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

Percentage Accurate: 96.3% → 99.6%

Time: 18.0s

Alternatives: 19

Speedup: 1.0×

• 20.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. +-commutative 97.3%

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

   2. fma-def 97.7%

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

   3. sub-neg 97.7%

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

   4. log1p-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

2. Final simplification 97.3%

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

Alternative 3: 91.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+33} \lor \neg \left(y \leq 1.95 \cdot 10^{+58}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right) - y \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.8e+33) (not (<= y 1.95e+58)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (- (* a (- b)) (* y t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.8e+33) || !(y <= 1.95e+58)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp(((a * -b) - (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 ((y <= (-1.8d+33)) .or. (.not. (y <= 1.95d+58))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp(((a * -b) - (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 ((y <= -1.8e+33) || !(y <= 1.95e+58)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp(((a * -b) - (y * t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.8e+33) or not (y <= 1.95e+58):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp(((a * -b) - (y * t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.8e+33) || !(y <= 1.95e+58))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(Float64(a * Float64(-b)) - Float64(y * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.8e+33) || ~((y <= 1.95e+58)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp(((a * -b) - (y * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.8e+33], N[Not[LessEqual[y, 1.95e+58]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[(a * (-b)), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8000000000000001e33 or 1.95000000000000005e58 < y

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 95.1%

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

   if -1.8000000000000001e33 < y < 1.95000000000000005e58

   1. Initial program 95.7%

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

   2. Taylor expanded in z around 0 94.9%

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

   3. Taylor expanded in t around inf 93.5%

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

      1. neg-mul-1 93.5%

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

   5. Simplified 93.5%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+33} \lor \neg \left(y \leq 1.95 \cdot 10^{+58}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right) - y \cdot t}\\ \end{array} \]

Alternative 4: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

2. Taylor expanded in z around 0 96.9%

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

3. Final simplification 96.9%

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

Alternative 5: 72.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(-b\right)}\\ t_2 := x \cdot e^{y \cdot \left(-t\right)}\\ t_3 := x \cdot {z}^{y}\\ \mathbf{if}\;t \leq -2.6 \cdot 10^{+48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.25 \cdot 10^{-166}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+37}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- b)))))
        (t_2 (* x (exp (* y (- t)))))
        (t_3 (* x (pow z y))))
   (if (<= t -2.6e+48)
     t_2
     (if (<= t -1.6e-13)
       t_1
       (if (<= t -2.25e-166)
         t_3
         (if (<= t -3.5e-242) t_1 (if (<= t 2.5e+37) t_3 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * -b));
	double t_2 = x * exp((y * -t));
	double t_3 = x * pow(z, y);
	double tmp;
	if (t <= -2.6e+48) {
		tmp = t_2;
	} else if (t <= -1.6e-13) {
		tmp = t_1;
	} else if (t <= -2.25e-166) {
		tmp = t_3;
	} else if (t <= -3.5e-242) {
		tmp = t_1;
	} else if (t <= 2.5e+37) {
		tmp = t_3;
	} 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) :: t_3
    real(8) :: tmp
    t_1 = x * exp((a * -b))
    t_2 = x * exp((y * -t))
    t_3 = x * (z ** y)
    if (t <= (-2.6d+48)) then
        tmp = t_2
    else if (t <= (-1.6d-13)) then
        tmp = t_1
    else if (t <= (-2.25d-166)) then
        tmp = t_3
    else if (t <= (-3.5d-242)) then
        tmp = t_1
    else if (t <= 2.5d+37) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((a * -b));
	double t_2 = x * Math.exp((y * -t));
	double t_3 = x * Math.pow(z, y);
	double tmp;
	if (t <= -2.6e+48) {
		tmp = t_2;
	} else if (t <= -1.6e-13) {
		tmp = t_1;
	} else if (t <= -2.25e-166) {
		tmp = t_3;
	} else if (t <= -3.5e-242) {
		tmp = t_1;
	} else if (t <= 2.5e+37) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * -b))
	t_2 = x * math.exp((y * -t))
	t_3 = x * math.pow(z, y)
	tmp = 0
	if t <= -2.6e+48:
		tmp = t_2
	elif t <= -1.6e-13:
		tmp = t_1
	elif t <= -2.25e-166:
		tmp = t_3
	elif t <= -3.5e-242:
		tmp = t_1
	elif t <= 2.5e+37:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(-b))))
	t_2 = Float64(x * exp(Float64(y * Float64(-t))))
	t_3 = Float64(x * (z ^ y))
	tmp = 0.0
	if (t <= -2.6e+48)
		tmp = t_2;
	elseif (t <= -1.6e-13)
		tmp = t_1;
	elseif (t <= -2.25e-166)
		tmp = t_3;
	elseif (t <= -3.5e-242)
		tmp = t_1;
	elseif (t <= 2.5e+37)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * -b));
	t_2 = x * exp((y * -t));
	t_3 = x * (z ^ y);
	tmp = 0.0;
	if (t <= -2.6e+48)
		tmp = t_2;
	elseif (t <= -1.6e-13)
		tmp = t_1;
	elseif (t <= -2.25e-166)
		tmp = t_3;
	elseif (t <= -3.5e-242)
		tmp = t_1;
	elseif (t <= 2.5e+37)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.6e+48], t$95$2, If[LessEqual[t, -1.6e-13], t$95$1, If[LessEqual[t, -2.25e-166], t$95$3, If[LessEqual[t, -3.5e-242], t$95$1, If[LessEqual[t, 2.5e+37], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.59999999999999995e48 or 2.49999999999999994e37 < t

