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

Percentage Accurate: 96.5% → 99.5%

Time: 14.7s

Alternatives: 13

Speedup: 1.5×

• 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: 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: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

   1. fma-define 96.0%

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

   2. sub-neg 96.0%

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

   3. log1p-define 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 92.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+83} \lor \neg \left(y \leq 6.5 \cdot 10^{+98}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot \left(z + b\right) - y \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.05e+83) (not (<= y 6.5e+98)))
   (* x (pow (/ z (exp t)) y))
   (* x (exp (- (* (- a) (+ z b)) (* y t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.05e+83) || !(y <= 6.5e+98)) {
		tmp = x * pow((z / exp(t)), y);
	} else {
		tmp = x * exp(((-a * (z + b)) - (y * t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.05d+83)) .or. (.not. (y <= 6.5d+98))) then
        tmp = x * ((z / exp(t)) ** y)
    else
        tmp = x * exp(((-a * (z + b)) - (y * t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.05e+83) || !(y <= 6.5e+98)) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else {
		tmp = x * Math.exp(((-a * (z + b)) - (y * t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.05e+83) or not (y <= 6.5e+98):
		tmp = x * math.pow((z / math.exp(t)), y)
	else:
		tmp = x * math.exp(((-a * (z + b)) - (y * t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.05e+83) || !(y <= 6.5e+98))
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	else
		tmp = Float64(x * exp(Float64(Float64(Float64(-a) * Float64(z + b)) - Float64(y * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.05e+83) || ~((y <= 6.5e+98)))
		tmp = x * ((z / exp(t)) ^ y);
	else
		tmp = x * exp(((-a * (z + b)) - (y * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.05e+83], N[Not[LessEqual[y, 6.5e+98]], $MachinePrecision]], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[((-a) * N[(z + b), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05000000000000001e83 or 6.4999999999999999e98 < 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. Step-by-step derivation

      1. fma-define 98.8%

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

      2. sub-neg 98.8%

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

      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. Taylor expanded in a around 0 92.8%

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

      1. *-commutative 92.8%

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

      2. exp-prod 92.8%

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

      3. exp-diff 92.8%

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

      4. rem-exp-log 92.8%

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

   7. Simplified 92.8%

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

   if -1.05000000000000001e83 < y < 6.4999999999999999e98

   1. Initial program 94.6%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*r* 99.9%

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

      4. distribute-lft-out 99.9%

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

      5. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in t around inf 96.7%

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

      1. neg-mul-1 96.7%

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

   8. Simplified 96.7%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+83} \lor \neg \left(y \leq 6.5 \cdot 10^{+98}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot \left(z + b\right) - y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.5× speedup

Math

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

FPCore

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

Python

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) - (a * (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[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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 z around 0 99.2%

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

   1. +-commutative 99.2%

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

   2. associate-*r* 99.2%

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

   3. associate-*r* 99.2%

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

   4. distribute-lft-out 99.2%

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

   5. mul-1-neg 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 72.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -35000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.95 \cdot 10^{-106}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -35000000000.0)
     t_1
     (if (<= y -3.95e-106)
       (* x (exp (* y (- t))))
       (if (<= y 2.8e+15) (* x (exp (* a (- b)))) 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 <= -35000000000.0) {
		tmp = t_1;
	} else if (y <= -3.95e-106) {
		tmp = x * exp((y * -t));
	} else if (y <= 2.8e+15) {
		tmp = x * exp((a * -b));
	} 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 <= (-35000000000.0d0)) then
        tmp = t_1
    else if (y <= (-3.95d-106)) then
        tmp = x * exp((y * -t))
    else if (y <= 2.8d+15) then
        tmp = x * exp((a * -b))
    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 <= -35000000000.0) {
		tmp = t_1;
	} else if (y <= -3.95e-106) {
		tmp = x * Math.exp((y * -t));
	} else if (y <= 2.8e+15) {
		tmp = x * Math.exp((a * -b));
	} 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 <= -35000000000.0:
		tmp = t_1
	elif y <= -3.95e-106:
		tmp = x * math.exp((y * -t))
	elif y <= 2.8e+15:
		tmp = x * math.exp((a * -b))
	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 <= -35000000000.0)
		tmp = t_1;
	elseif (y <= -3.95e-106)
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	elseif (y <= 2.8e+15)
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	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 <= -35000000000.0)
		tmp = t_1;
	elseif (y <= -3.95e-106)
		tmp = x * exp((y * -t));
	elseif (y <= 2.8e+15)
		tmp = x * exp((a * -b));
	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, -35000000000.0], t$95$1, If[LessEqual[y, -3.95e-106], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e+15], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.5e10 or 2.8e15 < 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. Step-by-step derivation

