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

Percentage Accurate: 96.5% → 99.4%

Time: 21.4s

Alternatives: 18

Speedup: 1.0×

• 40.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

   1. fma-define 96.5%

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

   2. sub-neg 96.5%

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

   3. log1p-define 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Final simplification 96.5%

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

Alternative 3: 87.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -7 \cdot 10^{+183}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1 \cdot 10^{+109}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{elif}\;t \leq -7400000000000 \lor \neg \left(t \leq 2.15 \cdot 10^{+176}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \log z - a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- t))))))
   (if (<= t -7e+183)
     t_1
     (if (<= t -1e+109)
       (* x (exp (* a (- (- b) z))))
       (if (or (<= t -7400000000000.0) (not (<= t 2.15e+176)))
         t_1
         (* x (exp (- (* y (log z)) (* a b)))))))))

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 <= -7e+183) {
		tmp = t_1;
	} else if (t <= -1e+109) {
		tmp = x * exp((a * (-b - z)));
	} else if ((t <= -7400000000000.0) || !(t <= 2.15e+176)) {
		tmp = t_1;
	} else {
		tmp = x * exp(((y * log(z)) - (a * b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((y * -t))
    if (t <= (-7d+183)) then
        tmp = t_1
    else if (t <= (-1d+109)) then
        tmp = x * exp((a * (-b - z)))
    else if ((t <= (-7400000000000.0d0)) .or. (.not. (t <= 2.15d+176))) then
        tmp = t_1
    else
        tmp = x * exp(((y * log(z)) - (a * b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * -t));
	double tmp;
	if (t <= -7e+183) {
		tmp = t_1;
	} else if (t <= -1e+109) {
		tmp = x * Math.exp((a * (-b - z)));
	} else if ((t <= -7400000000000.0) || !(t <= 2.15e+176)) {
		tmp = t_1;
	} else {
		tmp = x * Math.exp(((y * Math.log(z)) - (a * b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * -t))
	tmp = 0
	if t <= -7e+183:
		tmp = t_1
	elif t <= -1e+109:
		tmp = x * math.exp((a * (-b - z)))
	elif (t <= -7400000000000.0) or not (t <= 2.15e+176):
		tmp = t_1
	else:
		tmp = x * math.exp(((y * math.log(z)) - (a * b)))
	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 <= -7e+183)
		tmp = t_1;
	elseif (t <= -1e+109)
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-b) - z))));
	elseif ((t <= -7400000000000.0) || !(t <= 2.15e+176))
		tmp = t_1;
	else
		tmp = Float64(x * exp(Float64(Float64(y * log(z)) - Float64(a * b))));
	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 <= -7e+183)
		tmp = t_1;
	elseif (t <= -1e+109)
		tmp = x * exp((a * (-b - z)));
	elseif ((t <= -7400000000000.0) || ~((t <= 2.15e+176)))
		tmp = t_1;
	else
		tmp = x * exp(((y * log(z)) - (a * b)));
	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, -7e+183], t$95$1, If[LessEqual[t, -1e+109], N[(x * N[Exp[N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -7400000000000.0], N[Not[LessEqual[t, 2.15e+176]], $MachinePrecision]], t$95$1, N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - N[(a * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.99999999999999974e183 or -9.99999999999999982e108 < t < -7.4e12 or 2.15000000000000013e176 < t

