 
(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))))))