   1. Initial program 97.5%

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

   2. Taylor expanded in t around inf 87.9%

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

      1. mul-1-neg 87.9%

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

      2. distribute-rgt-neg-out 87.9%

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

   4. Simplified 87.9%

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

   if -2.59999999999999995e48 < t < -1.6e-13 or -2.2499999999999999e-166 < t < -3.4999999999999999e-242

   1. Initial program 99.9%

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

   2. Taylor expanded in b around inf 88.9%

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

      1. associate-*r* 88.9%

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

      2. neg-mul-1 88.9%

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

      3. *-commutative 88.9%

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

   4. Simplified 88.9%

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

   if -1.6e-13 < t < -2.2499999999999999e-166 or -3.4999999999999999e-242 < t < 2.49999999999999994e37

   1. Initial program 96.5%

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

   2. Taylor expanded in y around inf 71.3%

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

   3. Taylor expanded in t around 0 71.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+48}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-13}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;t \leq -2.25 \cdot 10^{-166}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-242}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+37}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]

Alternative 6: 74.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ t_2 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;y \leq -9 \cdot 10^{+129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -330:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-25}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+194}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))) (t_2 (* x (exp (* y (- t))))))
   (if (<= y -9e+129)
     t_2
     (if (<= y -330.0)
       t_1
       (if (<= y 1.3e-25)
         (* x (exp (* a (- (- z) b))))
         (if (<= y 6.6e+194) t_2 t_1))))))

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 (y <= -9e+129) {
		tmp = t_2;
	} else if (y <= -330.0) {
		tmp = t_1;
	} else if (y <= 1.3e-25) {
		tmp = x * exp((a * (-z - b)));
	} else if (y <= 6.6e+194) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * (z ** y)
    t_2 = x * exp((y * -t))
    if (y <= (-9d+129)) then
        tmp = t_2
    else if (y <= (-330.0d0)) then
        tmp = t_1
    else if (y <= 1.3d-25) then
        tmp = x * exp((a * (-z - b)))
    else if (y <= 6.6d+194) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.pow(z, y);
	double t_2 = x * Math.exp((y * -t));
	double tmp;
	if (y <= -9e+129) {
		tmp = t_2;
	} else if (y <= -330.0) {
		tmp = t_1;
	} else if (y <= 1.3e-25) {
		tmp = x * Math.exp((a * (-z - b)));
	} else if (y <= 6.6e+194) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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 y <= -9e+129:
		tmp = t_2
	elif y <= -330.0:
		tmp = t_1
	elif y <= 1.3e-25:
		tmp = x * math.exp((a * (-z - b)))
	elif y <= 6.6e+194:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	t_2 = Float64(x * exp(Float64(y * Float64(-t))))
	tmp = 0.0
	if (y <= -9e+129)
		tmp = t_2;
	elseif (y <= -330.0)
		tmp = t_1;
	elseif (y <= 1.3e-25)
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))));
	elseif (y <= 6.6e+194)
		tmp = t_2;
	else
		tmp = t_1;
	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 (y <= -9e+129)
		tmp = t_2;
	elseif (y <= -330.0)
		tmp = t_1;
	elseif (y <= 1.3e-25)
		tmp = x * exp((a * (-z - b)));
	elseif (y <= 6.6e+194)
		tmp = t_2;
	else
		tmp = t_1;
	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[y, -9e+129], t$95$2, If[LessEqual[y, -330.0], t$95$1, If[LessEqual[y, 1.3e-25], N[(x * N[Exp[N[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.6e+194], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.0000000000000003e129 or 1.3e-25 < y < 6.59999999999999967e194

   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 78.2%

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

      1. mul-1-neg 78.2%

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

      2. distribute-rgt-neg-out 78.2%

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

   4. Simplified 78.2%

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

   if -9.0000000000000003e129 < y < -330 or 6.59999999999999967e194 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 93.9%

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

   3. Taylor expanded in t around 0 78.8%

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

   if -330 < y < 1.3e-25

   1. Initial program 94.7%

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

   2. Taylor expanded in y around 0 79.0%

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

      1. sub-neg 79.0%

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

      2. neg-mul-1 79.0%

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

      3. log1p-def 84.2%

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

      4. neg-mul-1 84.2%

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

   4. Simplified 84.2%

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

   5. Taylor expanded in z around 0 84.2%

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

      1. +-commutative 84.2%

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

      2. associate-*r* 84.2%

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

      3. associate-*r* 84.2%

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

      4. distribute-lft-out 84.2%

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

      5. neg-mul-1 84.2%

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

   7. Simplified 84.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+129}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq -330:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-25}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+194}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]