      1. fma-define 98.2%

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

      2. sub-neg 98.2%

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

      3. log1p-define 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in a around 0 89.5%

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

      1. *-commutative 89.5%

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

      2. exp-prod 89.5%

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

      3. exp-diff 89.5%

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

      4. rem-exp-log 89.5%

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

   7. Simplified 89.5%

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

   8. Taylor expanded in t around 0 74.5%

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

   if -3.5e10 < y < -3.94999999999999993e-106

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 70.9%

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

      1. mul-1-neg 70.9%

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

      2. distribute-lft-neg-out 70.9%

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

      3. *-commutative 70.9%

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

   5. Simplified 70.9%

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

   if -3.94999999999999993e-106 < y < 2.8e15

   1. Initial program 92.7%

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

   3. Taylor expanded in b around inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. distribute-rgt-neg-out 86.5%

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

   5. Simplified 86.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -35000000000:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq -3.95 \cdot 10^{-106}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.9 \cdot 10^{+22} \lor \neg \left(b \leq 2.1 \cdot 10^{+95}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right) - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -6.9e+22) (not (<= b 2.1e+95)))
   (* x (exp (* a (- b))))
   (* x (exp (- (* y (- t)) (* z a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -6.9e+22) || !(b <= 2.1e+95)) {
		tmp = x * exp((a * -b));
	} else {
		tmp = x * exp(((y * -t) - (z * 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 ((b <= (-6.9d+22)) .or. (.not. (b <= 2.1d+95))) then
        tmp = x * exp((a * -b))
    else
        tmp = x * exp(((y * -t) - (z * 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 ((b <= -6.9e+22) || !(b <= 2.1e+95)) {
		tmp = x * Math.exp((a * -b));
	} else {
		tmp = x * Math.exp(((y * -t) - (z * a)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -6.9e+22) or not (b <= 2.1e+95):
		tmp = x * math.exp((a * -b))
	else:
		tmp = x * math.exp(((y * -t) - (z * a)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -6.9e+22) || !(b <= 2.1e+95))
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	else
		tmp = Float64(x * exp(Float64(Float64(y * Float64(-t)) - Float64(z * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -6.9e+22) || ~((b <= 2.1e+95)))
		tmp = x * exp((a * -b));
	else
		tmp = x * exp(((y * -t) - (z * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -6.9e+22], N[Not[LessEqual[b, 2.1e+95]], $MachinePrecision]], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[(y * (-t)), $MachinePrecision] - N[(z * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.8999999999999998e22 or 2.1e95 < b

   1. Initial program 98.0%

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

   3. Taylor expanded in b around inf 78.7%

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

      1. mul-1-neg 78.7%

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

      2. distribute-rgt-neg-out 78.7%

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

   5. Simplified 78.7%

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

   if -6.8999999999999998e22 < b < 2.1e95

   1. Initial program 94.0%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*r* 99.9%

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

      4. distribute-lft-out 99.9%

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

      5. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in t around inf 88.7%

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

      1. neg-mul-1 88.7%

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

   8. Simplified 88.7%

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

   9. Taylor expanded in b around 0 82.4%

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

       1. distribute-lft-out 82.4%

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

   11. Simplified 82.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.9 \cdot 10^{+22} \lor \neg \left(b \leq 2.1 \cdot 10^{+95}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right) - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -40000 \lor \neg \left(y \leq 3.8 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -40000.0) (not (<= y 3.8e+15)))
   (* 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 <= -40000.0) || !(y <= 3.8e+15)) {
		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 <= (-40000.0d0)) .or. (.not. (y <= 3.8d+15))) 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 <= -40000.0) || !(y <= 3.8e+15)) {
		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 <= -40000.0) or not (y <= 3.8e+15):
		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 <= -40000.0) || !(y <= 3.8e+15))
		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 <= -40000.0) || ~((y <= 3.8e+15)))
		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, -40000.0], N[Not[LessEqual[y, 3.8e+15]], $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 < -4e4 or 3.8e15 < 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. Step-by-step derivation

