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

Percentage Accurate: 98.0% → 98.0%

Time: 21.2s

Alternatives: 12

Speedup: 1.0×

• 41.3% 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: 98.0% 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: 98.0% 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 98.1%

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

3. Final simplification 98.1%

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

Alternative 2: 86.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-69} \lor \neg \left(y \leq 6.9 \cdot 10^{-28}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.9e-69) (not (<= y 6.9e-28)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (- (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.9e-69) || !(y <= 6.9e-28)) {
		tmp = x * exp((y * (log(z) - t)));
	} 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 <= (-1.9d-69)) .or. (.not. (y <= 6.9d-28))) then
        tmp = x * exp((y * (log(z) - t)))
    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 <= -1.9e-69) || !(y <= 6.9e-28)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp(-(a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.9e-69) or not (y <= 6.9e-28):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp(-(a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.9e-69) || !(y <= 6.9e-28))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(-Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.9e-69) || ~((y <= 6.9e-28)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp(-(a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.9e-69], N[Not[LessEqual[y, 6.9e-28]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[(-N[(a * b), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8999999999999999e-69 or 6.90000000000000001e-28 < y

   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 y around inf 92.9%

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

   if -1.8999999999999999e-69 < y < 6.90000000000000001e-28

   1. Initial program 99.1%

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

   3. Taylor expanded in b around inf 85.7%

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

      1. mul-1-neg 85.7%

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

      2. distribute-rgt-neg-out 85.7%

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

   5. Simplified 85.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-69} \lor \neg \left(y \leq 6.9 \cdot 10^{-28}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+24} \lor \neg \left(y \leq 7\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -5.6e+24) (not (<= y 7.0)))
   (* x (pow z y))
   (* x (exp (- (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.6e+24) || !(y <= 7.0)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp(-(a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-5.6d+24)) .or. (.not. (y <= 7.0d0))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp(-(a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.6e+24) || !(y <= 7.0)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp(-(a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -5.6e+24) or not (y <= 7.0):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp(-(a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -5.6e+24) || !(y <= 7.0))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(-Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -5.6e+24) || ~((y <= 7.0)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp(-(a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -5.6e+24], N[Not[LessEqual[y, 7.0]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[(-N[(a * b), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.6000000000000003e24 or 7 < y

   1. Initial program 97.6%

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

   3. Taylor expanded in y around inf 96.8%

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

   4. Taylor expanded in t around 0 79.4%

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

   if -5.6000000000000003e24 < y < 7

   1. Initial program 98.5%

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

   3. Taylor expanded in b around inf 80.1%

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

      1. mul-1-neg 80.1%

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

      2. distribute-rgt-neg-out 80.1%

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

   5. Simplified 80.1%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+24} \lor \neg \left(y \leq 7\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{-17} \lor \neg \left(y \leq 4.5 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.08e-17) (not (<= y 4.5e-5)))
   (* x (pow z y))
   (* x (- 1.0 (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.08e-17) || !(y <= 4.5e-5)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.08d-17)) .or. (.not. (y <= 4.5d-5))) then
        tmp = x * (z ** y)
    else
        tmp = x * (1.0d0 - (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.08e-17) || !(y <= 4.5e-5)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.08e-17) or not (y <= 4.5e-5):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * (1.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.08e-17) || !(y <= 4.5e-5))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.08e-17) || ~((y <= 4.5e-5)))
		tmp = x * (z ^ y);
	else
		tmp = x * (1.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.08e-17], N[Not[LessEqual[y, 4.5e-5]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.07999999999999995e-17 or 4.50000000000000028e-5 < y

   1. Initial program 97.7%

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

   3. Taylor expanded in y around inf 96.3%

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

   4. Taylor expanded in t around 0 76.3%

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

   if -1.07999999999999995e-17 < y < 4.50000000000000028e-5

   1. Initial program 98.4%

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

   3. Taylor expanded in b around inf 81.4%

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

      1. mul-1-neg 81.4%

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

      2. distribute-rgt-neg-out 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in a around 0 49.9%