Alternative 7: 72.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \lor \neg \left(y \leq 9.8 \cdot 10^{+58}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.6) (not (<= y 9.8e+58)))
   (* x (pow z y))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.6) || !(y <= 9.8e+58)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-6.6d0)) .or. (.not. (y <= 9.8d+58))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.6) || !(y <= 9.8e+58)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6.6) or not (y <= 9.8e+58):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.6) || !(y <= 9.8e+58))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6.6) || ~((y <= 9.8e+58)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.6], N[Not[LessEqual[y, 9.8e+58]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5999999999999996 or 9.80000000000000037e58 < y

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 93.9%

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

   3. Taylor expanded in t around 0 71.0%

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

   if -6.5999999999999996 < y < 9.80000000000000037e58

   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 b around inf 74.0%

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

      1. associate-*r* 74.0%

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

      2. neg-mul-1 74.0%

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

      3. *-commutative 74.0%

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

   4. Simplified 74.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \lor \neg \left(y \leq 9.8 \cdot 10^{+58}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]

Alternative 8: 84.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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(((a * -b) - (y * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

2. Taylor expanded in z around 0 96.9%

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

3. Taylor expanded in t around inf 84.4%

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

   1. neg-mul-1 84.4%

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

5. Simplified 84.4%

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

6. Final simplification 84.4%

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

Alternative 9: 53.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -3e-13) (* x (* y (- t))) (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -3e-13) {
		tmp = x * (y * -t);
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-3d-13)) then
        tmp = x * (y * -t)
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -3e-13) {
		tmp = x * (y * -t);
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -3e-13:
		tmp = x * (y * -t)
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -3e-13)
		tmp = Float64(x * Float64(y * Float64(-t)));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -3e-13)
		tmp = x * (y * -t);
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -3e-13], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.99999999999999984e-13

   1. Initial program 97.1%

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

   2. Taylor expanded in t around inf 80.9%

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

      1. mul-1-neg 80.9%

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

      2. distribute-rgt-neg-out 80.9%

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

   4. Simplified 80.9%

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

   5. Taylor expanded in y around 0 26.7%

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

   6. Taylor expanded in y around inf 28.6%

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

      1. neg-mul-1 28.6%

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

      2. associate-*r* 31.5%

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

      3. distribute-rgt-neg-in 31.5%

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

   8. Simplified 31.5%

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

   if -2.99999999999999984e-13 < t

   1. Initial program 97.4%

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

   2. Taylor expanded in y around inf 75.3%

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

   3. Taylor expanded in t around 0 67.3%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]

Alternative 10: 33.9% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.15e+50)
   (* x (* y (- t)))
   (if (<= y 4.6e-14) (* x (- 1.0 (* a b))) (* (- y) (* x t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.15e+50) {
		tmp = x * (y * -t);
	} else if (y <= 4.6e-14) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -y * (x * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-1.15d+50)) then
        tmp = x * (y * -t)
    else if (y <= 4.6d-14) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = -y * (x * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.15e+50) {
		tmp = x * (y * -t);
	} else if (y <= 4.6e-14) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -y * (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.15e+50:
		tmp = x * (y * -t)
	elif y <= 4.6e-14:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = -y * (x * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.15e+50)
		tmp = Float64(x * Float64(y * Float64(-t)));
	elseif (y <= 4.6e-14)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(Float64(-y) * Float64(x * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.15e+50)
		tmp = x * (y * -t);
	elseif (y <= 4.6e-14)
		tmp = x * (1.0 - (a * b));
	else
		tmp = -y * (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.15e+50], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e-14], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-y) * N[(x * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.14999999999999998e50

   1. Initial program 98.2%

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

   2. Taylor expanded in t around inf 72.2%

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

      1. mul-1-neg 72.2%

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

      2. distribute-rgt-neg-out 72.2%

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

   4. Simplified 72.2%

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

   5. Taylor expanded in y around 0 20.5%

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

   6. Taylor expanded in y around inf 23.6%

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

      1. neg-mul-1 23.6%

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

      2. associate-*r* 27.1%

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

      3. distribute-rgt-neg-in 27.1%

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

   8. Simplified 27.1%

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

   if -1.14999999999999998e50 < y < 4.59999999999999996e-14

   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 b around inf 71.9%

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

      1. associate-*r* 71.9%

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

      2. neg-mul-1 71.9%

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

      3. *-commutative 71.9%

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

   4. Simplified 71.9%

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

   5. Taylor expanded in b around 0 37.3%

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

      1. +-commutative 37.3%

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

      2. mul-1-neg 37.3%

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

      3. unsub-neg 37.3%

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

   7. Simplified 37.3%

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

   8. Taylor expanded in x around 0 37.8%

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

   if 4.59999999999999996e-14 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 69.1%