   1. Initial program 94.1%

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

   3. Taylor expanded in t around inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. distribute-lft-neg-out 89.8%

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

      3. *-commutative 89.8%

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

   5. Simplified 89.8%

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

   if -6.99999999999999974e183 < t < -9.99999999999999982e108

   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 y around 0 77.4%

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

      1. sub-neg 77.4%

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

      2. log1p-define 85.0%

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

   5. Simplified 85.0%

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

   6. Taylor expanded in z around 0 85.0%

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

      1. +-commutative 85.0%

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

      2. associate-*r* 85.0%

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

      3. associate-*r* 85.0%

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

      4. distribute-lft-out 85.0%

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

      5. mul-1-neg 85.0%

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

   8. Simplified 85.0%

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

   if -7.4e12 < t < 2.15000000000000013e176

   1. Initial program 97.2%

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

   3. Taylor expanded in z around 0 96.6%

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

   4. Taylor expanded in z around inf 96.6%

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

      1. +-commutative 96.6%

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

      2. fma-define 96.6%

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

      3. mul-1-neg 96.6%

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

      4. log-rec 96.6%

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

      5. remove-double-neg 96.6%

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

      6. mul-1-neg 96.6%

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

      7. fma-neg 96.6%

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

   6. Simplified 96.6%

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

   7. Taylor expanded in t around 0 93.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+183}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{+109}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{elif}\;t \leq -7400000000000 \lor \neg \left(t \leq 2.15 \cdot 10^{+176}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \log z - a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.46 \lor \neg \left(y \leq 2.95 \cdot 10^{-83}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -0.46) (not (<= y 2.95e-83)))
   (* x (pow (/ z (exp t)) y))
   (* x (exp (* a (- (- b) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -0.46) || !(y <= 2.95e-83)) {
		tmp = x * pow((z / exp(t)), y);
	} else {
		tmp = x * exp((a * (-b - z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-0.46d0)) .or. (.not. (y <= 2.95d-83))) then
        tmp = x * ((z / exp(t)) ** y)
    else
        tmp = x * exp((a * (-b - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -0.46) || !(y <= 2.95e-83)) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else {
		tmp = x * Math.exp((a * (-b - z)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -0.46) or not (y <= 2.95e-83):
		tmp = x * math.pow((z / math.exp(t)), y)
	else:
		tmp = x * math.exp((a * (-b - z)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -0.46) || !(y <= 2.95e-83))
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-b) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -0.46) || ~((y <= 2.95e-83)))
		tmp = x * ((z / exp(t)) ^ y);
	else
		tmp = x * exp((a * (-b - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -0.46], N[Not[LessEqual[y, 2.95e-83]], $MachinePrecision]], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.46000000000000002 or 2.9499999999999998e-83 < 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 a around 0 87.9%

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

      1. *-commutative 87.9%

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

      2. exp-prod 86.5%

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

      3. exp-diff 86.5%

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

      4. rem-exp-log 86.5%

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

   5. Simplified 86.5%

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

   if -0.46000000000000002 < y < 2.9499999999999998e-83

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0 85.2%

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

      1. sub-neg 85.2%

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

      2. log1p-define 89.3%

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

   5. Simplified 89.3%

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

   6. Taylor expanded in z around 0 89.3%

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

      1. +-commutative 89.3%

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

      2. associate-*r* 89.3%

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

      3. associate-*r* 89.3%

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

      4. distribute-lft-out 89.3%

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

      5. mul-1-neg 89.3%

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

   8. Simplified 89.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.46 \lor \neg \left(y \leq 2.95 \cdot 10^{-83}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Taylor expanded in z around 0 95.7%

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

4. Taylor expanded in z around inf 95.7%

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

   1. +-commutative 95.7%

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

   2. fma-define 95.8%

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

   3. mul-1-neg 95.8%

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

   4. log-rec 95.8%

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

   5. remove-double-neg 95.8%

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

   6. mul-1-neg 95.8%

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

   7. fma-neg 95.7%

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

6. Simplified 95.7%

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

7. Final simplification 95.7%

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

Alternative 6: 72.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ t_2 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -7 \cdot 10^{+183}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-260}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-213}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- (- b) z))))) (t_2 (* x (exp (* y (- t))))))
   (if (<= t -7e+183)
     t_2
     (if (<= t -2.9e-260)
       t_1
       (if (<= t 8.8e-213) (* x (pow z y)) (if (<= t 6.8e-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 * (-b - z)));
	double t_2 = x * exp((y * -t));
	double tmp;
	if (t <= -7e+183) {
		tmp = t_2;
	} else if (t <= -2.9e-260) {
		tmp = t_1;
	} else if (t <= 8.8e-213) {
		tmp = x * pow(z, y);
	} else if (t <= 6.8e-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 * (-b - z)))
    t_2 = x * exp((y * -t))
    if (t <= (-7d+183)) then
        tmp = t_2
    else if (t <= (-2.9d-260)) then
        tmp = t_1
    else if (t <= 8.8d-213) then
        tmp = x * (z ** y)
    else if (t <= 6.8d-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 * (-b - z)));
	double t_2 = x * Math.exp((y * -t));
	double tmp;
	if (t <= -7e+183) {
		tmp = t_2;
	} else if (t <= -2.9e-260) {
		tmp = t_1;
	} else if (t <= 8.8e-213) {
		tmp = x * Math.pow(z, y);
	} else if (t <= 6.8e-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 * (-b - z)))
	t_2 = x * math.exp((y * -t))
	tmp = 0
	if t <= -7e+183:
		tmp = t_2
	elif t <= -2.9e-260:
		tmp = t_1
	elif t <= 8.8e-213:
		tmp = x * math.pow(z, y)
	elif t <= 6.8e-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(a * Float64(Float64(-b) - z))))
	t_2 = Float64(x * exp(Float64(y * Float64(-t))))
	tmp = 0.0
	if (t <= -7e+183)
		tmp = t_2;
	elseif (t <= -2.9e-260)
		tmp = t_1;
	elseif (t <= 8.8e-213)
		tmp = Float64(x * (z ^ y));
	elseif (t <= 6.8e-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 * (-b - z)));
	t_2 = x * exp((y * -t));
	tmp = 0.0;
	if (t <= -7e+183)
		tmp = t_2;
	elseif (t <= -2.9e-260)
		tmp = t_1;
	elseif (t <= 8.8e-213)
		tmp = x * (z ^ y);
	elseif (t <= 6.8e-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[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7e+183], t$95$2, If[LessEqual[t, -2.9e-260], t$95$1, If[LessEqual[t, 8.8e-213], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.8e-35], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.99999999999999974e183 or 6.8000000000000005e-35 < t

