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

Percentage Accurate: 96.6% → 99.6%

Time: 20.3s

Alternatives: 19

Speedup: 1.0×

• 41.2% 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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

   1. fma-define 95.7%

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

   2. sub-neg 95.7%

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

   3. log1p-define 99.2%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Final simplification 95.7%

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

Alternative 3: 87.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-41} \lor \neg \left(y \leq 8.4 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot \left(z + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.45e-41) (not (<= y 8.4e-19)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (* (- a) (+ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.45e-41) || !(y <= 8.4e-19)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((-a * (z + b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.45d-41)) .or. (.not. (y <= 8.4d-19))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp((-a * (z + b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.45e-41) || !(y <= 8.4e-19)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((-a * (z + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.45e-41) or not (y <= 8.4e-19):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((-a * (z + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.45e-41) || !(y <= 8.4e-19))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(Float64(-a) * Float64(z + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.45e-41) || ~((y <= 8.4e-19)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp((-a * (z + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.45e-41], N[Not[LessEqual[y, 8.4e-19]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-a) * N[(z + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.44999999999999989e-41 or 8.3999999999999996e-19 < y

   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. Add Preprocessing

   3. Taylor expanded in y around inf 87.3%

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

   if -1.44999999999999989e-41 < y < 8.3999999999999996e-19

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0 83.1%

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

      1. sub-neg 83.1%

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

      2. log1p-define 90.8%

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

   5. Simplified 90.8%

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

   6. Taylor expanded in z around 0 90.8%

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

      1. associate-*r* 90.8%

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

      2. associate-*r* 90.8%

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

      3. distribute-lft-out 90.8%

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

      4. mul-1-neg 90.8%

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

   8. Simplified 90.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-41} \lor \neg \left(y \leq 8.4 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot \left(z + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{\left(-a\right) \cdot \left(z + b\right)}\\ t_2 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -2.45 \cdot 10^{+114}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-285}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-190}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* (- a) (+ z b))))) (t_2 (* x (exp (* y (- t))))))
   (if (<= t -2.45e+114)
     t_2
     (if (<= t -1.1e-285)
       t_1
       (if (<= t 2.9e-190) (* x (pow z y)) (if (<= t 5.8e+35) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((-a * (z + b)));
	double t_2 = x * exp((y * -t));
	double tmp;
	if (t <= -2.45e+114) {
		tmp = t_2;
	} else if (t <= -1.1e-285) {
		tmp = t_1;
	} else if (t <= 2.9e-190) {
		tmp = x * pow(z, y);
	} else if (t <= 5.8e+35) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * exp((-a * (z + b)))
    t_2 = x * exp((y * -t))
    if (t <= (-2.45d+114)) then
        tmp = t_2
    else if (t <= (-1.1d-285)) then
        tmp = t_1
    else if (t <= 2.9d-190) then
        tmp = x * (z ** y)
    else if (t <= 5.8d+35) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((-a * (z + b)));
	double t_2 = x * Math.exp((y * -t));
	double tmp;
	if (t <= -2.45e+114) {
		tmp = t_2;
	} else if (t <= -1.1e-285) {
		tmp = t_1;
	} else if (t <= 2.9e-190) {
		tmp = x * Math.pow(z, y);
	} else if (t <= 5.8e+35) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((-a * (z + b)))
	t_2 = x * math.exp((y * -t))
	tmp = 0
	if t <= -2.45e+114:
		tmp = t_2
	elif t <= -1.1e-285:
		tmp = t_1
	elif t <= 2.9e-190:
		tmp = x * math.pow(z, y)
	elif t <= 5.8e+35:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(Float64(-a) * Float64(z + b))))
	t_2 = Float64(x * exp(Float64(y * Float64(-t))))
	tmp = 0.0
	if (t <= -2.45e+114)
		tmp = t_2;
	elseif (t <= -1.1e-285)
		tmp = t_1;
	elseif (t <= 2.9e-190)
		tmp = Float64(x * (z ^ y));
	elseif (t <= 5.8e+35)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((-a * (z + b)));
	t_2 = x * exp((y * -t));
	tmp = 0.0;
	if (t <= -2.45e+114)
		tmp = t_2;
	elseif (t <= -1.1e-285)
		tmp = t_1;
	elseif (t <= 2.9e-190)
		tmp = x * (z ^ y);
	elseif (t <= 5.8e+35)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[((-a) * N[(z + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.45e+114], t$95$2, If[LessEqual[t, -1.1e-285], t$95$1, If[LessEqual[t, 2.9e-190], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.8e+35], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.45e114 or 5.79999999999999989e35 < t

   1. Initial program 95.5%

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

   3. Taylor expanded in t around inf 83.0%

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

      1. mul-1-neg 83.0%

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

      2. distribute-lft-neg-out 83.0%

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

      3. *-commutative 83.0%

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

   5. Simplified 83.0%

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

   if -2.45e114 < t < -1.1e-285 or 2.9000000000000002e-190 < t < 5.79999999999999989e35