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

      1. mul-1-neg 49.9%

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

      2. unsub-neg 49.9%

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

   8. Simplified 49.9%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{-17} \lor \neg \left(y \leq 4.5 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 33.6% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+44}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -9.5e+44)
   (* t (* x (- y)))
   (if (<= y 1.8e-51) (* x (- 1.0 (* a b))) (* x (- (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9.5e+44) {
		tmp = t * (x * -y);
	} else if (y <= 1.8e-51) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * -(a * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-9.5d+44)) then
        tmp = t * (x * -y)
    else if (y <= 1.8d-51) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = x * -(a * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9.5e+44) {
		tmp = t * (x * -y);
	} else if (y <= 1.8e-51) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * -(a * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -9.5e+44:
		tmp = t * (x * -y)
	elif y <= 1.8e-51:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = x * -(a * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -9.5e+44)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 1.8e-51)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(x * Float64(-Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -9.5e+44)
		tmp = t * (x * -y);
	elseif (y <= 1.8e-51)
		tmp = x * (1.0 - (a * b));
	else
		tmp = x * -(a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -9.5e+44], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e-51], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * (-N[(a * b), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5000000000000004e44

   1. Initial program 98.4%

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

   3. Taylor expanded in t around inf 62.6%

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

      1. associate-*r* 62.6%

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

      2. mul-1-neg 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in t around 0 34.0%

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

      1. mul-1-neg 34.0%

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

      2. unsub-neg 34.0%

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

   8. Simplified 34.0%

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

   9. Taylor expanded in t around inf 35.3%

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

   if -9.5000000000000004e44 < y < 1.8e-51

   1. Initial program 98.4%

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

   3. Taylor expanded in b around inf 81.4%

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

      1. mul-1-neg 81.4%

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

      2. distribute-rgt-neg-out 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in a around 0 48.2%

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

      1. mul-1-neg 48.2%

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

      2. unsub-neg 48.2%

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

   8. Simplified 48.2%

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

   if 1.8e-51 < y

   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 42.4%

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

      1. mul-1-neg 42.4%

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

      2. distribute-rgt-neg-out 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in a around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. unsub-neg 9.7%

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

      3. *-commutative 9.7%

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

   8. Simplified 9.7%

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

   9. Taylor expanded in a around inf 24.3%

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

       1. mul-1-neg 24.3%

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

       2. associate-*r* 30.8%

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

       3. *-commutative 30.8%

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

       4. distribute-rgt-neg-in 30.8%

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

       5. distribute-rgt-neg-in 30.8%

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

   11. Simplified 30.8%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+44}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.6% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+42}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.2e+42)
   (- x (* t (* x y)))
   (if (<= y 1.8e-51) (* x (- 1.0 (* a b))) (* x (- (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.2e+42) {
		tmp = x - (t * (x * y));
	} else if (y <= 1.8e-51) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * -(a * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-4.2d+42)) then
        tmp = x - (t * (x * y))
    else if (y <= 1.8d-51) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = x * -(a * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.2e+42) {
		tmp = x - (t * (x * y));
	} else if (y <= 1.8e-51) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * -(a * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4.2e+42:
		tmp = x - (t * (x * y))
	elif y <= 1.8e-51:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = x * -(a * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.2e+42)
		tmp = Float64(x - Float64(t * Float64(x * y)));
	elseif (y <= 1.8e-51)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(x * Float64(-Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4.2e+42)
		tmp = x - (t * (x * y));
	elseif (y <= 1.8e-51)
		tmp = x * (1.0 - (a * b));
	else
		tmp = x * -(a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.2e+42], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e-51], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * (-N[(a * b), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.19999999999999991e42

   1. Initial program 98.4%

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

   3. Taylor expanded in t around inf 62.6%

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

      1. associate-*r* 62.6%

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

      2. mul-1-neg 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in t around 0 35.4%

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

      1. mul-1-neg 35.4%

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

      2. unsub-neg 35.4%

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

      3. *-commutative 35.4%

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

   8. Simplified 35.4%

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

   if -4.19999999999999991e42 < y < 1.8e-51

   1. Initial program 98.4%

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

   3. Taylor expanded in b around inf 81.4%

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

      1. mul-1-neg 81.4%

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

      2. distribute-rgt-neg-out 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in a around 0 48.2%