   1. Initial program 97.0%

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

   3. Taylor expanded in t around inf 88.4%

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

      1. mul-1-neg 88.4%

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

      2. distribute-lft-neg-out 88.4%

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

      3. *-commutative 88.4%

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

   5. Simplified 88.4%

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

   if -6.99999999999999974e183 < t < -2.8999999999999999e-260 or 8.80000000000000039e-213 < t < 6.8000000000000005e-35

   1. Initial program 95.3%

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

   3. Taylor expanded in y around 0 75.7%

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

      1. sub-neg 75.7%

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

      2. log1p-define 81.1%

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

   5. Simplified 81.1%

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

   6. Taylor expanded in z around 0 81.1%

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

      1. +-commutative 81.1%

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

      2. associate-*r* 81.1%

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

      3. associate-*r* 81.1%

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

      4. distribute-lft-out 81.1%

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

      5. mul-1-neg 81.1%

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

   8. Simplified 81.1%

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

   if -2.8999999999999999e-260 < t < 8.80000000000000039e-213

   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 a around 0 87.0%

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

      1. *-commutative 87.0%

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

      2. exp-prod 87.0%

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

      3. exp-diff 87.0%

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

      4. rem-exp-log 87.0%

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

   5. Simplified 87.0%

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

   6. Taylor expanded in t around 0 87.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+183}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-260}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-213}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-35}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -1.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-88}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;y \leq 0.037:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -1.2)
     t_1
     (if (<= y 1.06e-88)
       (* x (exp (* a (- b))))
       (if (<= y 0.037) (* x (exp (* y (- t)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double tmp;
	if (y <= -1.2) {
		tmp = t_1;
	} else if (y <= 1.06e-88) {
		tmp = x * exp((a * -b));
	} else if (y <= 0.037) {
		tmp = x * exp((y * -t));
	} 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 * (z ** y)
    if (y <= (-1.2d0)) then
        tmp = t_1
    else if (y <= 1.06d-88) then
        tmp = x * exp((a * -b))
    else if (y <= 0.037d0) then
        tmp = x * exp((y * -t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.pow(z, y);
	double tmp;
	if (y <= -1.2) {
		tmp = t_1;
	} else if (y <= 1.06e-88) {
		tmp = x * Math.exp((a * -b));
	} else if (y <= 0.037) {
		tmp = x * Math.exp((y * -t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	tmp = 0
	if y <= -1.2:
		tmp = t_1
	elif y <= 1.06e-88:
		tmp = x * math.exp((a * -b))
	elif y <= 0.037:
		tmp = x * math.exp((y * -t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -1.2)
		tmp = t_1;
	elseif (y <= 1.06e-88)
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	elseif (y <= 0.037)
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	tmp = 0.0;
	if (y <= -1.2)
		tmp = t_1;
	elseif (y <= 1.06e-88)
		tmp = x * exp((a * -b));
	elseif (y <= 0.037)
		tmp = x * exp((y * -t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.2], t$95$1, If[LessEqual[y, 1.06e-88], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.037], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.19999999999999996 or 0.0369999999999999982 < 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 a around 0 90.6%

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

      1. *-commutative 90.6%

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

      2. exp-prod 90.6%

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

      3. exp-diff 90.6%

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

      4. rem-exp-log 90.6%

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

   5. Simplified 90.6%

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

   6. Taylor expanded in t around 0 76.0%

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

   if -1.19999999999999996 < y < 1.06e-88

   1. Initial program 96.6%

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

   3. Taylor expanded in b around inf 84.3%

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

      1. mul-1-neg 84.3%

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

      2. distribute-rgt-neg-out 84.3%

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

   5. Simplified 84.3%

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

   if 1.06e-88 < y < 0.0369999999999999982

   1. Initial program 91.3%

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

   3. Taylor expanded in t around inf 74.4%

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

      1. mul-1-neg 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. *-commutative 74.4%

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

   5. Simplified 74.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-88}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;y \leq 0.037:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \lor \neg \left(y \leq 1.8\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.5) (not (<= y 1.8)))
   (* x (pow z y))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.5) || !(y <= 1.8)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-6.5d0)) .or. (.not. (y <= 1.8d0))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.5) || !(y <= 1.8)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6.5) or not (y <= 1.8):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.5) || !(y <= 1.8))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6.5) || ~((y <= 1.8)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.5], N[Not[LessEqual[y, 1.8]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5 or 1.80000000000000004 < 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 a around 0 90.6%