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0 73.8%

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

      1. sub-neg 73.8%

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

      2. log1p-define 79.8%

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

   5. Simplified 79.8%

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

   6. Taylor expanded in z around 0 79.8%

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

      1. associate-*r* 79.8%

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

      2. associate-*r* 79.8%

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

      3. distribute-lft-out 79.8%

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

      4. mul-1-neg 79.8%

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

   8. Simplified 79.8%

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

   if -1.1e-285 < t < 2.9000000000000002e-190

   1. Initial program 96.8%

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

   3. Taylor expanded in y around inf 82.0%

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

   4. Taylor expanded in t around 0 82.0%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   5. Step-by-step derivation

      1. *-commutative 82.0%

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

   6. Simplified 82.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{+114}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-285}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot \left(z + b\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-190}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+35}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-285}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-89}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- t))))))
   (if (<= t -1.95e+116)
     t_1
     (if (<= t -1.5e-285)
       (* x (exp (* a (- b))))
       (if (<= t 5.5e-89) (* x (pow z y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * -t));
	double tmp;
	if (t <= -1.95e+116) {
		tmp = t_1;
	} else if (t <= -1.5e-285) {
		tmp = x * exp((a * -b));
	} else if (t <= 5.5e-89) {
		tmp = x * pow(z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((y * -t))
    if (t <= (-1.95d+116)) then
        tmp = t_1
    else if (t <= (-1.5d-285)) then
        tmp = x * exp((a * -b))
    else if (t <= 5.5d-89) then
        tmp = x * (z ** y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * -t));
	double tmp;
	if (t <= -1.95e+116) {
		tmp = t_1;
	} else if (t <= -1.5e-285) {
		tmp = x * Math.exp((a * -b));
	} else if (t <= 5.5e-89) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * -t))
	tmp = 0
	if t <= -1.95e+116:
		tmp = t_1
	elif t <= -1.5e-285:
		tmp = x * math.exp((a * -b))
	elif t <= 5.5e-89:
		tmp = x * math.pow(z, y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(-t))))
	tmp = 0.0
	if (t <= -1.95e+116)
		tmp = t_1;
	elseif (t <= -1.5e-285)
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	elseif (t <= 5.5e-89)
		tmp = Float64(x * (z ^ y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * -t));
	tmp = 0.0;
	if (t <= -1.95e+116)
		tmp = t_1;
	elseif (t <= -1.5e-285)
		tmp = x * exp((a * -b));
	elseif (t <= 5.5e-89)
		tmp = x * (z ^ y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.95e+116], t$95$1, If[LessEqual[t, -1.5e-285], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e-89], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.95000000000000016e116 or 5.50000000000000012e-89 < t

   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. Add Preprocessing

   3. Taylor expanded in t around inf 79.6%

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

      1. mul-1-neg 79.6%

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

      2. distribute-lft-neg-out 79.6%

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

      3. *-commutative 79.6%

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

   5. Simplified 79.6%

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

   if -1.95000000000000016e116 < t < -1.50000000000000002e-285

   1. Initial program 96.9%

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

   3. Taylor expanded in b around inf 79.1%

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

      1. associate-*r* 79.1%

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

      2. mul-1-neg 79.1%

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

   5. Simplified 79.1%

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

   if -1.50000000000000002e-285 < t < 5.50000000000000012e-89

   1. Initial program 96.4%

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

   3. Taylor expanded in y around inf 73.8%

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

   4. Taylor expanded in t around 0 73.8%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   5. Step-by-step derivation

      1. *-commutative 73.8%

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

   6. Simplified 73.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{+116}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-285}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-89}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{+35} \lor \neg \left(t \leq 6.6 \cdot 10^{-89}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -9.8e+35) (not (<= t 6.6e-89)))
   (* x (exp (* y (- t))))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -9.8e+35) || !(t <= 6.6e-89)) {
		tmp = x * exp((y * -t));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-9.8d+35)) .or. (.not. (t <= 6.6d-89))) then
        tmp = x * exp((y * -t))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -9.8e+35) || !(t <= 6.6e-89)) {
		tmp = x * Math.exp((y * -t));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -9.8e+35) or not (t <= 6.6e-89):
		tmp = x * math.exp((y * -t))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -9.8e+35) || !(t <= 6.6e-89))
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -9.8e+35) || ~((t <= 6.6e-89)))
		tmp = x * exp((y * -t));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -9.8e+35], N[Not[LessEqual[t, 6.6e-89]], $MachinePrecision]], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.8000000000000005e35 or 6.5999999999999993e-89 < t

   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. Add Preprocessing

   3. Taylor expanded in t around inf 78.5%

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

      1. mul-1-neg 78.5%

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

      2. distribute-lft-neg-out 78.5%

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

      3. *-commutative 78.5%

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

   5. Simplified 78.5%

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

   if -9.8000000000000005e35 < t < 6.5999999999999993e-89

   1. Initial program 96.3%

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

   3. Taylor expanded in y around inf 66.6%

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

   4. Taylor expanded in t around 0 66.6%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   5. Step-by-step derivation

      1. *-commutative 66.6%

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

   6. Simplified 66.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{+35} \lor \neg \left(t \leq 6.6 \cdot 10^{-89}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -3.7e+36) (* x (- 1.0 (* 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 <= -3.7e+36) {
		tmp = x * (1.0 - (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 <= (-3.7d+36)) then
        tmp = x * (1.0d0 - (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 <= -3.7e+36) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -3.7e+36:
		tmp = x * (1.0 - (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 <= -3.7e+36)
		tmp = Float64(x * Float64(1.0 - Float64(y * 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 <= -3.7e+36)
		tmp = x * (1.0 - (y * t));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -3.7e+36], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.70000000000000029e36