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

      1. mul-1-neg 69.1%

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

      2. distribute-rgt-neg-out 69.1%

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

   4. Simplified 69.1%

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

   5. Taylor expanded in y around 0 23.6%

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

   6. Taylor expanded in y around inf 39.1%

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

      1. neg-mul-1 39.1%

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

      2. distribute-lft-neg-in 39.1%

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

   8. Simplified 39.1%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 11: 32.8% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-171}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -7.4e-171)
   (* x (- 1.0 (* y t)))
   (if (<= y 4.8e-14) (* x (- 1.0 (* a b))) (* (- y) (* x t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7.4e-171) {
		tmp = x * (1.0 - (y * t));
	} else if (y <= 4.8e-14) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -y * (x * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-7.4d-171)) then
        tmp = x * (1.0d0 - (y * t))
    else if (y <= 4.8d-14) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = -y * (x * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7.4e-171) {
		tmp = x * (1.0 - (y * t));
	} else if (y <= 4.8e-14) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -y * (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -7.4e-171:
		tmp = x * (1.0 - (y * t))
	elif y <= 4.8e-14:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = -y * (x * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -7.4e-171)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	elseif (y <= 4.8e-14)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(Float64(-y) * Float64(x * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -7.4e-171)
		tmp = x * (1.0 - (y * t));
	elseif (y <= 4.8e-14)
		tmp = x * (1.0 - (a * b));
	else
		tmp = -y * (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -7.4e-171], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e-14], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-y) * N[(x * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.40000000000000024e-171

   1. Initial program 97.2%

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

   2. Taylor expanded in t around inf 65.3%

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

      1. mul-1-neg 65.3%

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

      2. distribute-rgt-neg-out 65.3%

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

   4. Simplified 65.3%

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

   5. Taylor expanded in y around 0 19.2%

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

      1. +-commutative 19.2%

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

      2. mul-1-neg 19.2%

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

      3. unsub-neg 19.2%

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

      4. associate-*r* 22.7%

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

      5. *-commutative 22.7%

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

      6. associate-*l* 23.6%

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

   7. Simplified 23.6%

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

   8. Taylor expanded in x around 0 22.7%

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

   if -7.40000000000000024e-171 < y < 4.8e-14

   1. Initial program 94.9%

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

   2. Taylor expanded in b around inf 84.8%

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

      1. associate-*r* 84.8%

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

      2. neg-mul-1 84.8%

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

      3. *-commutative 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in b around 0 51.1%

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

      1. +-commutative 51.1%

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

      2. mul-1-neg 51.1%

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

      3. unsub-neg 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in x around 0 52.1%

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

   if 4.8e-14 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 69.1%

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

      1. mul-1-neg 69.1%

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

      2. distribute-rgt-neg-out 69.1%

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

   4. Simplified 69.1%

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

   5. Taylor expanded in y around 0 23.6%

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

   6. Taylor expanded in y around inf 39.1%

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

      1. neg-mul-1 39.1%

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

      2. distribute-lft-neg-in 39.1%

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

   8. Simplified 39.1%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-171}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 12: 32.3% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-173}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.8e-173)
   (- x (* t (* x y)))
   (if (<= y 4.8e-14) (* x (- 1.0 (* a b))) (* (- y) (* x t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.8e-173) {
		tmp = x - (t * (x * y));
	} else if (y <= 4.8e-14) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -y * (x * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.8d-173)) then
        tmp = x - (t * (x * y))
    else if (y <= 4.8d-14) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = -y * (x * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.8e-173) {
		tmp = x - (t * (x * y));
	} else if (y <= 4.8e-14) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -y * (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.8e-173:
		tmp = x - (t * (x * y))
	elif y <= 4.8e-14:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = -y * (x * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.8e-173)
		tmp = Float64(x - Float64(t * Float64(x * y)));
	elseif (y <= 4.8e-14)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(Float64(-y) * Float64(x * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.8e-173)
		tmp = x - (t * (x * y));
	elseif (y <= 4.8e-14)
		tmp = x * (1.0 - (a * b));
	else
		tmp = -y * (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.8e-173], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e-14], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-y) * N[(x * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7999999999999999e-173

   1. Initial program 97.2%

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

   2. Taylor expanded in t around inf 65.3%

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

      1. mul-1-neg 65.3%

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

      2. distribute-rgt-neg-out 65.3%

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

   4. Simplified 65.3%

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

   5. Taylor expanded in y around 0 19.2%

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

      1. +-commutative 19.2%

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

      2. mul-1-neg 19.2%

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

      3. unsub-neg 19.2%

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

      4. associate-*r* 22.7%

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

      5. *-commutative 22.7%

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

      6. associate-*l* 23.6%

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

   7. Simplified 23.6%

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

   if -2.7999999999999999e-173 < y < 4.8e-14

   1. Initial program 94.9%

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

   2. Taylor expanded in b around inf 84.8%

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

      1. associate-*r* 84.8%

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

      2. neg-mul-1 84.8%

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

      3. *-commutative 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in b around 0 51.1%