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

      1. mul-1-neg 48.2%

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

      2. unsub-neg 48.2%

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

   8. Simplified 48.2%

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

   if 1.8e-51 < y

   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 42.4%

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

      1. mul-1-neg 42.4%

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

      2. distribute-rgt-neg-out 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in a around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. unsub-neg 9.7%

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

      3. *-commutative 9.7%

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

   8. Simplified 9.7%

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

   9. Taylor expanded in a around inf 24.3%

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

       1. mul-1-neg 24.3%

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

       2. associate-*r* 30.8%

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

       3. *-commutative 30.8%

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

       4. distribute-rgt-neg-in 30.8%

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

       5. distribute-rgt-neg-in 30.8%

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

   11. Simplified 30.8%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+42}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 25.8% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+24}:\\ \;\;\;\;b \cdot \left(x \cdot a\right)\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-53}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.3e+24) (* b (* x a)) (if (<= y 1.02e-53) x (* b (* x (- a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.3e+24) {
		tmp = b * (x * a);
	} else if (y <= 1.02e-53) {
		tmp = x;
	} else {
		tmp = b * (x * -a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.3d+24)) then
        tmp = b * (x * a)
    else if (y <= 1.02d-53) then
        tmp = x
    else
        tmp = b * (x * -a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.3e+24) {
		tmp = b * (x * a);
	} else if (y <= 1.02e-53) {
		tmp = x;
	} else {
		tmp = b * (x * -a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.3e+24:
		tmp = b * (x * a)
	elif y <= 1.02e-53:
		tmp = x
	else:
		tmp = b * (x * -a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.3e+24)
		tmp = Float64(b * Float64(x * a));
	elseif (y <= 1.02e-53)
		tmp = x;
	else
		tmp = Float64(b * Float64(x * Float64(-a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.3e+24)
		tmp = b * (x * a);
	elseif (y <= 1.02e-53)
		tmp = x;
	else
		tmp = b * (x * -a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.3e+24], N[(b * N[(x * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.02e-53], x, N[(b * N[(x * (-a)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2999999999999999e24

   1. Initial program 98.5%

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

   3. Taylor expanded in b around inf 25.6%

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

      1. mul-1-neg 25.6%

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

      2. distribute-rgt-neg-out 25.6%

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

   5. Simplified 25.6%

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

   6. Taylor expanded in a around 0 12.1%

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

      1. mul-1-neg 12.1%

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

      2. unsub-neg 12.1%

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

      3. *-commutative 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in a around inf 18.1%

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

       1. mul-1-neg 18.1%

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

       2. associate-*r* 15.4%

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

       3. *-commutative 15.4%

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

       4. distribute-rgt-neg-in 15.4%

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

       5. distribute-rgt-neg-in 15.4%

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

   11. Simplified 15.4%

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

       1. expm1-log1p-u 8.9%

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

       2. expm1-udef 16.0%

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

       3. add-sqr-sqrt 1.9%

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

       4. sqrt-unprod 16.1%

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

       5. sqr-neg 16.1%

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

       6. sqrt-unprod 14.1%

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

       7. add-sqr-sqrt 17.8%

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

   13. Applied egg-rr 17.8%

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

       1. expm1-def 10.7%

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

       2. expm1-log1p 12.8%

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

       3. associate-*r* 17.1%

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

   15. Simplified 17.1%

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

   if -2.2999999999999999e24 < y < 1.02000000000000002e-53

   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 84.0%

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

      1. mul-1-neg 84.0%

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

      2. distribute-rgt-neg-out 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in a around 0 37.7%

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

   if 1.02000000000000002e-53 < y

   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 42.4%

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

      1. mul-1-neg 42.4%

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

      2. distribute-rgt-neg-out 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in a around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. unsub-neg 9.7%