   1. Initial program 93.8%

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

   3. Taylor expanded in t around inf 79.7%

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

      1. mul-1-neg 79.7%

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

      2. distribute-lft-neg-out 79.7%

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

      3. *-commutative 79.7%

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

   5. Simplified 79.7%

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

   6. Taylor expanded in y around 0 36.3%

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

      1. mul-1-neg 36.3%

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

      2. unsub-neg 36.3%

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

      3. *-commutative 36.3%

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

   8. Simplified 36.3%

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

   9. Taylor expanded in x around 0 42.4%

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

   if -3.70000000000000029e36 < t

   1. Initial program 96.3%

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

   3. Taylor expanded in y around inf 71.0%

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

   4. Taylor expanded in t around 0 65.1%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   5. Step-by-step derivation

      1. *-commutative 65.1%

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

   6. Simplified 65.1%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 27.5% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{x}{t}\\ t_2 := \left(y \cdot t\right) \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -2 \cdot 10^{+59}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{-221}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 3.35 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (/ x t))) (t_2 (* (* y t) (- x))))
   (if (<= y -2e+59)
     (* t (* x (- y)))
     (if (<= y -2.5e-155)
       t_1
       (if (<= y -8.8e-221)
         t_2
         (if (<= y 3.35e-40) x (if (<= y 6.5e-10) t_1 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (x / t);
	double t_2 = (y * t) * -x;
	double tmp;
	if (y <= -2e+59) {
		tmp = t * (x * -y);
	} else if (y <= -2.5e-155) {
		tmp = t_1;
	} else if (y <= -8.8e-221) {
		tmp = t_2;
	} else if (y <= 3.35e-40) {
		tmp = x;
	} else if (y <= 6.5e-10) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t * (x / t)
    t_2 = (y * t) * -x
    if (y <= (-2d+59)) then
        tmp = t * (x * -y)
    else if (y <= (-2.5d-155)) then
        tmp = t_1
    else if (y <= (-8.8d-221)) then
        tmp = t_2
    else if (y <= 3.35d-40) then
        tmp = x
    else if (y <= 6.5d-10) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (x / t);
	double t_2 = (y * t) * -x;
	double tmp;
	if (y <= -2e+59) {
		tmp = t * (x * -y);
	} else if (y <= -2.5e-155) {
		tmp = t_1;
	} else if (y <= -8.8e-221) {
		tmp = t_2;
	} else if (y <= 3.35e-40) {
		tmp = x;
	} else if (y <= 6.5e-10) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t * (x / t)
	t_2 = (y * t) * -x
	tmp = 0
	if y <= -2e+59:
		tmp = t * (x * -y)
	elif y <= -2.5e-155:
		tmp = t_1
	elif y <= -8.8e-221:
		tmp = t_2
	elif y <= 3.35e-40:
		tmp = x
	elif y <= 6.5e-10:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(x / t))
	t_2 = Float64(Float64(y * t) * Float64(-x))
	tmp = 0.0
	if (y <= -2e+59)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= -2.5e-155)
		tmp = t_1;
	elseif (y <= -8.8e-221)
		tmp = t_2;
	elseif (y <= 3.35e-40)
		tmp = x;
	elseif (y <= 6.5e-10)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t * (x / t);
	t_2 = (y * t) * -x;
	tmp = 0.0;
	if (y <= -2e+59)
		tmp = t * (x * -y);
	elseif (y <= -2.5e-155)
		tmp = t_1;
	elseif (y <= -8.8e-221)
		tmp = t_2;
	elseif (y <= 3.35e-40)
		tmp = x;
	elseif (y <= 6.5e-10)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * t), $MachinePrecision] * (-x)), $MachinePrecision]}, If[LessEqual[y, -2e+59], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.5e-155], t$95$1, If[LessEqual[y, -8.8e-221], t$95$2, If[LessEqual[y, 3.35e-40], x, If[LessEqual[y, 6.5e-10], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.99999999999999994e59

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 64.5%

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

      1. mul-1-neg 64.5%

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

      2. distribute-lft-neg-out 64.5%

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

      3. *-commutative 64.5%

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

   5. Simplified 64.5%

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

   6. Taylor expanded in y around 0 30.3%

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

      1. mul-1-neg 30.3%

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

      2. unsub-neg 30.3%

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

      3. *-commutative 30.3%

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

   8. Simplified 30.3%

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

   9. Taylor expanded in t around inf 31.6%

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

       1. mul-1-neg 31.6%

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

       2. *-commutative 31.6%

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

       3. distribute-rgt-neg-in 31.6%

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

   11. Simplified 31.6%

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

   if -1.99999999999999994e59 < y < -2.4999999999999999e-155 or 3.3499999999999999e-40 < y < 6.5000000000000003e-10

   1. Initial program 95.2%

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

   3. Taylor expanded in t around inf 54.1%

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

      1. mul-1-neg 54.1%

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

      2. distribute-lft-neg-out 54.1%

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

      3. *-commutative 54.1%

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

   5. Simplified 54.1%

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

   6. Taylor expanded in y around 0 26.1%

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

      1. mul-1-neg 26.1%

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

      2. unsub-neg 26.1%

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

      3. *-commutative 26.1%

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

   8. Simplified 26.1%

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

   9. Taylor expanded in t around inf 25.6%

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

   10. Taylor expanded in t around 0 30.6%

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

   if -2.4999999999999999e-155 < y < -8.80000000000000005e-221 or 6.5000000000000003e-10 < y

