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

Percentage Accurate: 96.6% → 99.2%

Time: 16.4s

Alternatives: 16

Speedup: 1.5×

• 41.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% 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 96.9%

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

3. Taylor expanded in 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. Final simplification 100.0%

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

Alternative 2: 86.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+15} \lor \neg \left(y \leq 1.18 \cdot 10^{-55}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.8e+15) (not (<= y 1.18e-55)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (* a (- (- z) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.8e+15) || !(y <= 1.18e-55)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((a * (-z - b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.8d+15)) .or. (.not. (y <= 1.18d-55))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp((a * (-z - b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.8e+15) || !(y <= 1.18e-55)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((a * (-z - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.8e+15) or not (y <= 1.18e-55):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((a * (-z - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.8e+15) || !(y <= 1.18e-55))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.8e+15) || ~((y <= 1.18e-55)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp((a * (-z - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.8e+15], N[Not[LessEqual[y, 1.18e-55]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e15 or 1.18e-55 < 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. Step-by-step derivation

      1. fma-define 100.0%

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

      2. sub-neg 100.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 90.4%

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

   if -1.8e15 < y < 1.18e-55

   1. Initial program 93.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 y around 0 89.0%

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

      1. associate-*r* 89.0%

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

      2. mul-1-neg 89.0%

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

      3. +-commutative 89.0%

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

   8. Simplified 89.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+15} \lor \neg \left(y \leq 1.18 \cdot 10^{-55}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+75} \lor \neg \left(t \leq 6.5 \cdot 10^{-79}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -4.4e+75) (not (<= t 6.5e-79)))
   (* x (exp (* t (- y))))
   (* x (exp (* a (- (- z) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -4.4e+75) || !(t <= 6.5e-79)) {
		tmp = x * exp((t * -y));
	} else {
		tmp = x * exp((a * (-z - b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-4.4d+75)) .or. (.not. (t <= 6.5d-79))) then
        tmp = x * exp((t * -y))
    else
        tmp = x * exp((a * (-z - b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -4.4e+75) || !(t <= 6.5e-79)) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = x * Math.exp((a * (-z - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -4.4e+75) or not (t <= 6.5e-79):
		tmp = x * math.exp((t * -y))
	else:
		tmp = x * math.exp((a * (-z - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -4.4e+75) || !(t <= 6.5e-79))
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	else
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -4.4e+75) || ~((t <= 6.5e-79)))
		tmp = x * exp((t * -y));
	else
		tmp = x * exp((a * (-z - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -4.4e+75], N[Not[LessEqual[t, 6.5e-79]], $MachinePrecision]], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.40000000000000024e75 or 6.5000000000000003e-79 < t

   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. Step-by-step derivation

      1. fma-define 97.7%

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

      2. sub-neg 97.7%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 82.8%

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

   6. Taylor expanded in t around inf 82.7%

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

      1. mul-1-neg 82.7%

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

      2. distribute-lft-neg-out 82.7%

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

      3. *-commutative 82.7%

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

   8. Simplified 82.7%

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

   if -4.40000000000000024e75 < t < 6.5000000000000003e-79

   1. Initial program 96.1%

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

   3. Taylor expanded in 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 y around 0 73.0%

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

      1. associate-*r* 73.0%

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

      2. mul-1-neg 73.0%

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

      3. +-commutative 73.0%

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

   8. Simplified 73.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+75} \lor \neg \left(t \leq 6.5 \cdot 10^{-79}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{+74} \lor \neg \left(t \leq 5.8 \cdot 10^{-79}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -7.8e+74) (not (<= t 5.8e-79)))
   (* x (exp (* t (- y))))
   (* x (exp (* (- a) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -7.8e+74) || !(t <= 5.8e-79)) {
		tmp = x * exp((t * -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 ((t <= (-7.8d+74)) .or. (.not. (t <= 5.8d-79))) then
        tmp = x * exp((t * -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 ((t <= -7.8e+74) || !(t <= 5.8e-79)) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = x * Math.exp((-a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -7.8e+74) or not (t <= 5.8e-79):
		tmp = x * math.exp((t * -y))
	else:
		tmp = x * math.exp((-a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -7.8e+74) || !(t <= 5.8e-79))
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	else
		tmp = Float64(x * exp(Float64(Float64(-a) * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -7.8e+74) || ~((t <= 5.8e-79)))
		tmp = x * exp((t * -y));
	else
		tmp = x * exp((-a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -7.8e+74], N[Not[LessEqual[t, 5.8e-79]], $MachinePrecision]], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.80000000000000015e74 or 5.8000000000000001e-79 < t