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

      1. *-commutative 90.6%

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

      2. exp-prod 90.6%

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

      3. exp-diff 90.6%

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

      4. rem-exp-log 90.6%

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

   5. Simplified 90.6%

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

   6. Taylor expanded in t around 0 76.0%

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

   if -6.5 < y < 1.80000000000000004

   1. Initial program 95.8%

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

   3. Taylor expanded in b around inf 78.5%

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

      1. mul-1-neg 78.5%

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

      2. distribute-rgt-neg-out 78.5%

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

   5. Simplified 78.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \lor \neg \left(y \leq 1.8\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+45}:\\ \;\;\;\;b \cdot \left(a \cdot \left(\frac{x}{a \cdot b} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.5e+45) (* b (* a (- (/ x (* a b)) x))) (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.5e+45) {
		tmp = b * (a * ((x / (a * b)) - x));
	} 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 <= (-2.5d+45)) then
        tmp = b * (a * ((x / (a * b)) - x))
    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 <= -2.5e+45) {
		tmp = b * (a * ((x / (a * b)) - x));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -2.5e+45:
		tmp = b * (a * ((x / (a * b)) - x))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.5e+45)
		tmp = Float64(b * Float64(a * Float64(Float64(x / Float64(a * b)) - x)));
	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 <= -2.5e+45)
		tmp = b * (a * ((x / (a * b)) - x));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.5e+45], N[(b * N[(a * N[(N[(x / N[(a * b), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.5e45

   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 b around inf 46.2%

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

      1. mul-1-neg 46.2%

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

      2. distribute-rgt-neg-out 46.2%

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

   5. Simplified 46.2%

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

   6. Taylor expanded in a around 0 22.2%

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

      1. +-commutative 22.2%

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

      2. associate-*r* 22.2%

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

      3. neg-mul-1 22.2%

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

   8. Simplified 22.2%

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

   9. Taylor expanded in b around inf 24.5%

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

       1. +-commutative 24.5%

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

       2. mul-1-neg 24.5%

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

       3. unsub-neg 24.5%

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

   11. Simplified 24.5%

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

   12. Taylor expanded in a around inf 35.4%

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

   if -2.5e45 < t

   1. Initial program 96.6%

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

   3. Taylor expanded in a around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. exp-prod 67.0%

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

      3. exp-diff 67.0%

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

      4. rem-exp-log 67.0%

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

   5. Simplified 67.0%

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

   6. Taylor expanded in t around 0 66.1%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+45}:\\ \;\;\;\;b \cdot \left(a \cdot \left(\frac{x}{a \cdot b} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 28.7% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \frac{x}{b}\\ \mathbf{if}\;y \leq -0.24:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-210}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \left(x \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (/ x b))))
   (if (<= y -0.24)
     t_1
     (if (<= y 1.1e-210) x (if (<= y 4e-7) t_1 (- (* z (* x a))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (x / b);
	double tmp;
	if (y <= -0.24) {
		tmp = t_1;
	} else if (y <= 1.1e-210) {
		tmp = x;
	} else if (y <= 4e-7) {
		tmp = t_1;
	} else {
		tmp = -(z * (x * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = b * (x / b)
    if (y <= (-0.24d0)) then
        tmp = t_1
    else if (y <= 1.1d-210) then
        tmp = x
    else if (y <= 4d-7) then
        tmp = t_1
    else
        tmp = -(z * (x * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (x / b);
	double tmp;
	if (y <= -0.24) {
		tmp = t_1;
	} else if (y <= 1.1e-210) {
		tmp = x;
	} else if (y <= 4e-7) {
		tmp = t_1;
	} else {
		tmp = -(z * (x * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (x / b)
	tmp = 0
	if y <= -0.24:
		tmp = t_1
	elif y <= 1.1e-210:
		tmp = x
	elif y <= 4e-7:
		tmp = t_1
	else:
		tmp = -(z * (x * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(x / b))
	tmp = 0.0
	if (y <= -0.24)
		tmp = t_1;
	elseif (y <= 1.1e-210)
		tmp = x;
	elseif (y <= 4e-7)
		tmp = t_1;
	else
		tmp = Float64(-Float64(z * Float64(x * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (x / b);
	tmp = 0.0;
	if (y <= -0.24)
		tmp = t_1;
	elseif (y <= 1.1e-210)
		tmp = x;
	elseif (y <= 4e-7)
		tmp = t_1;
	else
		tmp = -(z * (x * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(x / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.24], t$95$1, If[LessEqual[y, 1.1e-210], x, If[LessEqual[y, 4e-7], t$95$1, (-N[(z * N[(x * a), $MachinePrecision]), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.23999999999999999 or 1.09999999999999995e-210 < y < 3.9999999999999998e-7

   1. Initial program 97.3%

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

   3. Taylor expanded in b around inf 50.7%

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

      1. mul-1-neg 50.7%

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

      2. distribute-rgt-neg-out 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in a around 0 25.4%