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

      3. *-commutative 9.7%

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

   8. Simplified 9.7%

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

   9. Taylor expanded in a around inf 24.3%

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

       1. mul-1-neg 24.3%

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

       2. associate-*r* 30.8%

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

       3. *-commutative 30.8%

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

       4. distribute-rgt-neg-in 30.8%

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

       5. distribute-rgt-neg-in 30.8%

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

   11. Simplified 30.8%

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

   12. Taylor expanded in x around 0 24.3%

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

       1. mul-1-neg 24.3%

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

       2. associate-*r* 30.8%

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

       3. *-commutative 30.8%

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

       4. associate-*r* 25.5%

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

   14. Simplified 25.5%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+24}:\\ \;\;\;\;b \cdot \left(x \cdot a\right)\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-53}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.0% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+24}:\\ \;\;\;\;b \cdot \left(x \cdot a\right)\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.6e+24) (* b (* x a)) (if (<= y 7.4e-52) x (* x (- (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.6e+24) {
		tmp = b * (x * a);
	} else if (y <= 7.4e-52) {
		tmp = x;
	} else {
		tmp = x * -(a * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-1.6d+24)) then
        tmp = b * (x * a)
    else if (y <= 7.4d-52) then
        tmp = x
    else
        tmp = x * -(a * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.6e+24) {
		tmp = b * (x * a);
	} else if (y <= 7.4e-52) {
		tmp = x;
	} else {
		tmp = x * -(a * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.6e+24:
		tmp = b * (x * a)
	elif y <= 7.4e-52:
		tmp = x
	else:
		tmp = x * -(a * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.6e+24)
		tmp = Float64(b * Float64(x * a));
	elseif (y <= 7.4e-52)
		tmp = x;
	else
		tmp = Float64(x * Float64(-Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.6e+24)
		tmp = b * (x * a);
	elseif (y <= 7.4e-52)
		tmp = x;
	else
		tmp = x * -(a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.6e+24], N[(b * N[(x * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.4e-52], x, N[(x * (-N[(a * b), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5999999999999999e24

   1. Initial program 98.5%

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

   3. Taylor expanded in b around inf 25.6%

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

      1. mul-1-neg 25.6%

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

      2. distribute-rgt-neg-out 25.6%

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

   5. Simplified 25.6%

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

   6. Taylor expanded in a around 0 12.1%

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

      1. mul-1-neg 12.1%

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

      2. unsub-neg 12.1%

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

      3. *-commutative 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in a around inf 18.1%

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

       1. mul-1-neg 18.1%

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

       2. associate-*r* 15.4%

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

       3. *-commutative 15.4%

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

       4. distribute-rgt-neg-in 15.4%

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

       5. distribute-rgt-neg-in 15.4%

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

   11. Simplified 15.4%

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

       1. expm1-log1p-u 8.9%

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

       2. expm1-udef 16.0%

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

       3. add-sqr-sqrt 1.9%

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

       4. sqrt-unprod 16.1%

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

       5. sqr-neg 16.1%

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

       6. sqrt-unprod 14.1%

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

       7. add-sqr-sqrt 17.8%

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

   13. Applied egg-rr 17.8%

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

       1. expm1-def 10.7%

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

       2. expm1-log1p 12.8%

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

       3. associate-*r* 17.1%

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

   15. Simplified 17.1%

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

   if -1.5999999999999999e24 < y < 7.3999999999999995e-52

   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 84.0%

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

      1. mul-1-neg 84.0%

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

      2. distribute-rgt-neg-out 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in a around 0 37.7%

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

   if 7.3999999999999995e-52 < y

   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 42.4%

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

      1. mul-1-neg 42.4%

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

      2. distribute-rgt-neg-out 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in a around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. unsub-neg 9.7%