      1. fma-define 98.3%

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

      2. sub-neg 98.3%

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

      3. log1p-define 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in a around 0 89.7%

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

      1. *-commutative 89.7%

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

      2. exp-prod 89.7%

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

      3. exp-diff 89.7%

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

      4. rem-exp-log 89.7%

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

   7. Simplified 89.7%

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

   8. Taylor expanded in t around 0 73.5%

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

   if -4e4 < y < 3.8e15

   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 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 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -40000 \lor \neg \left(y \leq 3.8 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 z around 0 99.2%

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

   1. +-commutative 99.2%

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

   2. associate-*r* 99.2%

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

   3. associate-*r* 99.2%

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

   4. distribute-lft-out 99.2%

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

   5. mul-1-neg 99.2%

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

5. Simplified 99.2%

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

6. Taylor expanded in t around inf 90.4%

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

   1. neg-mul-1 90.4%

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

8. Simplified 90.4%

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

9. Final simplification 90.4%

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

Alternative 8: 54.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+100}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.25e+100) (- x (* t (* x y))) (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.25e+100) {
		tmp = x - (t * (x * y));
	} 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 <= (-1.25d+100)) then
        tmp = x - (t * (x * y))
    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 <= -1.25e+100) {
		tmp = x - (t * (x * y));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.25e+100:
		tmp = x - (t * (x * y))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.25e+100)
		tmp = Float64(x - Float64(t * Float64(x * y)));
	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 <= -1.25e+100)
		tmp = x - (t * (x * y));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.25e+100], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.25e100

   1. Initial program 96.8%

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

   3. Taylor expanded in t around inf 93.5%

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

      1. mul-1-neg 93.5%

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

      2. distribute-lft-neg-out 93.5%

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

      3. *-commutative 93.5%

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

   5. Simplified 93.5%

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

   6. Taylor expanded in y around 0 38.9%

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

      1. mul-1-neg 38.9%

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

      2. unsub-neg 38.9%

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

      3. associate-*r* 38.4%

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

   8. Simplified 38.4%

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

   9. Taylor expanded in t around 0 38.9%

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

       1. *-commutative 38.9%

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

   11. Simplified 38.9%

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

   if -1.25e100 < t

   1. Initial program 95.4%

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

      1. fma-define 95.9%

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

      2. sub-neg 95.9%

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

      3. log1p-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around 0 69.8%

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

      1. *-commutative 69.8%

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

      2. exp-prod 67.6%

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

      3. exp-diff 67.6%

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

      4. rem-exp-log 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in t around 0 63.8%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+100}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.2% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-69}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+207}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -8.5e-69)
   (* x (- 1.0 (* a b)))
   (if (<= a 4.8e+207) (- x (* t (* x y))) (* a (- (* x b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -8.5e-69) {
		tmp = x * (1.0 - (a * b));
	} else if (a <= 4.8e+207) {
		tmp = x - (t * (x * y));
	} 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 (a <= (-8.5d-69)) then
        tmp = x * (1.0d0 - (a * b))
    else if (a <= 4.8d+207) then
        tmp = x - (t * (x * y))
    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 (a <= -8.5e-69) {
		tmp = x * (1.0 - (a * b));
	} else if (a <= 4.8e+207) {
		tmp = x - (t * (x * y));
	} else {
		tmp = a * -(x * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -8.5e-69:
		tmp = x * (1.0 - (a * b))
	elif a <= 4.8e+207:
		tmp = x - (t * (x * y))
	else:
		tmp = a * -(x * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -8.5e-69)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	elseif (a <= 4.8e+207)
		tmp = Float64(x - Float64(t * Float64(x * y)));
	else
		tmp = Float64(a * Float64(-Float64(x * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -8.5e-69)
		tmp = x * (1.0 - (a * b));
	elseif (a <= 4.8e+207)
		tmp = x - (t * (x * y));
	else
		tmp = a * -(x * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -8.5e-69], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.8e+207], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * (-N[(x * b), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.50000000000000046e-69

