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

Percentage Accurate: 96.3% → 99.2%

Time: 17.1s

Alternatives: 11

Speedup: 1.5×

• 40.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% 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 96.5%

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

3. Taylor expanded in z around 0 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. associate-*r* 99.9%

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

   2. associate-*r* 99.9%

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

   3. distribute-lft-out 99.9%

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

   4. mul-1-neg 99.9%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 85.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-20} \lor \neg \left(y \leq 6.4 \cdot 10^{+52}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -9e-20) (not (<= y 6.4e+52)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (* a (- (- b) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -9e-20) || !(y <= 6.4e+52)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((a * (-b - z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-9d-20)) .or. (.not. (y <= 6.4d+52))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp((a * (-b - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -9e-20) || !(y <= 6.4e+52)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((a * (-b - z)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -9e-20) or not (y <= 6.4e+52):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((a * (-b - z)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -9e-20) || !(y <= 6.4e+52))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-b) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -9e-20) || ~((y <= 6.4e+52)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp((a * (-b - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -9e-20], N[Not[LessEqual[y, 6.4e+52]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.0000000000000003e-20 or 6.4e52 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0 94.9%

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

   if -9.0000000000000003e-20 < y < 6.4e52

   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 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. associate-*r* 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-lft-out 99.9%

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

      4. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 91.5%

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

      1. mul-1-neg 91.5%

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

      2. +-commutative 91.5%

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

      3. distribute-lft-neg-in 91.5%

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

      4. +-commutative 91.5%

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

   8. Simplified 91.5%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-20} \lor \neg \left(y \leq 6.4 \cdot 10^{+52}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -16500000000000 \lor \neg \left(y \leq 4 \cdot 10^{+60}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -16500000000000.0) (not (<= y 4e+60)))
   (* x (pow z y))
   (* x (exp (* a (- (- b) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -16500000000000.0) || !(y <= 4e+60)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((a * (-b - z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-16500000000000.0d0)) .or. (.not. (y <= 4d+60))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((a * (-b - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -16500000000000.0) || !(y <= 4e+60)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((a * (-b - z)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -16500000000000.0) or not (y <= 4e+60):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((a * (-b - z)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -16500000000000.0) || !(y <= 4e+60))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-b) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -16500000000000.0) || ~((y <= 4e+60)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((a * (-b - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -16500000000000.0], N[Not[LessEqual[y, 4e+60]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[((-b) - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65e13 or 3.9999999999999998e60 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0 95.4%

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

   4. Taylor expanded in t around 0 78.8%

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

      1. *-commutative 78.8%

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

   6. Simplified 78.8%

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

   if -1.65e13 < y < 3.9999999999999998e60

   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. associate-*r* 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-lft-out 99.9%

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

      4. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 88.3%

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

      1. mul-1-neg 88.3%

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

      2. +-commutative 88.3%

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

      3. distribute-lft-neg-in 88.3%

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

      4. +-commutative 88.3%

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

   8. Simplified 88.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -16500000000000 \lor \neg \left(y \leq 4 \cdot 10^{+60}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-b\right) - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -16000000000000 \lor \neg \left(y \leq 7.2 \cdot 10^{+60}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -16000000000000.0) (not (<= y 7.2e+60)))
   (* 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 <= -16000000000000.0) || !(y <= 7.2e+60)) {
		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 <= (-16000000000000.0d0)) .or. (.not. (y <= 7.2d+60))) 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 <= -16000000000000.0) || !(y <= 7.2e+60)) {
		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 <= -16000000000000.0) or not (y <= 7.2e+60):
		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 <= -16000000000000.0) || !(y <= 7.2e+60))
		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 <= -16000000000000.0) || ~((y <= 7.2e+60)))
		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, -16000000000000.0], N[Not[LessEqual[y, 7.2e+60]], $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 < -1.6e13 or 7.19999999999999935e60 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0 95.4%