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

      3. *-commutative 9.7%

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

   8. Simplified 9.7%

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

   9. Taylor expanded in a around inf 24.3%

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

       1. mul-1-neg 24.3%

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

       2. associate-*r* 30.8%

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

       3. *-commutative 30.8%

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

       4. distribute-rgt-neg-in 30.8%

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

       5. distribute-rgt-neg-in 30.8%

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

   11. Simplified 30.8%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+24}:\\ \;\;\;\;b \cdot \left(x \cdot a\right)\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 28.8% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.32e+39)
   (* x (* y (- t)))
   (if (<= y 5.5e-54) x (* x (- (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.32e+39) {
		tmp = x * (y * -t);
	} else if (y <= 5.5e-54) {
		tmp = x;
	} else {
		tmp = x * -(a * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-1.32d+39)) then
        tmp = x * (y * -t)
    else if (y <= 5.5d-54) then
        tmp = x
    else
        tmp = x * -(a * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.32e+39) {
		tmp = x * (y * -t);
	} else if (y <= 5.5e-54) {
		tmp = x;
	} else {
		tmp = x * -(a * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.32e+39:
		tmp = x * (y * -t)
	elif y <= 5.5e-54:
		tmp = x
	else:
		tmp = x * -(a * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.32e+39)
		tmp = Float64(x * Float64(y * Float64(-t)));
	elseif (y <= 5.5e-54)
		tmp = x;
	else
		tmp = Float64(x * Float64(-Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.32e+39)
		tmp = x * (y * -t);
	elseif (y <= 5.5e-54)
		tmp = x;
	else
		tmp = x * -(a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.32e+39], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e-54], x, N[(x * (-N[(a * b), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.32e39

   1. Initial program 98.4%

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

   3. Taylor expanded in t around inf 62.6%

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

      1. associate-*r* 62.6%

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

      2. mul-1-neg 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in t around 0 34.0%

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

      1. mul-1-neg 34.0%

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

      2. unsub-neg 34.0%

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

   8. Simplified 34.0%

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

   9. Taylor expanded in t around inf 35.3%

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

       1. mul-1-neg 35.3%

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

       2. *-commutative 35.3%

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

       3. associate-*r* 33.7%

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

       4. distribute-lft-neg-in 33.7%

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

       5. *-commutative 33.7%

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

   11. Simplified 33.7%

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

   if -1.32e39 < y < 5.50000000000000046e-54

   1. Initial program 98.4%

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

   3. Taylor expanded in b around inf 81.4%

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

      1. mul-1-neg 81.4%

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

      2. distribute-rgt-neg-out 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in a around 0 36.6%

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

   if 5.50000000000000046e-54 < y

   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 42.4%

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

      1. mul-1-neg 42.4%

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

      2. distribute-rgt-neg-out 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in a around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. unsub-neg 9.7%

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

      3. *-commutative 9.7%

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

   8. Simplified 9.7%

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

   9. Taylor expanded in a around inf 24.3%

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

       1. mul-1-neg 24.3%

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

       2. associate-*r* 30.8%

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

       3. *-commutative 30.8%

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

       4. distribute-rgt-neg-in 30.8%

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

       5. distribute-rgt-neg-in 30.8%

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

   11. Simplified 30.8%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 28.6% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+37}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -5e+37) (* t (* x (- y))) (if (<= y 3.1e-52) x (* x (- (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5e+37) {
		tmp = t * (x * -y);
	} else if (y <= 3.1e-52) {
		tmp = x;
	} else {
		tmp = x * -(a * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-5d+37)) then
        tmp = t * (x * -y)
    else if (y <= 3.1d-52) then
        tmp = x
    else
        tmp = x * -(a * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5e+37) {
		tmp = t * (x * -y);
	} else if (y <= 3.1e-52) {
		tmp = x;
	} else {
		tmp = x * -(a * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -5e+37:
		tmp = t * (x * -y)
	elif y <= 3.1e-52:
		tmp = x
	else:
		tmp = x * -(a * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -5e+37)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 3.1e-52)
		tmp = x;
	else
		tmp = Float64(x * Float64(-Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -5e+37)
		tmp = t * (x * -y);
	elseif (y <= 3.1e-52)
		tmp = x;
	else
		tmp = x * -(a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -5e+37], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e-52], x, N[(x * (-N[(a * b), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.99999999999999989e37

   1. Initial program 98.4%

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

   3. Taylor expanded in t around inf 62.6%

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

      1. associate-*r* 62.6%

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

      2. mul-1-neg 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in t around 0 34.0%