   1. Initial program 94.2%

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

   3. Taylor expanded in b around inf 71.3%

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

      1. mul-1-neg 71.3%

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

      2. distribute-rgt-neg-out 71.3%

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

   5. Simplified 71.3%

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

   6. Taylor expanded in a around 0 30.0%

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

      1. mul-1-neg 30.0%

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

      2. unsub-neg 30.0%

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

   8. Simplified 30.0%

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

   9. Taylor expanded in x around 0 34.2%

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

   if -8.50000000000000046e-69 < a < 4.8000000000000002e207

   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 t around inf 71.1%

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

      1. mul-1-neg 71.1%

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

      2. distribute-lft-neg-out 71.1%

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

      3. *-commutative 71.1%

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

   5. Simplified 71.1%

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

   6. Taylor expanded in y around 0 43.1%

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

      1. mul-1-neg 43.1%

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

      2. unsub-neg 43.1%

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

      3. associate-*r* 40.8%

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

   8. Simplified 40.8%

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

   9. Taylor expanded in t around 0 43.1%

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

       1. *-commutative 43.1%

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

   11. Simplified 43.1%

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

   if 4.8000000000000002e207 < a

   1. Initial program 84.6%

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

   3. Taylor expanded in b around inf 79.6%

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

      1. mul-1-neg 79.6%

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

      2. distribute-rgt-neg-out 79.6%

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

   5. Simplified 79.6%

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

   6. Taylor expanded in a around 0 24.4%

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

      1. mul-1-neg 24.4%

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

      2. unsub-neg 24.4%

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

   8. Simplified 24.4%

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

   9. Taylor expanded in a around inf 39.1%

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

       1. mul-1-neg 39.1%

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

       2. *-commutative 39.1%

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

       3. distribute-rgt-neg-in 39.1%

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

       4. distribute-rgt-neg-in 39.1%

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

   11. Simplified 39.1%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-69}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+207}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 28.0% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-25} \lor \neg \left(y \leq 0.00074\right):\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.6e-25) (not (<= y 0.00074))) (* a (- (* x b))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.6e-25) || !(y <= 0.00074)) {
		tmp = a * -(x * b);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.6d-25)) .or. (.not. (y <= 0.00074d0))) then
        tmp = a * -(x * b)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.6e-25) || !(y <= 0.00074)) {
		tmp = a * -(x * b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.6e-25) or not (y <= 0.00074):
		tmp = a * -(x * b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.6e-25) || !(y <= 0.00074))
		tmp = Float64(a * Float64(-Float64(x * b)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.6e-25) || ~((y <= 0.00074)))
		tmp = a * -(x * b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.6e-25], N[Not[LessEqual[y, 0.00074]], $MachinePrecision]], N[(a * (-N[(x * b), $MachinePrecision])), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.6000000000000001e-25 or 7.3999999999999999e-4 < 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 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 13.6%