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

      1. +-commutative 25.4%

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

      2. associate-*r* 25.4%

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

      3. neg-mul-1 25.4%

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

   8. Simplified 25.4%

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

   9. Taylor expanded in b around inf 30.4%

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

       1. +-commutative 30.4%

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

       2. mul-1-neg 30.4%

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

       3. unsub-neg 30.4%

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

   11. Simplified 30.4%

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

   12. Taylor expanded in b around 0 23.5%

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

   if -0.23999999999999999 < y < 1.09999999999999995e-210

   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 a around 0 54.8%

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

      1. *-commutative 54.8%

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

      2. exp-prod 44.3%

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

      3. exp-diff 44.3%

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

      4. rem-exp-log 44.3%

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

   5. Simplified 44.3%

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

   6. Taylor expanded in y around 0 43.9%

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

   if 3.9999999999999998e-7 < y

   1. Initial program 96.6%

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

   3. Taylor expanded in y around 0 36.3%

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

      1. sub-neg 36.3%

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

      2. log1p-define 41.2%

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

   5. Simplified 41.2%

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

   6. Taylor expanded in b around 0 5.6%

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

   7. Taylor expanded in z around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

      3. associate-*r* 3.4%

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

      4. *-commutative 3.4%

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

      5. associate-*l* 3.5%

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

   9. Simplified 3.5%

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

   10. Taylor expanded in a around inf 27.6%

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

       1. associate-*r* 24.4%

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

       2. *-commutative 24.4%

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

       3. neg-mul-1 24.4%

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

       4. distribute-rgt-neg-in 24.4%

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

       5. mul-1-neg 24.4%

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

       6. associate-*r* 24.4%

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

       7. mul-1-neg 24.4%

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

   12. Simplified 24.4%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.24:\\ \;\;\;\;b \cdot \frac{x}{b}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-210}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-7}:\\ \;\;\;\;b \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \left(x \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.4% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.26:\\ \;\;\;\;b \cdot \left(x \cdot \left(\frac{1}{b} - a\right)\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -0.26)
   (* b (* x (- (/ 1.0 b) a)))
   (if (<= y 8.2e-17) (* x (- 1.0 (* a b))) (* (- b) (* x a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -0.26) {
		tmp = b * (x * ((1.0 / b) - a));
	} else if (y <= 8.2e-17) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -b * (x * a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-0.26d0)) then
        tmp = b * (x * ((1.0d0 / b) - a))
    else if (y <= 8.2d-17) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = -b * (x * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -0.26) {
		tmp = b * (x * ((1.0 / b) - a));
	} else if (y <= 8.2e-17) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -b * (x * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -0.26:
		tmp = b * (x * ((1.0 / b) - a))
	elif y <= 8.2e-17:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = -b * (x * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -0.26)
		tmp = Float64(b * Float64(x * Float64(Float64(1.0 / b) - a)));
	elseif (y <= 8.2e-17)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(Float64(-b) * Float64(x * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -0.26)
		tmp = b * (x * ((1.0 / b) - a));
	elseif (y <= 8.2e-17)
		tmp = x * (1.0 - (a * b));
	else
		tmp = -b * (x * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -0.26], N[(b * N[(x * N[(N[(1.0 / b), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-17], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) * N[(x * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.26000000000000001

   1. Initial program 98.3%

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

   3. Taylor expanded in b around inf 38.3%

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

      1. mul-1-neg 38.3%

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

      2. distribute-rgt-neg-out 38.3%

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

   5. Simplified 38.3%

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

   6. Taylor expanded in a around 0 14.8%

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

      1. +-commutative 14.8%

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

      2. associate-*r* 14.8%

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

      3. neg-mul-1 14.8%

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

   8. Simplified 14.8%

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

   9. Taylor expanded in b around inf 26.1%

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

       1. +-commutative 26.1%

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

       2. mul-1-neg 26.1%

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

       3. unsub-neg 26.1%

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

   11. Simplified 26.1%

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

   12. Taylor expanded in x around 0 26.1%

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

   if -0.26000000000000001 < y < 8.2000000000000001e-17

   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 b around inf 80.5%

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

      1. mul-1-neg 80.5%

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

      2. distribute-rgt-neg-out 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in a around 0 49.4%