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

      1. +-commutative 51.1%

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

      2. mul-1-neg 51.1%

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

      3. unsub-neg 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in x around 0 52.1%

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

   if 4.8e-14 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 69.1%

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

      1. mul-1-neg 69.1%

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

      2. distribute-rgt-neg-out 69.1%

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

   4. Simplified 69.1%

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

   5. Taylor expanded in y around 0 23.6%

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

   6. Taylor expanded in y around inf 39.1%

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

      1. neg-mul-1 39.1%

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

      2. distribute-lft-neg-in 39.1%

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

   8. Simplified 39.1%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-173}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 13: 27.4% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-94} \lor \neg \left(y \leq 2.45 \cdot 10^{-63}\right):\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3.2e-94) (not (<= y 2.45e-63))) (* (* a b) (- x)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.2e-94) || !(y <= 2.45e-63)) {
		tmp = (a * b) * -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-3.2d-94)) .or. (.not. (y <= 2.45d-63))) then
        tmp = (a * b) * -x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.2e-94) || !(y <= 2.45e-63)) {
		tmp = (a * b) * -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -3.2e-94) or not (y <= 2.45e-63):
		tmp = (a * b) * -x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3.2e-94) || !(y <= 2.45e-63))
		tmp = Float64(Float64(a * b) * Float64(-x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -3.2e-94) || ~((y <= 2.45e-63)))
		tmp = (a * b) * -x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3.2e-94], N[Not[LessEqual[y, 2.45e-63]], $MachinePrecision]], N[(N[(a * b), $MachinePrecision] * (-x)), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -3.19999999999999997e-94 or 2.45000000000000008e-63 < 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 38.1%

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

      1. associate-*r* 38.1%

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

      2. neg-mul-1 38.1%

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

      3. *-commutative 38.1%

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

   4. Simplified 38.1%

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

   5. Taylor expanded in b around 0 10.7%

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

      1. +-commutative 10.7%

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

      2. mul-1-neg 10.7%

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

      3. unsub-neg 10.7%

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

   7. Simplified 10.7%

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

   8. Taylor expanded in a around inf 15.6%

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

      1. neg-mul-1 15.6%

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

      2. distribute-rgt-neg-in 15.6%

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

      3. *-commutative 15.6%

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

      4. distribute-rgt-neg-in 15.6%

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

   10. Simplified 15.6%

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

       1. associate-*r* 18.8%

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

       2. distribute-rgt-neg-out 18.8%

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

       3. add-sqr-sqrt 10.7%

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

       4. sqrt-unprod 20.8%

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

       5. sqr-neg 20.8%

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

       6. sqrt-unprod 4.5%

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

       7. add-sqr-sqrt 11.8%

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

       8. associate-*r* 12.4%

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

       9. *-commutative 12.4%

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

       10. associate-*l* 11.8%

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

       11. add-sqr-sqrt 3.0%

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

       12. sqrt-unprod 21.1%

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

       13. sqr-neg 21.1%

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

       14. sqrt-unprod 10.6%

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

       15. add-sqr-sqrt 17.1%

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

   12. Applied egg-rr 17.1%

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

   if -3.19999999999999997e-94 < y < 2.45000000000000008e-63

   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 b around inf 86.1%

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

      1. associate-*r* 86.1%

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

      2. neg-mul-1 86.1%

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

      3. *-commutative 86.1%

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

   4. Simplified 86.1%

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

   5. Taylor expanded in b around 0 41.8%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-94} \lor \neg \left(y \leq 2.45 \cdot 10^{-63}\right):\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14: 28.3% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+34} \lor \neg \left(y \leq 4.6 \cdot 10^{-14}\right):\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.2e+34) (not (<= y 4.6e-14))) (* (- y) (* x t)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.2e+34) || !(y <= 4.6e-14)) {
		tmp = -y * (x * t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-6.2d+34)) .or. (.not. (y <= 4.6d-14))) then
        tmp = -y * (x * t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.2e+34) || !(y <= 4.6e-14)) {
		tmp = -y * (x * t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6.2e+34) or not (y <= 4.6e-14):
		tmp = -y * (x * t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.2e+34) || !(y <= 4.6e-14))
		tmp = Float64(Float64(-y) * Float64(x * t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6.2e+34) || ~((y <= 4.6e-14)))
		tmp = -y * (x * t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.2e+34], N[Not[LessEqual[y, 4.6e-14]], $MachinePrecision]], N[((-y) * N[(x * t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -6.19999999999999955e34 or 4.59999999999999996e-14 < 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 t around inf 69.4%