   1. Initial program 95.6%

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

   3. Taylor expanded in t around inf 54.3%

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

      1. mul-1-neg 54.3%

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

      2. distribute-lft-neg-out 54.3%

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

      3. *-commutative 54.3%

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

   5. Simplified 54.3%

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

   6. Taylor expanded in y around 0 20.2%

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

      1. mul-1-neg 20.2%

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

      2. unsub-neg 20.2%

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

      3. *-commutative 20.2%

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

   8. Simplified 20.2%

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

   9. Taylor expanded in x around 0 24.4%

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

   10. Taylor expanded in t around inf 27.5%

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

       1. mul-1-neg 27.5%

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

       2. associate-*r* 30.7%

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

       3. *-commutative 30.7%

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

       4. distribute-rgt-neg-in 30.7%

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

       5. associate-*r* 32.7%

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

   12. Simplified 32.7%

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

   if -8.80000000000000005e-221 < y < 3.3499999999999999e-40

   1. Initial program 93.5%

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

   3. Taylor expanded in y around inf 56.3%

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

   4. Taylor expanded in y around 0 41.5%

      \[\leadsto \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+59}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{-221}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 3.35 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-10}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.7% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.35:\\ \;\;\;\;x + b \cdot \left(a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot b\right)\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 0.35)
   (+ x (* b (* a (- (* 0.5 (* a (* x b))) x))))
   (* (* y t) (- x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 0.35) {
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	} else {
		tmp = (y * t) * -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 <= 0.35d0) then
        tmp = x + (b * (a * ((0.5d0 * (a * (x * b))) - x)))
    else
        tmp = (y * t) * -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 <= 0.35) {
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	} else {
		tmp = (y * t) * -x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 0.35:
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)))
	else:
		tmp = (y * t) * -x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 0.35)
		tmp = Float64(x + Float64(b * Float64(a * Float64(Float64(0.5 * Float64(a * Float64(x * b))) - x))));
	else
		tmp = Float64(Float64(y * t) * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 0.35)
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	else
		tmp = (y * t) * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 0.35], N[(x + N[(b * N[(a * N[(N[(0.5 * N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * t), $MachinePrecision] * (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.34999999999999998

   1. Initial program 95.4%

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

   3. Taylor expanded in b around inf 66.2%

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

      1. associate-*r* 66.2%

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

      2. mul-1-neg 66.2%

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

   5. Simplified 66.2%

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

   6. Taylor expanded in b around 0 39.2%

      \[\leadsto \color{blue}{x + b \cdot \left(-1 \cdot \left(a \cdot x\right) + 0.5 \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 39.2%

         \[\leadsto x + b \cdot \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) + -1 \cdot \left(a \cdot x\right)\right)} \]

      2. mul-1-neg 39.2%

         \[\leadsto x + b \cdot \left(0.5 \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) + \color{blue}{\left(-a \cdot x\right)}\right) \]

      3. unsub-neg 39.2%

         \[\leadsto x + b \cdot \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) - a \cdot x\right)} \]

      4. associate-*r* 39.2%

         \[\leadsto x + b \cdot \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot \left(b \cdot x\right)} - a \cdot x\right) \]

      5. *-commutative 39.2%

         \[\leadsto x + b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(x \cdot b\right)} - a \cdot x\right) \]

      6. *-commutative 39.2%

         \[\leadsto x + b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot \left(x \cdot b\right) - \color{blue}{x \cdot a}\right) \]

   8. Simplified 39.2%

      \[\leadsto \color{blue}{x + b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot \left(x \cdot b\right) - x \cdot a\right)} \]

   9. Taylor expanded in a around 0 41.7%

      \[\leadsto x + b \cdot \color{blue}{\left(a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot x\right)\right) - x\right)\right)} \]

   if 0.34999999999999998 < y

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf 59.8%

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

      1. mul-1-neg 59.8%

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

      2. distribute-lft-neg-out 59.8%

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

      3. *-commutative 59.8%

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

   5. Simplified 59.8%

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

   6. Taylor expanded in y around 0 22.0%

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

      1. mul-1-neg 22.0%

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

      2. unsub-neg 22.0%

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

      3. *-commutative 22.0%

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

   8. Simplified 22.0%

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

   9. Taylor expanded in x around 0 26.6%

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

   10. Taylor expanded in t around inf 26.6%

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

       1. mul-1-neg 26.6%

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

       2. associate-*r* 30.1%

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

       3. *-commutative 30.1%

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

       4. distribute-rgt-neg-in 30.1%

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

       5. associate-*r* 31.2%

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

   12. Simplified 31.2%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.35:\\ \;\;\;\;x + b \cdot \left(a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot b\right)\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.7% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-42}:\\ \;\;\;\;t \cdot \left(\frac{x}{t} - x \cdot y\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.7e-42)
   (* t (- (/ x t) (* x y)))
   (if (<= y 5.5e-11) (* x (- 1.0 (* a (+ z b)))) (* (* y t) (- x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.7e-42) {
		tmp = t * ((x / t) - (x * y));
	} else if (y <= 5.5e-11) {
		tmp = x * (1.0 - (a * (z + b)));
	} else {
		tmp = (y * t) * -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 <= (-4.7d-42)) then
        tmp = t * ((x / t) - (x * y))
    else if (y <= 5.5d-11) then
        tmp = x * (1.0d0 - (a * (z + b)))
    else
        tmp = (y * t) * -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 <= -4.7e-42) {
		tmp = t * ((x / t) - (x * y));
	} else if (y <= 5.5e-11) {
		tmp = x * (1.0 - (a * (z + b)));
	} else {
		tmp = (y * t) * -x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4.7e-42:
		tmp = t * ((x / t) - (x * y))
	elif y <= 5.5e-11:
		tmp = x * (1.0 - (a * (z + b)))
	else:
		tmp = (y * t) * -x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.7e-42)
		tmp = Float64(t * Float64(Float64(x / t) - Float64(x * y)));
	elseif (y <= 5.5e-11)
		tmp = Float64(x * Float64(1.0 - Float64(a * Float64(z + b))));
	else
		tmp = Float64(Float64(y * t) * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4.7e-42)
		tmp = t * ((x / t) - (x * y));
	elseif (y <= 5.5e-11)
		tmp = x * (1.0 - (a * (z + b)));
	else
		tmp = (y * t) * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.7e-42], N[(t * N[(N[(x / t), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e-11], N[(x * N[(1.0 - N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * t), $MachinePrecision] * (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.7000000000000001e-42