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

      1. mul-1-neg 13.6%

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

      2. unsub-neg 13.6%

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

   8. Simplified 13.6%

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

   9. Taylor expanded in a around inf 18.5%

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

       1. mul-1-neg 18.5%

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

       2. *-commutative 18.5%

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

       3. distribute-rgt-neg-in 18.5%

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

       4. distribute-rgt-neg-in 18.5%

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

   11. Simplified 18.5%

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

   if -1.6000000000000001e-25 < y < 7.3999999999999999e-4

   1. Initial program 93.5%

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

   3. Taylor expanded in b around inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. distribute-rgt-neg-out 81.2%

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

   5. Simplified 81.2%

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

   6. Taylor expanded in a around 0 47.5%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-25} \lor \neg \left(y \leq 0.00074\right):\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.7% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-25}:\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \mathbf{elif}\;y \leq 16500000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.7e-25)
   (* a (- (* x b)))
   (if (<= y 16500000000000.0) x (* a (- (* x z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.7e-25) {
		tmp = a * -(x * b);
	} else if (y <= 16500000000000.0) {
		tmp = x;
	} else {
		tmp = a * -(x * 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 <= (-1.7d-25)) then
        tmp = a * -(x * b)
    else if (y <= 16500000000000.0d0) then
        tmp = x
    else
        tmp = a * -(x * 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 <= -1.7e-25) {
		tmp = a * -(x * b);
	} else if (y <= 16500000000000.0) {
		tmp = x;
	} else {
		tmp = a * -(x * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.7e-25:
		tmp = a * -(x * b)
	elif y <= 16500000000000.0:
		tmp = x
	else:
		tmp = a * -(x * z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.7e-25)
		tmp = Float64(a * Float64(-Float64(x * b)));
	elseif (y <= 16500000000000.0)
		tmp = x;
	else
		tmp = Float64(a * Float64(-Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.7e-25)
		tmp = a * -(x * b);
	elseif (y <= 16500000000000.0)
		tmp = x;
	else
		tmp = a * -(x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.7e-25], N[(a * (-N[(x * b), $MachinePrecision])), $MachinePrecision], If[LessEqual[y, 16500000000000.0], x, N[(a * (-N[(x * z), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.70000000000000001e-25

   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 39.6%

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

      1. mul-1-neg 39.6%

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

      2. distribute-rgt-neg-out 39.6%

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

   5. Simplified 39.6%

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

   6. Taylor expanded in a around 0 13.0%

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

      1. mul-1-neg 13.0%

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

      2. unsub-neg 13.0%

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

   8. Simplified 13.0%

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

   9. Taylor expanded in a around inf 18.4%

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

       1. mul-1-neg 18.4%

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

       2. *-commutative 18.4%

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

       3. distribute-rgt-neg-in 18.4%

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

       4. distribute-rgt-neg-in 18.4%

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

   11. Simplified 18.4%

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

   if -1.70000000000000001e-25 < y < 1.65e13

   1. Initial program 93.6%

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

   3. Taylor expanded in b around inf 81.7%

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

      1. mul-1-neg 81.7%

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

      2. distribute-rgt-neg-out 81.7%

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

   5. Simplified 81.7%

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

   6. Taylor expanded in a around 0 46.5%

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

   if 1.65e13 < 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 z around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. associate-*r* 98.3%

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

      3. associate-*r* 98.3%

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

      4. distribute-lft-out 98.3%

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

      5. mul-1-neg 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in z around inf 26.6%

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

      1. associate-*r* 26.6%

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

      2. mul-1-neg 26.6%

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

   8. Simplified 26.6%

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

   9. Taylor expanded in a around 0 3.6%

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

       1. mul-1-neg 3.6%

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

       2. unsub-neg 3.6%

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

       3. associate-*r* 5.0%

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

       4. *-commutative 5.0%

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

   11. Simplified 5.0%

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

   12. Taylor expanded in z around inf 19.7%

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

       1. associate-*r* 19.7%

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

       2. neg-mul-1 19.7%

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

       3. *-commutative 19.7%

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

   14. Simplified 19.7%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-25}:\\ \;\;\;\;a \cdot \left(-x \cdot b\right)\\ \mathbf{elif}\;y \leq 16500000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 28.6% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - a \cdot b\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * (1.0 - (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 * (1.0d0 - (a * b))
end function

Java

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

Python

def code(x, y, z, t, a, b):
	return x * (1.0 - (a * b))

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 b around inf 61.8%

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

   1. mul-1-neg 61.8%

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

   2. distribute-rgt-neg-out 61.8%

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

5. Simplified 61.8%

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

6. Taylor expanded in a around 0 33.8%

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

   1. mul-1-neg 33.8%

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

   2. unsub-neg 33.8%

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

8. Simplified 33.8%

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

9. Taylor expanded in x around 0 34.9%

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

10. Final simplification 34.9%

    \[\leadsto x \cdot \left(1 - a \cdot b\right) \]
11. Add Preprocessing

Alternative 13: 19.6% accurate, 315.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in b around inf 61.8%

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

   1. mul-1-neg 61.8%

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

   2. distribute-rgt-neg-out 61.8%

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

5. Simplified 61.8%

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

6. Taylor expanded in a around 0 26.0%

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

7. Final simplification 26.0%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(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))))))