   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. Step-by-step derivation

      1. fma-define 97.7%

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

      2. sub-neg 97.7%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 82.8%

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

   6. Taylor expanded in t around inf 82.7%

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

      1. mul-1-neg 82.7%

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

      2. distribute-lft-neg-out 82.7%

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

      3. *-commutative 82.7%

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

   8. Simplified 82.7%

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

   if -7.80000000000000015e74 < t < 5.8000000000000001e-79

   1. Initial program 96.1%

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

      1. fma-define 96.1%

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

      2. sub-neg 96.1%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 67.5%

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

   6. Taylor expanded in z around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. mul-1-neg 66.7%

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

   8. Simplified 66.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{+74} \lor \neg \left(t \leq 5.8 \cdot 10^{-79}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -13500000000 \lor \neg \left(t \leq 4 \cdot 10^{-151}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -13500000000.0) (not (<= t 4e-151)))
   (* x (exp (* t (- y))))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -13500000000.0) || !(t <= 4e-151)) {
		tmp = x * exp((t * -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 <= (-13500000000.0d0)) .or. (.not. (t <= 4d-151))) then
        tmp = x * exp((t * -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 <= -13500000000.0) || !(t <= 4e-151)) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -13500000000.0) or not (t <= 4e-151):
		tmp = x * math.exp((t * -y))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -13500000000.0) || !(t <= 4e-151))
		tmp = Float64(x * exp(Float64(t * Float64(-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 <= -13500000000.0) || ~((t <= 4e-151)))
		tmp = x * exp((t * -y));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -13500000000.0], N[Not[LessEqual[t, 4e-151]], $MachinePrecision]], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35e10 or 3.9999999999999998e-151 < t

   1. Initial program 96.3%

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

      1. fma-define 96.3%

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

      2. sub-neg 96.3%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 76.2%

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

   6. Taylor expanded in t around inf 76.1%

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

      1. mul-1-neg 76.1%

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

      2. distribute-lft-neg-out 76.1%

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

      3. *-commutative 76.1%

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

   8. Simplified 76.1%

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

   if -1.35e10 < t < 3.9999999999999998e-151

   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. Step-by-step derivation

      1. fma-define 98.0%

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

      2. sub-neg 98.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 66.3%

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

   6. Taylor expanded in t around 0 66.3%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -13500000000 \lor \neg \left(t \leq 4 \cdot 10^{-151}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{+192} \lor \neg \left(a \leq 2.8 \cdot 10^{+30}\right):\\ \;\;\;\;x \cdot {\left(-z\right)}^{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -4.1e+192) (not (<= a 2.8e+30)))
   (* x (pow (- z) a))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -4.1e+192) || !(a <= 2.8e+30)) {
		tmp = x * pow(-z, a);
	} 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 ((a <= (-4.1d+192)) .or. (.not. (a <= 2.8d+30))) then
        tmp = x * (-z ** a)
    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 ((a <= -4.1e+192) || !(a <= 2.8e+30)) {
		tmp = x * Math.pow(-z, a);
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -4.1e+192) or not (a <= 2.8e+30):
		tmp = x * math.pow(-z, a)
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -4.1e+192) || !(a <= 2.8e+30))
		tmp = Float64(x * (Float64(-z) ^ a));
	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 ((a <= -4.1e+192) || ~((a <= 2.8e+30)))
		tmp = x * (-z ^ a);
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -4.1e+192], N[Not[LessEqual[a, 2.8e+30]], $MachinePrecision]], N[(x * N[Power[(-z), a], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.10000000000000003e192 or 2.79999999999999983e30 < a

   1. Initial program 93.4%

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

      1. fma-define 93.4%

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

      2. sub-neg 93.4%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 70.3%

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

   6. Taylor expanded in b around 0 12.4%

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

   7. Taylor expanded in z around inf 70.1%

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

      1. mul-1-neg 70.1%

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

   9. Simplified 70.1%

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

   if -4.10000000000000003e192 < a < 2.79999999999999983e30

   1. Initial program 98.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 86.2%

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

   6. Taylor expanded in t around 0 63.4%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{+192} \lor \neg \left(a \leq 2.8 \cdot 10^{+30}\right):\\ \;\;\;\;x \cdot {\left(-z\right)}^{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+259}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;t \leq -4000000000000:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.8e+259)
   (* a (* x (- z)))
   (if (<= t -4000000000000.0) (* x (- 1.0 (* y t))) (* x (pow z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.8e+259) {
		tmp = a * (x * -z);
	} else if (t <= -4000000000000.0) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-2.8d+259)) then
        tmp = a * (x * -z)
    else if (t <= (-4000000000000.0d0)) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.8e+259) {
		tmp = a * (x * -z);
	} else if (t <= -4000000000000.0) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -2.8e+259:
		tmp = a * (x * -z)
	elif t <= -4000000000000.0:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.8e+259)
		tmp = Float64(a * Float64(x * Float64(-z)));
	elseif (t <= -4000000000000.0)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -2.8e+259)
		tmp = a * (x * -z);
	elseif (t <= -4000000000000.0)
		tmp = x * (1.0 - (y * t));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.8e+259], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4000000000000.0], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.8000000000000001e259