   1. Initial program 98.6%

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

   3. Taylor expanded in t around inf 57.6%

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

      1. mul-1-neg 57.6%

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

      2. distribute-lft-neg-out 57.6%

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

      3. *-commutative 57.6%

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

   5. Simplified 57.6%

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

   6. Taylor expanded in y around 0 26.5%

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

      1. mul-1-neg 26.5%

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

      2. unsub-neg 26.5%

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

      3. *-commutative 26.5%

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

   8. Simplified 26.5%

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

   9. Taylor expanded in t around inf 29.0%

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

   if -4.7000000000000001e-42 < y < 5.49999999999999975e-11

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0 83.1%

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

      1. sub-neg 83.1%

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

      2. log1p-define 90.8%

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

   5. Simplified 90.8%

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

   6. Taylor expanded in z around 0 90.8%

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

      1. associate-*r* 90.8%

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

      2. associate-*r* 90.8%

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

      3. distribute-lft-out 90.8%

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

      4. mul-1-neg 90.8%

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

   8. Simplified 90.8%

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

   9. Taylor expanded in a around 0 47.5%

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

       1. neg-mul-1 47.5%

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

       2. unsub-neg 47.5%

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

   11. Simplified 47.5%

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

   if 5.49999999999999975e-11 < y

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf 59.8%

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

      1. mul-1-neg 59.8%

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

      2. distribute-lft-neg-out 59.8%

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

      3. *-commutative 59.8%

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

   5. Simplified 59.8%

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

   6. Taylor expanded in y around 0 22.0%

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

      1. mul-1-neg 22.0%

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

      2. unsub-neg 22.0%

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

      3. *-commutative 22.0%

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

   8. Simplified 22.0%

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

   9. Taylor expanded in x around 0 26.6%

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

   10. Taylor expanded in t around inf 26.6%

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

       1. mul-1-neg 26.6%

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

       2. associate-*r* 30.1%

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

       3. *-commutative 30.1%

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

       4. distribute-rgt-neg-in 30.1%

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

       5. associate-*r* 31.2%

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

   12. Simplified 31.2%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-42}:\\ \;\;\;\;t \cdot \left(\frac{x}{t} - x \cdot y\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 31.8% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+23}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 0.001:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.5e+23)
   (* t (* x (- y)))
   (if (<= y 0.001) (- x (* a (* x b))) (* (* y t) (- x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.5e+23) {
		tmp = t * (x * -y);
	} else if (y <= 0.001) {
		tmp = x - (a * (x * b));
	} else {
		tmp = (y * t) * -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-1.5d+23)) then
        tmp = t * (x * -y)
    else if (y <= 0.001d0) then
        tmp = x - (a * (x * b))
    else
        tmp = (y * t) * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.5e+23) {
		tmp = t * (x * -y);
	} else if (y <= 0.001) {
		tmp = x - (a * (x * b));
	} else {
		tmp = (y * t) * -x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.5e+23:
		tmp = t * (x * -y)
	elif y <= 0.001:
		tmp = x - (a * (x * b))
	else:
		tmp = (y * t) * -x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.5e+23)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 0.001)
		tmp = Float64(x - Float64(a * Float64(x * b)));
	else
		tmp = Float64(Float64(y * t) * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.5e+23)
		tmp = t * (x * -y);
	elseif (y <= 0.001)
		tmp = x - (a * (x * b));
	else
		tmp = (y * t) * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.5e+23], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.001], N[(x - N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * t), $MachinePrecision] * (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e23