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

   4. Taylor expanded in t around 0 78.8%

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

      1. *-commutative 78.8%

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

   6. Simplified 78.8%

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

   if -1.6e13 < y < 7.19999999999999935e60

   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. associate-*r* 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-lft-out 99.9%

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

      4. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in b around inf 82.4%

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

      1. associate-*r* 82.4%

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

      2. mul-1-neg 82.4%

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

   8. Simplified 82.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -16000000000000 \lor \neg \left(y \leq 7.2 \cdot 10^{+60}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-21} \lor \neg \left(y \leq 6.4 \cdot 10^{+52}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -5e-21) (not (<= y 6.4e+52)))
   (* x (pow z y))
   (* x (- 1.0 (* a (+ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5e-21) || !(y <= 6.4e+52)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * (1.0 - (a * (z + b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-5d-21)) .or. (.not. (y <= 6.4d+52))) then
        tmp = x * (z ** y)
    else
        tmp = x * (1.0d0 - (a * (z + b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5e-21) || !(y <= 6.4e+52)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * (1.0 - (a * (z + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -5e-21) or not (y <= 6.4e+52):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * (1.0 - (a * (z + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -5e-21) || !(y <= 6.4e+52))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * Float64(1.0 - Float64(a * Float64(z + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -5e-21) || ~((y <= 6.4e+52)))
		tmp = x * (z ^ y);
	else
		tmp = x * (1.0 - (a * (z + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -5e-21], N[Not[LessEqual[y, 6.4e+52]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999973e-21 or 6.4e52 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0 95.0%

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

   4. Taylor expanded in t around 0 74.9%

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

      1. *-commutative 74.9%

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

   6. Simplified 74.9%

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

   if -4.99999999999999973e-21 < y < 6.4e52

   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 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. associate-*r* 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-lft-out 99.9%

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

      4. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 91.4%

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

      1. mul-1-neg 91.4%

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

      2. +-commutative 91.4%

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

      3. distribute-lft-neg-in 91.4%

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

      4. +-commutative 91.4%

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

   8. Simplified 91.4%

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

   9. Taylor expanded in a around 0 44.1%

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

       1. mul-1-neg 44.1%

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

       2. unsub-neg 44.1%

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

   11. Simplified 44.1%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-21} \lor \neg \left(y \leq 6.4 \cdot 10^{+52}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.7% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00066 \lor \neg \left(y \leq 6.6 \cdot 10^{-29}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -0.00066) (not (<= y 6.6e-29))) (* a (* x (- b))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -0.00066) || !(y <= 6.6e-29)) {
		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 <= (-0.00066d0)) .or. (.not. (y <= 6.6d-29))) 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 <= -0.00066) || !(y <= 6.6e-29)) {
		tmp = a * (x * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -0.00066) or not (y <= 6.6e-29):
		tmp = a * (x * -b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -0.00066) || !(y <= 6.6e-29))
		tmp = Float64(a * Float64(x * Float64(-b)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -0.00066) || ~((y <= 6.6e-29)))
		tmp = a * (x * -b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -0.00066], N[Not[LessEqual[y, 6.6e-29]], $MachinePrecision]], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -6.6e-4 or 6.60000000000000055e-29 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 37.2%