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

      1. mul-1-neg 69.4%

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

      2. distribute-rgt-neg-out 69.4%

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

   4. Simplified 69.4%

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

   5. Taylor expanded in y around 0 22.2%

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

   6. Taylor expanded in y around inf 31.9%

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

      1. neg-mul-1 31.9%

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

      2. distribute-lft-neg-in 31.9%

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

   8. Simplified 31.9%

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

   if -6.19999999999999955e34 < y < 4.59999999999999996e-14

   1. Initial program 95.2%

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

   2. Taylor expanded in b around inf 73.2%

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

      1. associate-*r* 73.2%

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

      2. neg-mul-1 73.2%

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

      3. *-commutative 73.2%

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

   4. Simplified 73.2%

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

   5. Taylor expanded in b around 0 30.4%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+34} \lor \neg \left(y \leq 4.6 \cdot 10^{-14}\right):\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15: 29.2% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.2e+34)
   (* x (* y (- t)))
   (if (<= y 4.5e-14) x (* (- y) (* x t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.2e+34) {
		tmp = x * (y * -t);
	} else if (y <= 4.5e-14) {
		tmp = x;
	} else {
		tmp = -y * (x * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-6.2d+34)) then
        tmp = x * (y * -t)
    else if (y <= 4.5d-14) then
        tmp = x
    else
        tmp = -y * (x * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.2e+34) {
		tmp = x * (y * -t);
	} else if (y <= 4.5e-14) {
		tmp = x;
	} else {
		tmp = -y * (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.2e+34:
		tmp = x * (y * -t)
	elif y <= 4.5e-14:
		tmp = x
	else:
		tmp = -y * (x * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.2e+34)
		tmp = Float64(x * Float64(y * Float64(-t)));
	elseif (y <= 4.5e-14)
		tmp = x;
	else
		tmp = Float64(Float64(-y) * Float64(x * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6.2e+34)
		tmp = x * (y * -t);
	elseif (y <= 4.5e-14)
		tmp = x;
	else
		tmp = -y * (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.2e+34], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e-14], x, N[((-y) * N[(x * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.19999999999999955e34

   1. Initial program 98.4%

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

   2. Taylor expanded in t around inf 69.7%

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

      1. mul-1-neg 69.7%

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

      2. distribute-rgt-neg-out 69.7%

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

   4. Simplified 69.7%

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

   5. Taylor expanded in y around 0 20.6%

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

   6. Taylor expanded in y around inf 23.4%

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

      1. neg-mul-1 23.4%

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

      2. associate-*r* 26.6%

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

      3. distribute-rgt-neg-in 26.6%

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

   8. Simplified 26.6%

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

   if -6.19999999999999955e34 < y < 4.4999999999999998e-14

   1. Initial program 95.2%

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

   2. Taylor expanded in b around inf 73.2%

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

      1. associate-*r* 73.2%

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

      2. neg-mul-1 73.2%

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

      3. *-commutative 73.2%

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

   4. Simplified 73.2%

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

   5. Taylor expanded in b around 0 30.4%

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

   if 4.4999999999999998e-14 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 69.1%

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

      1. mul-1-neg 69.1%

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

      2. distribute-rgt-neg-out 69.1%

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

   4. Simplified 69.1%

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

   5. Taylor expanded in y around 0 23.6%

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

   6. Taylor expanded in y around inf 39.1%

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

      1. neg-mul-1 39.1%

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

      2. distribute-lft-neg-in 39.1%

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

   8. Simplified 39.1%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 16: 24.9% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -60000 \lor \neg \left(y \leq 3.8 \cdot 10^{+28}\right):\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -60000.0) (not (<= y 3.8e+28))) (* a (* x b)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -60000.0) || !(y <= 3.8e+28)) {
		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 <= (-60000.0d0)) .or. (.not. (y <= 3.8d+28))) 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 <= -60000.0) || !(y <= 3.8e+28)) {
		tmp = a * (x * b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -60000.0) or not (y <= 3.8e+28):
		tmp = a * (x * b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -60000.0) || !(y <= 3.8e+28))
		tmp = 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 <= -60000.0) || ~((y <= 3.8e+28)))
		tmp = a * (x * b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -60000.0], N[Not[LessEqual[y, 3.8e+28]], $MachinePrecision]], N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -6e4 or 3.7999999999999999e28 < 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 33.2%