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

      1. +-commutative 49.4%

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

      2. associate-*r* 49.4%

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

      3. neg-mul-1 49.4%

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

   8. Simplified 49.4%

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

   9. Taylor expanded in x around 0 49.4%

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

       1. neg-mul-1 49.4%

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

       2. unsub-neg 49.4%

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

   11. Simplified 49.4%

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

   if 8.2000000000000001e-17 < y

   1. Initial program 95.3%

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

   3. Taylor expanded in b around inf 34.7%

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

      1. mul-1-neg 34.7%

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

      2. distribute-rgt-neg-out 34.7%

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

   5. Simplified 34.7%

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

   6. Taylor expanded in a around 0 12.6%

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

      1. +-commutative 12.6%

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

      2. associate-*r* 12.6%

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

      3. neg-mul-1 12.6%

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

   8. Simplified 12.6%

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

   9. Taylor expanded in a around inf 29.0%

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

       1. *-commutative 29.0%

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

       2. associate-*r* 29.0%

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

       3. associate-*r* 29.0%

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

       4. associate-*l* 29.0%

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

       5. mul-1-neg 29.0%

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

       6. *-commutative 29.0%

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

   11. Simplified 29.0%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.26:\\ \;\;\;\;b \cdot \left(x \cdot \left(\frac{1}{b} - a\right)\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.3% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;b \cdot \left(a \cdot \left(\frac{x}{a \cdot b} - x\right)\right)\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -0.15)
   (* b (* a (- (/ x (* a b)) x)))
   (if (<= y 7.2e-17) (* x (- 1.0 (* a b))) (* (- b) (* x a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -0.15) {
		tmp = b * (a * ((x / (a * b)) - x));
	} else if (y <= 7.2e-17) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -b * (x * a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-0.15d0)) then
        tmp = b * (a * ((x / (a * b)) - x))
    else if (y <= 7.2d-17) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = -b * (x * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -0.15) {
		tmp = b * (a * ((x / (a * b)) - x));
	} else if (y <= 7.2e-17) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -b * (x * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -0.15:
		tmp = b * (a * ((x / (a * b)) - x))
	elif y <= 7.2e-17:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = -b * (x * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -0.15)
		tmp = Float64(b * Float64(a * Float64(Float64(x / Float64(a * b)) - x)));
	elseif (y <= 7.2e-17)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(Float64(-b) * Float64(x * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -0.15)
		tmp = b * (a * ((x / (a * b)) - x));
	elseif (y <= 7.2e-17)
		tmp = x * (1.0 - (a * b));
	else
		tmp = -b * (x * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -0.15], N[(b * N[(a * N[(N[(x / N[(a * b), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e-17], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) * N[(x * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.149999999999999994

   1. Initial program 98.3%

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

   3. Taylor expanded in b around inf 38.3%

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

      1. mul-1-neg 38.3%

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

      2. distribute-rgt-neg-out 38.3%

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

   5. Simplified 38.3%

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

   6. Taylor expanded in a around 0 14.8%

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

      1. +-commutative 14.8%

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

      2. associate-*r* 14.8%

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

      3. neg-mul-1 14.8%

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

   8. Simplified 14.8%

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

   9. Taylor expanded in b around inf 26.1%

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

       1. +-commutative 26.1%

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

       2. mul-1-neg 26.1%

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

       3. unsub-neg 26.1%

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

   11. Simplified 26.1%

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

   12. Taylor expanded in a around inf 31.1%

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

   if -0.149999999999999994 < y < 7.1999999999999999e-17

   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 b around inf 80.5%

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

      1. mul-1-neg 80.5%

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

      2. distribute-rgt-neg-out 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in a around 0 49.4%

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

      1. +-commutative 49.4%

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

      2. associate-*r* 49.4%

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

      3. neg-mul-1 49.4%

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

   8. Simplified 49.4%

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

   9. Taylor expanded in x around 0 49.4%

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

       1. neg-mul-1 49.4%

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

       2. unsub-neg 49.4%

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

   11. Simplified 49.4%

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

   if 7.1999999999999999e-17 < y

   1. Initial program 95.3%

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

   3. Taylor expanded in b around inf 34.7%

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

      1. mul-1-neg 34.7%

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

      2. distribute-rgt-neg-out 34.7%

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

   5. Simplified 34.7%

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

   6. Taylor expanded in a around 0 12.6%

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

      1. +-commutative 12.6%

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

      2. associate-*r* 12.6%

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

      3. neg-mul-1 12.6%

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

   8. Simplified 12.6%

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

   9. Taylor expanded in a around inf 29.0%

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

       1. *-commutative 29.0%

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

       2. associate-*r* 29.0%

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

       3. associate-*r* 29.0%

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

       4. associate-*l* 29.0%

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

       5. mul-1-neg 29.0%

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

       6. *-commutative 29.0%

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

   11. Simplified 29.0%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;b \cdot \left(a \cdot \left(\frac{x}{a \cdot b} - x\right)\right)\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.0% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7800000000000 \lor \neg \left(y \leq 2.6 \cdot 10^{-20}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7800000000000.0) (not (<= y 2.6e-20))) (* a (* x (- b))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7800000000000.0) || !(y <= 2.6e-20)) {
		tmp = a * (x * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-7800000000000.0d0)) .or. (.not. (y <= 2.6d-20))) then
        tmp = a * (x * -b)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7800000000000.0) || !(y <= 2.6e-20)) {
		tmp = a * (x * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -7800000000000.0) or not (y <= 2.6e-20):
		tmp = a * (x * -b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -7800000000000.0) || !(y <= 2.6e-20))
		tmp = Float64(a * Float64(x * Float64(-b)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -7800000000000.0) || ~((y <= 2.6e-20)))
		tmp = a * (x * -b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -7800000000000.0], N[Not[LessEqual[y, 2.6e-20]], $MachinePrecision]], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -7.8e12 or 2.59999999999999995e-20 < y