   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. Step-by-step derivation

      1. fma-define 100.0%

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

      2. sub-neg 100.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 38.4%

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

   6. Taylor expanded in b around 0 3.9%

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

   7. Taylor expanded in z around 0 3.6%

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

      1. mul-1-neg 3.6%

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

      2. unsub-neg 3.6%

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

      3. *-commutative 3.6%

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

   9. Simplified 3.6%

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

   10. Taylor expanded in z around inf 47.2%

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

       1. associate-*r* 47.2%

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

       2. mul-1-neg 47.2%

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

   12. Simplified 47.2%

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

   if -2.8000000000000001e259 < t < -4e12

   1. Initial program 96.1%

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

      1. fma-define 96.1%

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

      2. sub-neg 96.1%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 75.8%

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

   6. Taylor expanded in t around inf 75.8%

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

      1. mul-1-neg 75.8%

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

      2. distribute-lft-neg-out 75.8%

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

      3. *-commutative 75.8%

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

   8. Simplified 75.8%

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

   9. Taylor expanded in y around 0 43.8%

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

       1. mul-1-neg 43.8%

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

       2. *-commutative 43.8%

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

       3. distribute-lft-neg-in 43.8%

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

   11. Simplified 43.8%

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

   12. Taylor expanded in x around 0 43.8%

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

       1. mul-1-neg 43.8%

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

       2. *-commutative 43.8%

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

       3. sub-neg 43.8%

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

   14. Simplified 43.8%

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

   if -4e12 < t

   1. Initial program 97.0%

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

      1. fma-define 97.0%

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

      2. sub-neg 97.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 70.4%

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

   6. Taylor expanded in t around 0 62.4%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+259}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;t \leq -4000000000000:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.1% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+155}:\\ \;\;\;\;x \cdot \frac{1 - \left(z \cdot a\right) \cdot \left(z \cdot a\right)}{z \cdot a + 1}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.35e+155)
   (* x (/ (- 1.0 (* (* z a) (* z a))) (+ (* z a) 1.0)))
   (if (<= a 1.8e+28) (* x (- 1.0 (* y t))) (* x (* t (- y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.35e+155) {
		tmp = x * ((1.0 - ((z * a) * (z * a))) / ((z * a) + 1.0));
	} else if (a <= 1.8e+28) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * (t * -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 (a <= (-1.35d+155)) then
        tmp = x * ((1.0d0 - ((z * a) * (z * a))) / ((z * a) + 1.0d0))
    else if (a <= 1.8d+28) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = x * (t * -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 (a <= -1.35e+155) {
		tmp = x * ((1.0 - ((z * a) * (z * a))) / ((z * a) + 1.0));
	} else if (a <= 1.8e+28) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * (t * -y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.35e+155:
		tmp = x * ((1.0 - ((z * a) * (z * a))) / ((z * a) + 1.0))
	elif a <= 1.8e+28:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = x * (t * -y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.35e+155)
		tmp = Float64(x * Float64(Float64(1.0 - Float64(Float64(z * a) * Float64(z * a))) / Float64(Float64(z * a) + 1.0)));
	elseif (a <= 1.8e+28)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = Float64(x * Float64(t * Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -1.35e+155)
		tmp = x * ((1.0 - ((z * a) * (z * a))) / ((z * a) + 1.0));
	elseif (a <= 1.8e+28)
		tmp = x * (1.0 - (y * t));
	else
		tmp = x * (t * -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.35e+155], N[(x * N[(N[(1.0 - N[(N[(z * a), $MachinePrecision] * N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(z * a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.8e+28], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.34999999999999997e155