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

      1. associate-*r* 37.2%

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

      2. mul-1-neg 37.2%

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

   8. Simplified 37.2%

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

   9. Taylor expanded in a around 0 13.0%

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

       1. mul-1-neg 13.0%

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

       2. unsub-neg 13.0%

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

   11. Simplified 13.0%

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

   12. Taylor expanded in a around inf 26.4%

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

       1. neg-mul-1 26.4%

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

       2. distribute-lft-neg-in 26.4%

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

   14. Simplified 26.4%

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

   if -6.6e-4 < y < 6.60000000000000055e-29

   1. Initial program 92.8%

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

   3. Taylor expanded in a around 0 49.5%

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

   4. Taylor expanded in y around 0 34.9%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00066 \lor \neg \left(y \leq 6.6 \cdot 10^{-29}\right):\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.7% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00082 \lor \neg \left(y \leq 2.45 \cdot 10^{-42}\right):\\ \;\;\;\;\left(x \cdot a\right) \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -0.00082) (not (<= y 2.45e-42))) (* (* x a) (- b)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -0.00082) || !(y <= 2.45e-42)) {
		tmp = (x * a) * -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 <= (-0.00082d0)) .or. (.not. (y <= 2.45d-42))) then
        tmp = (x * a) * -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 <= -0.00082) || !(y <= 2.45e-42)) {
		tmp = (x * a) * -b;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -0.00082) or not (y <= 2.45e-42):
		tmp = (x * a) * -b
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -0.00082) || !(y <= 2.45e-42))
		tmp = Float64(Float64(x * a) * Float64(-b));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -0.00082) || ~((y <= 2.45e-42)))
		tmp = (x * a) * -b;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -0.00082], N[Not[LessEqual[y, 2.45e-42]], $MachinePrecision]], N[(N[(x * a), $MachinePrecision] * (-b)), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -8.1999999999999998e-4 or 2.45e-42 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 37.2%

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

      1. associate-*r* 37.2%

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

      2. mul-1-neg 37.2%

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

   8. Simplified 37.2%

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

   9. Taylor expanded in a around 0 13.0%

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

       1. mul-1-neg 13.0%

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

       2. unsub-neg 13.0%

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

   11. Simplified 13.0%

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

   12. Taylor expanded in a around inf 26.4%

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

       1. neg-mul-1 26.4%

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

       2. *-commutative 26.4%

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

       3. associate-*r* 28.5%

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

       4. distribute-rgt-neg-in 28.5%

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

   14. Simplified 28.5%

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

   if -8.1999999999999998e-4 < y < 2.45e-42

   1. Initial program 92.8%

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

   3. Taylor expanded in a around 0 49.5%

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

   4. Taylor expanded in y around 0 34.9%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00082 \lor \neg \left(y \leq 2.45 \cdot 10^{-42}\right):\\ \;\;\;\;\left(x \cdot a\right) \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.7% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot a\right) \cdot \left(-b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 4e-28) (* x (- 1.0 (* a (+ z b)))) (* (* x a) (- b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 4e-28) {
		tmp = x * (1.0 - (a * (z + 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 <= 4d-28) then
        tmp = x * (1.0d0 - (a * (z + 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 <= 4e-28) {
		tmp = x * (1.0 - (a * (z + b)));
	} else {
		tmp = (x * a) * -b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 4e-28:
		tmp = x * (1.0 - (a * (z + b)))
	else:
		tmp = (x * a) * -b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 4e-28)
		tmp = Float64(x * Float64(1.0 - Float64(a * Float64(z + b))));
	else
		tmp = Float64(Float64(x * a) * Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 4e-28)
		tmp = x * (1.0 - (a * (z + b)));
	else
		tmp = (x * a) * -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 4e-28], N[(x * N[(1.0 - N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * a), $MachinePrecision] * (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999988e-28

   1. Initial program 95.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. associate-*r* 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-lft-out 99.9%

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

      4. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 75.8%

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

      1. mul-1-neg 75.8%

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

      2. +-commutative 75.8%

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

      3. distribute-lft-neg-in 75.8%

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

      4. +-commutative 75.8%

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

   8. Simplified 75.8%

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

   9. Taylor expanded in a around 0 36.5%

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

       1. mul-1-neg 36.5%

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

       2. unsub-neg 36.5%

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

   11. Simplified 36.5%

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

   if 3.99999999999999988e-28 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 37.9%