   1. Initial program 96.6%

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

   3. Taylor expanded in b around inf 38.8%

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

      1. mul-1-neg 38.8%

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

      2. distribute-rgt-neg-out 38.8%

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

   5. Simplified 38.8%

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

   6. Taylor expanded in a around 0 14.9%

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

      1. +-commutative 14.9%

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

      2. associate-*r* 14.9%

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

      3. neg-mul-1 14.9%

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

   8. Simplified 14.9%

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

   9. Taylor expanded in a around inf 24.9%

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

       1. associate-*r* 24.9%

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

       2. mul-1-neg 24.9%

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

   11. Simplified 24.9%

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

   if -7.8e12 < y < 2.59999999999999995e-20

   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 a around 0 61.4%

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

      1. *-commutative 61.4%

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

      2. exp-prod 51.8%

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

      3. exp-diff 51.8%

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

      4. rem-exp-log 51.8%

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

   5. Simplified 51.8%

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

   6. Taylor expanded in y around 0 39.8%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7800000000000 \lor \neg \left(y \leq 2.6 \cdot 10^{-20}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.9% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6600000000000 \lor \neg \left(y \leq 4.2 \cdot 10^{-24}\right):\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6600000000000.0) (not (<= y 4.2e-24))) (* (- b) (* x a)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6600000000000.0) || !(y <= 4.2e-24)) {
		tmp = -b * (x * a);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-6600000000000.0d0)) .or. (.not. (y <= 4.2d-24))) then
        tmp = -b * (x * a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6600000000000.0) || !(y <= 4.2e-24)) {
		tmp = -b * (x * a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6600000000000.0) or not (y <= 4.2e-24):
		tmp = -b * (x * a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6600000000000.0) || !(y <= 4.2e-24))
		tmp = Float64(Float64(-b) * Float64(x * a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6600000000000.0) || ~((y <= 4.2e-24)))
		tmp = -b * (x * a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6600000000000.0], N[Not[LessEqual[y, 4.2e-24]], $MachinePrecision]], N[((-b) * N[(x * a), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -6.6e12 or 4.1999999999999999e-24 < y

   1. Initial program 96.6%

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

   3. Taylor expanded in b around inf 38.8%

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

      1. mul-1-neg 38.8%

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

      2. distribute-rgt-neg-out 38.8%

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

   5. Simplified 38.8%

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

   6. Taylor expanded in a around 0 14.9%

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

      1. +-commutative 14.9%

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

      2. associate-*r* 14.9%

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

      3. neg-mul-1 14.9%

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

   8. Simplified 14.9%

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

   9. Taylor expanded in a around inf 24.9%

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

       1. *-commutative 24.9%

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

       2. associate-*r* 24.9%

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

       3. associate-*r* 24.9%

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

       4. associate-*l* 25.7%

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

       5. mul-1-neg 25.7%

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

       6. *-commutative 25.7%

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

   11. Simplified 25.7%

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

   if -6.6e12 < y < 4.1999999999999999e-24

   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 a around 0 61.4%

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

      1. *-commutative 61.4%

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

      2. exp-prod 51.8%

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

      3. exp-diff 51.8%

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

      4. rem-exp-log 51.8%

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

   5. Simplified 51.8%

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

   6. Taylor expanded in y around 0 39.8%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6600000000000 \lor \neg \left(y \leq 4.2 \cdot 10^{-24}\right):\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 25.1% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-274} \lor \neg \left(b \leq 8.2 \cdot 10^{-267}\right):\\ \;\;\;\;b \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -5.2e-274) (not (<= b 8.2e-267))) (* b (/ x b)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -5.2e-274) || !(b <= 8.2e-267)) {
		tmp = b * (x / b);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-5.2d-274)) .or. (.not. (b <= 8.2d-267))) then
        tmp = b * (x / b)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -5.2e-274) || !(b <= 8.2e-267)) {
		tmp = b * (x / b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -5.2e-274) or not (b <= 8.2e-267):
		tmp = b * (x / b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -5.2e-274) || !(b <= 8.2e-267))
		tmp = Float64(b * Float64(x / b));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -5.2e-274) || ~((b <= 8.2e-267)))
		tmp = b * (x / b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -5.2e-274], N[Not[LessEqual[b, 8.2e-267]], $MachinePrecision]], N[(b * N[(x / b), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if b < -5.2e-274 or 8.20000000000000022e-267 < b