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf 61.2%

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

      1. mul-1-neg 61.2%

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

      2. distribute-lft-neg-out 61.2%

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

      3. *-commutative 61.2%

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

   5. Simplified 61.2%

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

   6. Taylor expanded in y around 0 27.4%

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

      1. mul-1-neg 27.4%

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

      2. unsub-neg 27.4%

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

      3. *-commutative 27.4%

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

   8. Simplified 27.4%

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

   9. Taylor expanded in t around inf 30.3%

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

       1. mul-1-neg 30.3%

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

       2. *-commutative 30.3%

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

       3. distribute-rgt-neg-in 30.3%

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

   11. Simplified 30.3%

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

   if -1.5e23 < y < 1e-3

   1. Initial program 94.2%

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

   3. Taylor expanded in b around inf 79.4%

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

      1. associate-*r* 79.4%

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

      2. mul-1-neg 79.4%

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

   5. Simplified 79.4%

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

   6. Taylor expanded in a around 0 42.6%

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

      1. mul-1-neg 42.6%

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

      2. unsub-neg 42.6%

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

      3. *-commutative 42.6%

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

   8. Simplified 42.6%

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

   if 1e-3 < y

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf 59.8%

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

      1. mul-1-neg 59.8%

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

      2. distribute-lft-neg-out 59.8%

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

      3. *-commutative 59.8%

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

   5. Simplified 59.8%

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

   6. Taylor expanded in y around 0 22.0%

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

      1. mul-1-neg 22.0%

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

      2. unsub-neg 22.0%

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

      3. *-commutative 22.0%

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

   8. Simplified 22.0%

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

   9. Taylor expanded in x around 0 26.6%

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

   10. Taylor expanded in t around inf 26.6%

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

       1. mul-1-neg 26.6%

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

       2. associate-*r* 30.1%

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

       3. *-commutative 30.1%

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

       4. distribute-rgt-neg-in 30.1%

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

       5. associate-*r* 31.2%

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

   12. Simplified 31.2%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+23}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 0.001:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 32.8% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+22}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 0.0027:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.1e+22)
   (* t (* x (- y)))
   (if (<= y 0.0027) (- x (* x (* a b))) (* (* y t) (- x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.1e+22) {
		tmp = t * (x * -y);
	} else if (y <= 0.0027) {
		tmp = x - (x * (a * b));
	} else {
		tmp = (y * t) * -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 <= (-3.1d+22)) then
        tmp = t * (x * -y)
    else if (y <= 0.0027d0) then
        tmp = x - (x * (a * b))
    else
        tmp = (y * t) * -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 <= -3.1e+22) {
		tmp = t * (x * -y);
	} else if (y <= 0.0027) {
		tmp = x - (x * (a * b));
	} else {
		tmp = (y * t) * -x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.1e+22:
		tmp = t * (x * -y)
	elif y <= 0.0027:
		tmp = x - (x * (a * b))
	else:
		tmp = (y * t) * -x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.1e+22)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 0.0027)
		tmp = Float64(x - Float64(x * Float64(a * b)));
	else
		tmp = Float64(Float64(y * t) * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3.1e+22)
		tmp = t * (x * -y);
	elseif (y <= 0.0027)
		tmp = x - (x * (a * b));
	else
		tmp = (y * t) * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.1e+22], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0027], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * t), $MachinePrecision] * (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1000000000000002e22

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf 61.2%

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

      1. mul-1-neg 61.2%

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

      2. distribute-lft-neg-out 61.2%

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

      3. *-commutative 61.2%

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

   5. Simplified 61.2%

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

   6. Taylor expanded in y around 0 27.4%

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

      1. mul-1-neg 27.4%

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

      2. unsub-neg 27.4%

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

      3. *-commutative 27.4%

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

   8. Simplified 27.4%

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

   9. Taylor expanded in t around inf 30.3%

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

       1. mul-1-neg 30.3%

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

       2. *-commutative 30.3%

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

       3. distribute-rgt-neg-in 30.3%

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

   11. Simplified 30.3%

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

   if -3.1000000000000002e22 < y < 0.0027000000000000001

   1. Initial program 94.2%

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

   3. Taylor expanded in b around inf 79.4%

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

      1. associate-*r* 79.4%

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

      2. mul-1-neg 79.4%

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

   5. Simplified 79.4%

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

   6. Taylor expanded in a around 0 42.6%

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

      1. mul-1-neg 42.6%

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

      2. unsub-neg 42.6%

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

      3. *-commutative 42.6%

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

   8. Simplified 42.6%

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

   9. Taylor expanded in a around 0 42.6%

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

       1. associate-*r* 42.7%

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

   11. Simplified 42.7%

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

   if 0.0027000000000000001 < y

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf 59.8%

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

      1. mul-1-neg 59.8%

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

      2. distribute-lft-neg-out 59.8%

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

      3. *-commutative 59.8%

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

   5. Simplified 59.8%

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

   6. Taylor expanded in y around 0 22.0%

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

      1. mul-1-neg 22.0%

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

      2. unsub-neg 22.0%

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

      3. *-commutative 22.0%

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

   8. Simplified 22.0%

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

   9. Taylor expanded in x around 0 26.6%

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

   10. Taylor expanded in t around inf 26.6%

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

       1. mul-1-neg 26.6%

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

       2. associate-*r* 30.1%

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

       3. *-commutative 30.1%

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

       4. distribute-rgt-neg-in 30.1%

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

       5. associate-*r* 31.2%

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

   12. Simplified 31.2%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+22}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 0.0027:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.4% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-42}:\\ \;\;\;\;t \cdot \left(\frac{x}{t} - x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2e-42)
   (* t (- (/ x t) (* x y)))
   (if (<= y 1.5e-5) (- x (* x (* a b))) (* (* y t) (- x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2e-42) {
		tmp = t * ((x / t) - (x * y));
	} else if (y <= 1.5e-5) {
		tmp = x - (x * (a * b));
	} else {
		tmp = (y * t) * -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 <= (-2d-42)) then
        tmp = t * ((x / t) - (x * y))
    else if (y <= 1.5d-5) then
        tmp = x - (x * (a * b))
    else
        tmp = (y * t) * -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 <= -2e-42) {
		tmp = t * ((x / t) - (x * y));
	} else if (y <= 1.5e-5) {
		tmp = x - (x * (a * b));
	} else {
		tmp = (y * t) * -x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2e-42:
		tmp = t * ((x / t) - (x * y))
	elif y <= 1.5e-5:
		tmp = x - (x * (a * b))
	else:
		tmp = (y * t) * -x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2e-42)
		tmp = Float64(t * Float64(Float64(x / t) - Float64(x * y)));
	elseif (y <= 1.5e-5)
		tmp = Float64(x - Float64(x * Float64(a * b)));
	else
		tmp = Float64(Float64(y * t) * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2e-42)
		tmp = t * ((x / t) - (x * y));
	elseif (y <= 1.5e-5)
		tmp = x - (x * (a * b));
	else
		tmp = (y * t) * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2e-42], N[(t * N[(N[(x / t), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e-5], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * t), $MachinePrecision] * (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.00000000000000008e-42