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

      1. associate-*r* 37.9%

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

      2. mul-1-neg 37.9%

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

   8. Simplified 37.9%

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

   9. Taylor expanded in a around 0 10.0%

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

       1. mul-1-neg 10.0%

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

       2. unsub-neg 10.0%

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

   11. Simplified 10.0%

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

   12. Taylor expanded in a around inf 30.2%

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

       1. neg-mul-1 30.2%

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

       2. *-commutative 30.2%

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

       3. associate-*r* 33.9%

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

       4. distribute-rgt-neg-in 33.9%

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

   14. Simplified 33.9%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(1 - a \cdot \left(z + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot a\right) \cdot \left(-b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.3% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot a\right) \cdot \left(-b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 4e-28) (* 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 <= 4e-28) {
		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 <= 4d-28) 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 <= 4e-28) {
		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 <= 4e-28:
		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 <= 4e-28)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(Float64(x * a) * Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 4e-28)
		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, 4e-28], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * a), $MachinePrecision] * (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999988e-28

   1. Initial program 95.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. associate-*r* 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-lft-out 99.9%

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

      4. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in b around inf 69.3%

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

      1. associate-*r* 69.3%

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

      2. mul-1-neg 69.3%

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

   8. Simplified 69.3%

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

   9. Taylor expanded in a around 0 35.9%

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

       1. mul-1-neg 35.9%

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

       2. unsub-neg 35.9%

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

   11. Simplified 35.9%

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

   if 3.99999999999999988e-28 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 37.9%

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

      1. associate-*r* 37.9%

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

      2. mul-1-neg 37.9%

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

   8. Simplified 37.9%

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

   9. Taylor expanded in a around 0 10.0%

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

       1. mul-1-neg 10.0%

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

       2. unsub-neg 10.0%

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

   11. Simplified 10.0%

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

   12. Taylor expanded in a around inf 30.2%

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

       1. neg-mul-1 30.2%

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

       2. *-commutative 30.2%

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

       3. associate-*r* 33.9%

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

       4. distribute-rgt-neg-in 33.9%

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

   14. Simplified 33.9%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot a\right) \cdot \left(-b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.3% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-28}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot a\right) \cdot \left(-b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 4e-28) (- x (* x (* a b))) (* (* x a) (- b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 4e-28) {
		tmp = x - (x * (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 <= 4d-28) then
        tmp = x - (x * (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 <= 4e-28) {
		tmp = x - (x * (a * b));
	} else {
		tmp = (x * a) * -b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 4e-28:
		tmp = x - (x * (a * b))
	else:
		tmp = (x * a) * -b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 4e-28)
		tmp = Float64(x - Float64(x * Float64(a * b)));
	else
		tmp = Float64(Float64(x * a) * Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 4e-28)
		tmp = x - (x * (a * b));
	else
		tmp = (x * a) * -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 4e-28], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * a), $MachinePrecision] * (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999988e-28

   1. Initial program 95.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. associate-*r* 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-lft-out 99.9%

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

      4. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in b around inf 69.3%

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

      1. associate-*r* 69.3%

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

      2. mul-1-neg 69.3%

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

   8. Simplified 69.3%

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

   9. Taylor expanded in a around 0 33.8%

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

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

       3. associate-*r* 35.9%

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

   11. Simplified 35.9%

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

   if 3.99999999999999988e-28 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 37.9%

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

      1. associate-*r* 37.9%

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

      2. mul-1-neg 37.9%

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

   8. Simplified 37.9%

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

   9. Taylor expanded in a around 0 10.0%

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

       1. mul-1-neg 10.0%

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

       2. unsub-neg 10.0%

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

   11. Simplified 10.0%

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

   12. Taylor expanded in a around inf 30.2%

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

       1. neg-mul-1 30.2%

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

       2. *-commutative 30.2%

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

       3. associate-*r* 33.9%

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

       4. distribute-rgt-neg-in 33.9%

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

   14. Simplified 33.9%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-28}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot a\right) \cdot \left(-b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 19.6% accurate, 315.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Taylor expanded in a around 0 69.4%

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

4. Taylor expanded in y around 0 19.1%

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

5. Final simplification 19.1%

   \[\leadsto x \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(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))))))