   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 b around inf 60.3%

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

      1. mul-1-neg 60.3%

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

      2. distribute-rgt-neg-out 60.3%

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

   5. Simplified 60.3%

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

   6. Taylor expanded in a around 0 30.2%

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

      1. +-commutative 30.2%

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

      2. associate-*r* 30.2%

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

      3. neg-mul-1 30.2%

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

   8. Simplified 30.2%

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

   9. Taylor expanded in b around inf 32.6%

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

       1. +-commutative 32.6%

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

       2. mul-1-neg 32.6%

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

       3. unsub-neg 32.6%

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

   11. Simplified 32.6%

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

   12. Taylor expanded in b around 0 26.3%

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

   if -5.2e-274 < b < 8.20000000000000022e-267

   1. Initial program 89.6%

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

   3. Taylor expanded in a around 0 86.1%

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

      1. *-commutative 86.1%

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

      2. exp-prod 72.7%

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

      3. exp-diff 72.7%

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

      4. rem-exp-log 72.7%

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

   5. Simplified 72.7%

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

   6. Taylor expanded in y around 0 52.4%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-274} \lor \neg \left(b \leq 8.2 \cdot 10^{-267}\right):\\ \;\;\;\;b \cdot \frac{x}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 31.6% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 6e-17) (* x (- 1.0 (* a b))) (* (- b) (* x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 6e-17) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -b * (x * a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= 6d-17) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = -b * (x * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 6e-17) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -b * (x * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 6e-17:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = -b * (x * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 6e-17)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(Float64(-b) * Float64(x * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 6e-17)
		tmp = x * (1.0 - (a * b));
	else
		tmp = -b * (x * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 6e-17], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) * N[(x * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.00000000000000012e-17

   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 67.9%

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

      1. mul-1-neg 67.9%

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

      2. distribute-rgt-neg-out 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in a around 0 39.1%

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

      1. +-commutative 39.1%

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

      2. associate-*r* 39.1%

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

      3. neg-mul-1 39.1%

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

   8. Simplified 39.1%

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

   9. Taylor expanded in x around 0 39.1%

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

       1. neg-mul-1 39.1%

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

       2. unsub-neg 39.1%

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

   11. Simplified 39.1%

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

   if 6.00000000000000012e-17 < y

   1. Initial program 95.3%

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

   3. Taylor expanded in b around inf 34.7%

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

      1. mul-1-neg 34.7%

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

      2. distribute-rgt-neg-out 34.7%

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

   5. Simplified 34.7%

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

   6. Taylor expanded in a around 0 12.6%

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

      1. +-commutative 12.6%

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

      2. associate-*r* 12.6%

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

      3. neg-mul-1 12.6%

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

   8. Simplified 12.6%

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

   9. Taylor expanded in a around inf 29.0%

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

       1. *-commutative 29.0%

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

       2. associate-*r* 29.0%

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

       3. associate-*r* 29.0%

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

       4. associate-*l* 29.0%

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

       5. mul-1-neg 29.0%

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

       6. *-commutative 29.0%

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

   11. Simplified 29.0%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(x \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 23.5% accurate, 31.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= y 8.2e-17) x (* a (* x b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 8.2e-17) {
		tmp = x;
	} else {
		tmp = a * (x * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= 8.2d-17) then
        tmp = x
    else
        tmp = a * (x * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 8.2e-17) {
		tmp = x;
	} else {
		tmp = a * (x * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 8.2e-17:
		tmp = x
	else:
		tmp = a * (x * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 8.2e-17)
		tmp = x;
	else
		tmp = Float64(a * Float64(x * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 8.2e-17)
		tmp = x;
	else
		tmp = a * (x * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 8.2e-17], x, N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.2000000000000001e-17

   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 a around 0 68.9%

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

      1. *-commutative 68.9%

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

      2. exp-prod 62.0%

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

      3. exp-diff 62.0%

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

      4. rem-exp-log 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in y around 0 29.6%

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

   if 8.2000000000000001e-17 < y

   1. Initial program 95.3%

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

   3. Taylor expanded in b around inf 34.7%

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

      1. mul-1-neg 34.7%

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

      2. distribute-rgt-neg-out 34.7%

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

   5. Simplified 34.7%

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

   6. Taylor expanded in a around 0 12.6%

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

      1. +-commutative 12.6%

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

      2. associate-*r* 12.6%

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

      3. neg-mul-1 12.6%

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

   8. Simplified 12.6%

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

   9. Taylor expanded in a around inf 29.0%

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

       1. associate-*r* 29.0%

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

       2. mul-1-neg 29.0%

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

   11. Simplified 29.0%

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

       1. add-sqr-sqrt 8.7%

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

       2. sqrt-unprod 34.8%

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

       3. sqr-neg 34.8%

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

       4. sqrt-unprod 15.3%

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

       5. add-sqr-sqrt 22.4%

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

       6. pow1 22.4%

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

   13. Applied egg-rr 22.4%

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

       1. unpow1 22.4%

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

   15. Simplified 22.4%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 19.6% accurate, 315.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Taylor expanded in a around 0 73.8%

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

   1. *-commutative 73.8%

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

   2. exp-prod 68.5%

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

   3. exp-diff 68.5%

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

   4. rem-exp-log 68.5%

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

5. Simplified 68.5%

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

6. Taylor expanded in y around 0 23.4%

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

7. Final simplification 23.4%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(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))))))