   1. Initial program 98.6%

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

   3. Taylor expanded in t around inf 57.6%

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

      1. mul-1-neg 57.6%

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

      2. distribute-lft-neg-out 57.6%

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

      3. *-commutative 57.6%

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

   5. Simplified 57.6%

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

   6. Taylor expanded in y around 0 26.5%

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

      1. mul-1-neg 26.5%

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

      2. unsub-neg 26.5%

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

      3. *-commutative 26.5%

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

   8. Simplified 26.5%

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

   9. Taylor expanded in t around inf 29.0%

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

   if -2.00000000000000008e-42 < y < 1.50000000000000004e-5

   1. Initial program 93.3%

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

   3. Taylor expanded in b around inf 83.1%

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

      1. associate-*r* 83.1%

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

      2. mul-1-neg 83.1%

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

   5. Simplified 83.1%

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

   6. Taylor expanded in a around 0 46.2%

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

      1. mul-1-neg 46.2%

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

      2. unsub-neg 46.2%

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

      3. *-commutative 46.2%

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

   8. Simplified 46.2%

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

   9. Taylor expanded in a around 0 46.2%

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

       1. associate-*r* 46.2%

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

   11. Simplified 46.2%

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

   if 1.50000000000000004e-5 < y

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf 59.8%

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

      1. mul-1-neg 59.8%

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

      2. distribute-lft-neg-out 59.8%

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

      3. *-commutative 59.8%

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

   5. Simplified 59.8%

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

   6. Taylor expanded in y around 0 22.0%

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

      1. mul-1-neg 22.0%

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

      2. unsub-neg 22.0%

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

      3. *-commutative 22.0%

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

   8. Simplified 22.0%

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

   9. Taylor expanded in x around 0 26.6%

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

   10. Taylor expanded in t around inf 26.6%

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

       1. mul-1-neg 26.6%

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

       2. associate-*r* 30.1%

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

       3. *-commutative 30.1%

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

       4. distribute-rgt-neg-in 30.1%

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

       5. associate-*r* 31.2%

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

   12. Simplified 31.2%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-42}:\\ \;\;\;\;t \cdot \left(\frac{x}{t} - x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.5% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-126}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 1e-126) (* t (/ x t)) (* x (- 1.0 (* y t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 1e-126) {
		tmp = t * (x / t);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= 1d-126) then
        tmp = t * (x / t)
    else
        tmp = x * (1.0d0 - (y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 1e-126) {
		tmp = t * (x / t);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 1e-126:
		tmp = t * (x / t)
	else:
		tmp = x * (1.0 - (y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 1e-126)
		tmp = Float64(t * Float64(x / t));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 1e-126)
		tmp = t * (x / t);
	else
		tmp = x * (1.0 - (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 1e-126], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999995e-127

   1. Initial program 96.3%

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

   3. Taylor expanded in t around inf 56.4%

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

      1. mul-1-neg 56.4%

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

      2. distribute-lft-neg-out 56.4%

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

      3. *-commutative 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in y around 0 29.2%

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

      1. mul-1-neg 29.2%

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

      2. unsub-neg 29.2%

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

      3. *-commutative 29.2%

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

   8. Simplified 29.2%

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

   9. Taylor expanded in t around inf 29.7%

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

   10. Taylor expanded in t around 0 27.6%

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

   if 9.9999999999999995e-127 < x

   1. Initial program 94.5%

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

   3. Taylor expanded in t around inf 57.3%

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

      1. mul-1-neg 57.3%

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

      2. distribute-lft-neg-out 57.3%

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

      3. *-commutative 57.3%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in y around 0 31.0%

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

      1. mul-1-neg 31.0%

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

      2. unsub-neg 31.0%

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

      3. *-commutative 31.0%

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

   8. Simplified 31.0%

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

   9. Taylor expanded in x around 0 27.9%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-126}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 28.7% accurate, 28.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00155:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 0.00155) (* t (/ x t)) (* y (* x (- t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 0.00155) {
		tmp = t * (x / t);
	} 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 <= 0.00155d0) then
        tmp = t * (x / t)
    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 <= 0.00155) {
		tmp = t * (x / t);
	} else {
		tmp = y * (x * -t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 0.00155:
		tmp = t * (x / t)
	else:
		tmp = y * (x * -t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 0.00155)
		tmp = Float64(t * Float64(x / t));
	else
		tmp = Float64(y * Float64(x * Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 0.00155)
		tmp = t * (x / t);
	else
		tmp = y * (x * -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 0.00155], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * (-t)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.00154999999999999995