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

      1. mul-1-neg 34.0%

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

      2. unsub-neg 34.0%

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

   8. Simplified 34.0%

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

   9. Taylor expanded in t around inf 35.3%

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

   if -4.99999999999999989e37 < y < 3.0999999999999999e-52

   1. Initial program 98.4%

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

   3. Taylor expanded in b around inf 81.4%

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

      1. mul-1-neg 81.4%

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

      2. distribute-rgt-neg-out 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in a around 0 36.6%

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

   if 3.0999999999999999e-52 < y

   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 42.4%

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

      1. mul-1-neg 42.4%

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

      2. distribute-rgt-neg-out 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in a around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. unsub-neg 9.7%

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

      3. *-commutative 9.7%

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

   8. Simplified 9.7%

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

   9. Taylor expanded in a around inf 24.3%

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

       1. mul-1-neg 24.3%

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

       2. associate-*r* 30.8%

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

       3. *-commutative 30.8%

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

       4. distribute-rgt-neg-in 30.8%

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

       5. distribute-rgt-neg-in 30.8%

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

   11. Simplified 30.8%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+37}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 24.3% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+24} \lor \neg \left(y \leq 1.8 \cdot 10^{-51}\right):\\ \;\;\;\;b \cdot \left(x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.5e+24) (not (<= y 1.8e-51))) (* b (* x a)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.5e+24) || !(y <= 1.8e-51)) {
		tmp = b * (x * a);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.5d+24)) .or. (.not. (y <= 1.8d-51))) then
        tmp = b * (x * a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.5e+24) || !(y <= 1.8e-51)) {
		tmp = b * (x * a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.5e+24) or not (y <= 1.8e-51):
		tmp = b * (x * a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.5e+24) || !(y <= 1.8e-51))
		tmp = Float64(b * Float64(x * a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.5e+24) || ~((y <= 1.8e-51)))
		tmp = b * (x * a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.5e+24], N[Not[LessEqual[y, 1.8e-51]], $MachinePrecision]], N[(b * N[(x * a), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.49999999999999997e24 or 1.8e-51 < y

   1. Initial program 97.9%

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

   3. Taylor expanded in b around inf 34.5%

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

      1. mul-1-neg 34.5%

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

      2. distribute-rgt-neg-out 34.5%

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

   5. Simplified 34.5%

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

   6. Taylor expanded in a around 0 10.8%

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

      1. mul-1-neg 10.8%

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

      2. unsub-neg 10.8%

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

      3. *-commutative 10.8%

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

   8. Simplified 10.8%

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

   9. Taylor expanded in a around inf 21.4%

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

       1. mul-1-neg 21.4%

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

       2. associate-*r* 23.5%

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

       3. *-commutative 23.5%

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

       4. distribute-rgt-neg-in 23.5%

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

       5. distribute-rgt-neg-in 23.5%

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

   11. Simplified 23.5%

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

       1. expm1-log1p-u 19.6%

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

       2. expm1-udef 33.7%

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

       3. add-sqr-sqrt 16.1%

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

       4. sqrt-unprod 31.4%

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

       5. sqr-neg 31.4%

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

       6. sqrt-unprod 16.8%

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

       7. add-sqr-sqrt 29.3%

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

   13. Applied egg-rr 29.3%

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

       1. expm1-def 15.1%

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

       2. expm1-log1p 17.1%

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

       3. associate-*r* 19.1%

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

   15. Simplified 19.1%

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

   if -1.49999999999999997e24 < y < 1.8e-51

   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 84.0%

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

      1. mul-1-neg 84.0%

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

      2. distribute-rgt-neg-out 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in a around 0 37.7%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+24} \lor \neg \left(y \leq 1.8 \cdot 10^{-51}\right):\\ \;\;\;\;b \cdot \left(x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 19.8% 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 98.1%

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

3. Taylor expanded in b around inf 56.9%

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

   1. mul-1-neg 56.9%

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

   2. distribute-rgt-neg-out 56.9%

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

5. Simplified 56.9%

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

6. Taylor expanded in a around 0 19.6%

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

7. Final simplification 19.6%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))))))