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

      1. associate-*r* 33.2%

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

      2. neg-mul-1 33.2%

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

      3. *-commutative 33.2%

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

   4. Simplified 33.2%

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

   5. Taylor expanded in b around 0 9.2%

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

      1. +-commutative 9.2%

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

      2. mul-1-neg 9.2%

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

      3. unsub-neg 9.2%

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

   7. Simplified 9.2%

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

   8. Taylor expanded in a around inf 16.0%

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

      1. neg-mul-1 16.0%

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

      2. distribute-rgt-neg-in 16.0%

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

      3. *-commutative 16.0%

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

      4. distribute-rgt-neg-in 16.0%

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

   10. Simplified 16.0%

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

       1. expm1-log1p-u 12.5%

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

       2. expm1-udef 28.1%

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

       3. *-commutative 28.1%

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

       4. associate-*l* 28.8%

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

       5. add-sqr-sqrt 12.6%

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

       6. sqrt-unprod 29.3%

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

       7. sqr-neg 29.3%

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

       8. sqrt-unprod 14.7%

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

       9. add-sqr-sqrt 25.1%

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

   12. Applied egg-rr 25.1%

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

       1. expm1-def 11.6%

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

       2. expm1-log1p 13.4%

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

       3. associate-*r* 14.1%

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

       4. *-commutative 14.1%

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

   14. Simplified 14.1%

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

   if -6e4 < y < 3.7999999999999999e28

   1. Initial program 95.2%

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

   2. Taylor expanded in b around inf 74.7%

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

      1. associate-*r* 74.7%

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

      2. neg-mul-1 74.7%

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

      3. *-commutative 74.7%

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

   4. Simplified 74.7%

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

   5. Taylor expanded in b around 0 30.2%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -60000 \lor \neg \left(y \leq 3.8 \cdot 10^{+28}\right):\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17: 24.0% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -56000:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+108}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -56000.0) (* a (* x b)) (if (<= y 5.2e+108) x (* x (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -56000.0) {
		tmp = a * (x * b);
	} else if (y <= 5.2e+108) {
		tmp = x;
	} else {
		tmp = x * (a * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-56000.0d0)) then
        tmp = a * (x * b)
    else if (y <= 5.2d+108) then
        tmp = x
    else
        tmp = x * (a * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -56000.0) {
		tmp = a * (x * b);
	} else if (y <= 5.2e+108) {
		tmp = x;
	} else {
		tmp = x * (a * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -56000.0:
		tmp = a * (x * b)
	elif y <= 5.2e+108:
		tmp = x
	else:
		tmp = x * (a * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -56000.0)
		tmp = Float64(a * Float64(x * b));
	elseif (y <= 5.2e+108)
		tmp = x;
	else
		tmp = Float64(x * Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -56000.0)
		tmp = a * (x * b);
	elseif (y <= 5.2e+108)
		tmp = x;
	else
		tmp = x * (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -56000.0], N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e+108], x, N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -56000

   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 b around inf 32.3%

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

      1. associate-*r* 32.3%

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

      2. neg-mul-1 32.3%

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

      3. *-commutative 32.3%

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

   4. Simplified 32.3%

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

   5. Taylor expanded in b around 0 7.2%

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

      1. +-commutative 7.2%

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

      2. mul-1-neg 7.2%

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

      3. unsub-neg 7.2%

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

   7. Simplified 7.2%

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

   8. Taylor expanded in a around inf 9.2%

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

      1. neg-mul-1 9.2%

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

      2. distribute-rgt-neg-in 9.2%

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

      3. *-commutative 9.2%

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

      4. distribute-rgt-neg-in 9.2%

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

   10. Simplified 9.2%

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

       1. expm1-log1p-u 7.2%

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

       2. expm1-udef 15.4%

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

       3. *-commutative 15.4%

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

       4. associate-*l* 15.4%

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

       5. add-sqr-sqrt 4.9%

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

       6. sqrt-unprod 19.4%

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

       7. sqr-neg 19.4%

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

       8. sqrt-unprod 9.2%

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

       9. add-sqr-sqrt 12.7%

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

   12. Applied egg-rr 12.7%

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

       1. expm1-def 7.3%

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

       2. expm1-log1p 9.0%

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

       3. associate-*r* 10.3%

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

       4. *-commutative 10.3%

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

   14. Simplified 10.3%

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

   if -56000 < y < 5.2000000000000005e108

   1. Initial program 95.8%

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

   2. Taylor expanded in b around inf 70.7%

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

      1. associate-*r* 70.7%

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

      2. neg-mul-1 70.7%

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

      3. *-commutative 70.7%

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

   4. Simplified 70.7%

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

   5. Taylor expanded in b around 0 27.2%

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

   if 5.2000000000000005e108 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in b around inf 32.2%

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

      1. associate-*r* 32.2%

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

      2. neg-mul-1 32.2%

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

      3. *-commutative 32.2%

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

   4. Simplified 32.2%

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

   5. Taylor expanded in b around 0 10.1%

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

      1. +-commutative 10.1%

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

      2. mul-1-neg 10.1%

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

      3. unsub-neg 10.1%

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

   7. Simplified 10.1%

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

   8. Taylor expanded in a around inf 26.6%

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

      1. neg-mul-1 26.6%

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

      2. distribute-rgt-neg-in 26.6%

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

      3. *-commutative 26.6%

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

      4. distribute-rgt-neg-in 26.6%

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

   10. Simplified 26.6%

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

       1. expm1-log1p-u 22.0%

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

       2. expm1-udef 47.8%

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

       3. *-commutative 47.8%

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

       4. associate-*l* 51.7%

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

       5. add-sqr-sqrt 25.9%

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

       6. sqrt-unprod 47.3%

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

       7. sqr-neg 47.3%

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

       8. sqrt-unprod 23.6%

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

       9. add-sqr-sqrt 45.4%

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

   12. Applied egg-rr 45.4%

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

       1. expm1-def 21.4%

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

       2. expm1-log1p 23.6%

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

       3. *-commutative 23.6%

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

   14. Simplified 23.6%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -56000:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+108}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot b\right)\\ \end{array} \]