   1. Initial program 95.4%

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

   3. Taylor expanded in t around inf 55.2%

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

      1. mul-1-neg 55.2%

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

      2. distribute-lft-neg-out 55.2%

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

      3. *-commutative 55.2%

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

   5. Simplified 55.2%

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

   6. Taylor expanded in y around 0 33.5%

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

      1. mul-1-neg 33.5%

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

      2. unsub-neg 33.5%

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

      3. *-commutative 33.5%

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

   8. Simplified 33.5%

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

   9. Taylor expanded in t around inf 33.3%

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

   10. Taylor expanded in t around 0 30.8%

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

   if 0.00154999999999999995 < y

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf 59.8%

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

      1. mul-1-neg 59.8%

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

      2. distribute-lft-neg-out 59.8%

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

      3. *-commutative 59.8%

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

   5. Simplified 59.8%

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

   6. Taylor expanded in y around 0 22.0%

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

      1. mul-1-neg 22.0%

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

      2. unsub-neg 22.0%

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

      3. *-commutative 22.0%

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

   8. Simplified 22.0%

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

   9. Taylor expanded in t around inf 26.6%

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

       1. mul-1-neg 26.6%

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

       2. associate-*r* 30.1%

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

       3. *-commutative 30.1%

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

   11. Simplified 30.1%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00155:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 28.0% accurate, 28.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.058:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 0.058) (* t (/ x t)) (* (* y t) (- x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 0.058) {
		tmp = t * (x / t);
	} else {
		tmp = (y * t) * -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 <= 0.058d0) then
        tmp = t * (x / t)
    else
        tmp = (y * t) * -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 <= 0.058) {
		tmp = t * (x / t);
	} else {
		tmp = (y * t) * -x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 0.058:
		tmp = t * (x / t)
	else:
		tmp = (y * t) * -x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 0.058)
		tmp = Float64(t * Float64(x / t));
	else
		tmp = Float64(Float64(y * t) * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 0.058)
		tmp = t * (x / t);
	else
		tmp = (y * t) * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 0.058], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(N[(y * t), $MachinePrecision] * (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0580000000000000029

   1. Initial program 95.4%

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

   3. Taylor expanded in t around inf 55.2%

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

      1. mul-1-neg 55.2%

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

      2. distribute-lft-neg-out 55.2%

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

      3. *-commutative 55.2%

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

   5. Simplified 55.2%

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

   6. Taylor expanded in y around 0 33.5%

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

      1. mul-1-neg 33.5%

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

      2. unsub-neg 33.5%

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

      3. *-commutative 33.5%

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

   8. Simplified 33.5%

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

   9. Taylor expanded in t around inf 33.3%

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

   10. Taylor expanded in t around 0 30.8%

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

   if 0.0580000000000000029 < y

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf 59.8%

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

      1. mul-1-neg 59.8%

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

      2. distribute-lft-neg-out 59.8%

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

      3. *-commutative 59.8%

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

   5. Simplified 59.8%

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

   6. Taylor expanded in y around 0 22.0%

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

      1. mul-1-neg 22.0%

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

      2. unsub-neg 22.0%

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

      3. *-commutative 22.0%

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

   8. Simplified 22.0%

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

   9. Taylor expanded in x around 0 26.6%

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

   10. Taylor expanded in t around inf 26.6%

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

       1. mul-1-neg 26.6%

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

       2. associate-*r* 30.1%

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

       3. *-commutative 30.1%

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

       4. distribute-rgt-neg-in 30.1%

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

       5. associate-*r* 31.2%

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

   12. Simplified 31.2%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.058:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 27.0% accurate, 31.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-237}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 6.5e-237) (* t (/ x t)) (* y (/ x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 6.5e-237) {
		tmp = t * (x / t);
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= 6.5d-237) then
        tmp = t * (x / t)
    else
        tmp = y * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 6.5e-237) {
		tmp = t * (x / t);
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 6.5e-237:
		tmp = t * (x / t)
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 6.5e-237)
		tmp = Float64(t * Float64(x / t));
	else
		tmp = Float64(y * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 6.5e-237)
		tmp = t * (x / t);
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 6.5e-237], N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5000000000000001e-237

   1. Initial program 96.2%

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

   3. Taylor expanded in t around inf 51.6%

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

      1. mul-1-neg 51.6%

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

      2. distribute-lft-neg-out 51.6%

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

      3. *-commutative 51.6%

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

   5. Simplified 51.6%

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

   6. Taylor expanded in y around 0 30.3%

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

      1. mul-1-neg 30.3%

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

      2. unsub-neg 30.3%

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

      3. *-commutative 30.3%

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

   8. Simplified 30.3%

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

   9. Taylor expanded in t around inf 32.3%

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

   10. Taylor expanded in t around 0 28.8%

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

   if 6.5000000000000001e-237 < y

   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. Add Preprocessing

   3. Taylor expanded in t around inf 61.8%

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

      1. mul-1-neg 61.8%

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

      2. distribute-lft-neg-out 61.8%

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

      3. *-commutative 61.8%

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

   5. Simplified 61.8%

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

   6. Taylor expanded in y around 0 29.3%

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

      1. mul-1-neg 29.3%

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

      2. unsub-neg 29.3%

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

      3. *-commutative 29.3%

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

   8. Simplified 29.3%

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

   9. Taylor expanded in y around inf 31.4%

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

       1. *-commutative 31.4%

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

   11. Simplified 31.4%

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

   12. Taylor expanded in y around 0 23.4%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-237}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 25.6% accurate, 63.0× speedup

Math

\[\begin{array}{l} \\ t \cdot \frac{x}{t} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return t * (x / 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 = t * (x / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in t around inf 56.7%

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

   1. mul-1-neg 56.7%

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

   2. distribute-lft-neg-out 56.7%

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

   3. *-commutative 56.7%

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

5. Simplified 56.7%

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

6. Taylor expanded in y around 0 29.8%

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

   1. mul-1-neg 29.8%

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

   2. unsub-neg 29.8%

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

   3. *-commutative 29.8%

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

8. Simplified 29.8%

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

9. Taylor expanded in t around inf 30.1%

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

10. Taylor expanded in t around 0 23.9%

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

11. Final simplification 23.9%

    \[\leadsto t \cdot \frac{x}{t} \]
12. Add Preprocessing

Alternative 19: 19.6% accurate, 315.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in y around inf 73.1%

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

4. Taylor expanded in y around 0 18.2%

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

5. Final simplification 18.2%

   \[\leadsto x \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(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))))))