Alternative 18: 22.9% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -60000:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -60000.0) (* a (* x b)) (if (<= y 3.6e+93) x (* y (* x t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -60000.0) {
		tmp = a * (x * b);
	} else if (y <= 3.6e+93) {
		tmp = x;
	} else {
		tmp = y * (x * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-60000.0d0)) then
        tmp = a * (x * b)
    else if (y <= 3.6d+93) then
        tmp = x
    else
        tmp = y * (x * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -60000.0) {
		tmp = a * (x * b);
	} else if (y <= 3.6e+93) {
		tmp = x;
	} else {
		tmp = y * (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -60000.0:
		tmp = a * (x * b)
	elif y <= 3.6e+93:
		tmp = x
	else:
		tmp = y * (x * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -60000.0)
		tmp = Float64(a * Float64(x * b));
	elseif (y <= 3.6e+93)
		tmp = x;
	else
		tmp = Float64(y * Float64(x * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -60000.0)
		tmp = a * (x * b);
	elseif (y <= 3.6e+93)
		tmp = x;
	else
		tmp = y * (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -60000.0], N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e+93], x, N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6e4

   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 b around inf 32.3%

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

      1. associate-*r* 32.3%

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

      2. neg-mul-1 32.3%

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

      3. *-commutative 32.3%

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

   4. Simplified 32.3%

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

   5. Taylor expanded in b around 0 7.2%

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

      1. +-commutative 7.2%

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

      2. mul-1-neg 7.2%

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

      3. unsub-neg 7.2%

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

   7. Simplified 7.2%

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

   8. Taylor expanded in a around inf 9.2%

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

      1. neg-mul-1 9.2%

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

      2. distribute-rgt-neg-in 9.2%

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

      3. *-commutative 9.2%

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

      4. distribute-rgt-neg-in 9.2%

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

   10. Simplified 9.2%

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

       1. expm1-log1p-u 7.2%

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

       2. expm1-udef 15.4%

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

       3. *-commutative 15.4%

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

       4. associate-*l* 15.4%

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

       5. add-sqr-sqrt 4.9%

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

       6. sqrt-unprod 19.4%

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

       7. sqr-neg 19.4%

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

       8. sqrt-unprod 9.2%

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

       9. add-sqr-sqrt 12.7%

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

   12. Applied egg-rr 12.7%

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

       1. expm1-def 7.3%

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

       2. expm1-log1p 9.0%

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

       3. associate-*r* 10.3%

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

       4. *-commutative 10.3%

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

   14. Simplified 10.3%

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

   if -6e4 < y < 3.5999999999999999e93

   1. Initial program 95.7%

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

   2. Taylor expanded in b around inf 71.4%

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

      1. associate-*r* 71.4%

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

      2. neg-mul-1 71.4%

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

      3. *-commutative 71.4%

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

   4. Simplified 71.4%

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

   5. Taylor expanded in b around 0 27.7%

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

   if 3.5999999999999999e93 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 67.2%

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

      1. mul-1-neg 67.2%

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

      2. distribute-rgt-neg-out 67.2%

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

   4. Simplified 67.2%

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

   5. Taylor expanded in y around 0 22.8%

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

      1. expm1-log1p-u 10.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x\right)\right)} \]

      2. expm1-udef 34.8%

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

      3. +-commutative 34.8%

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

      4. add-sqr-sqrt 34.7%

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

      5. sqrt-unprod 38.9%

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

      6. mul-1-neg 38.9%

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

      7. mul-1-neg 38.9%

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

      8. sqr-neg 38.9%

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

      9. sqrt-unprod 24.4%

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

      10. add-sqr-sqrt 26.4%

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

   7. Applied egg-rr 26.4%

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

      1. expm1-def 2.1%

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

      2. expm1-log1p 2.6%

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

   9. Simplified 2.6%

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

   10. Taylor expanded in y around inf 25.1%

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

       1. *-commutative 25.1%

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

   12. Simplified 25.1%

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

4. Final simplification 22.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -60000:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 19: 19.7% accurate, 315.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

2. Taylor expanded in b around inf 53.1%

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

   1. associate-*r* 53.1%

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

   2. neg-mul-1 53.1%

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

   3. *-commutative 53.1%

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

4. Simplified 53.1%

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

5. Taylor expanded in b around 0 16.6%

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

6. Final simplification 16.6%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023187 
(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))))))