   1. Initial program 91.9%

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

      1. fma-define 91.9%

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

      2. sub-neg 91.9%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 67.5%

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

   6. Taylor expanded in b around 0 14.2%

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

   7. Taylor expanded in z around 0 14.6%

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

      1. mul-1-neg 14.6%

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

      2. unsub-neg 14.6%

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

      3. *-commutative 14.6%

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

   9. Simplified 14.6%

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

       1. sub-neg 14.6%

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

       2. flip-+ 35.5%

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

       3. metadata-eval 35.5%

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

   11. Applied egg-rr 35.5%

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

   if -1.34999999999999997e155 < a < 1.8e28

   1. Initial program 99.3%

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

      1. fma-define 99.3%

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

      2. sub-neg 99.3%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 87.2%

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

   6. Taylor expanded in t around inf 71.2%

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

      1. mul-1-neg 71.2%

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

      2. distribute-lft-neg-out 71.2%

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

      3. *-commutative 71.2%

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

   8. Simplified 71.2%

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

   9. Taylor expanded in y around 0 37.8%

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

       1. mul-1-neg 37.8%

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

       2. *-commutative 37.8%

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

       3. distribute-lft-neg-in 37.8%

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

   11. Simplified 37.8%

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

   12. Taylor expanded in x around 0 37.8%

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

       1. mul-1-neg 37.8%

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

       2. *-commutative 37.8%

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

       3. sub-neg 37.8%

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

   14. Simplified 37.8%

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

   if 1.8e28 < a

   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. Step-by-step derivation

      1. fma-define 93.6%

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

      2. sub-neg 93.6%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 48.4%

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

   6. Taylor expanded in t around inf 31.3%

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

      1. mul-1-neg 31.3%

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

      2. distribute-lft-neg-out 31.3%

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

      3. *-commutative 31.3%

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

   8. Simplified 31.3%

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

   9. Taylor expanded in y around 0 14.1%

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

       1. mul-1-neg 14.1%

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

       2. *-commutative 14.1%

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

       3. distribute-lft-neg-in 14.1%

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

   11. Simplified 14.1%

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

   12. Taylor expanded in y around inf 26.7%

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

       1. mul-1-neg 26.7%

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

       2. *-commutative 26.7%

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

       3. associate-*r* 34.1%

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

       4. distribute-rgt-neg-in 34.1%

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

       5. distribute-rgt-neg-in 34.1%

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

   14. Simplified 34.1%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+155}:\\ \;\;\;\;x \cdot \frac{1 - \left(z \cdot a\right) \cdot \left(z \cdot a\right)}{z \cdot a + 1}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.3% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+155}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2e+155)
   (- x (* x (* a b)))
   (if (<= a 5.6e+24) (* x (- 1.0 (* y t))) (* x (* t (- y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2e+155) {
		tmp = x - (x * (a * b));
	} else if (a <= 5.6e+24) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * (t * -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 (a <= (-2d+155)) then
        tmp = x - (x * (a * b))
    else if (a <= 5.6d+24) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = x * (t * -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 (a <= -2e+155) {
		tmp = x - (x * (a * b));
	} else if (a <= 5.6e+24) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * (t * -y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -2e+155:
		tmp = x - (x * (a * b))
	elif a <= 5.6e+24:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = x * (t * -y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2e+155)
		tmp = Float64(x - Float64(x * Float64(a * b)));
	elseif (a <= 5.6e+24)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = Float64(x * Float64(t * Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -2e+155)
		tmp = x - (x * (a * b));
	elseif (a <= 5.6e+24)
		tmp = x * (1.0 - (y * t));
	else
		tmp = x * (t * -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2e+155], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.6e+24], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.00000000000000001e155

   1. Initial program 91.9%

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

      1. fma-define 91.9%

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

      2. sub-neg 91.9%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 67.5%

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

   6. Taylor expanded in a around 0 24.6%

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

      1. associate-*r* 29.8%

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

      2. sub-neg 29.8%

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

      3. log1p-undefine 33.0%

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

   8. Simplified 33.0%

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

   9. Taylor expanded in z around 0 24.6%

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

       1. mul-1-neg 24.6%

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

       2. *-commutative 24.6%

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

       3. distribute-lft-neg-in 24.6%

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

       4. distribute-lft-neg-out 24.6%

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

       5. associate-*r* 29.8%

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

       6. *-commutative 29.8%

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

       7. associate-*l* 35.0%

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

   11. Simplified 35.0%

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

   if -2.00000000000000001e155 < a < 5.6000000000000003e24

   1. Initial program 99.3%

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

      1. fma-define 99.3%

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

      2. sub-neg 99.3%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 87.2%

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

   6. Taylor expanded in t around inf 71.2%

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

      1. mul-1-neg 71.2%

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

      2. distribute-lft-neg-out 71.2%

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

      3. *-commutative 71.2%

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

   8. Simplified 71.2%

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

   9. Taylor expanded in y around 0 37.8%

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

       1. mul-1-neg 37.8%

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

       2. *-commutative 37.8%

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

       3. distribute-lft-neg-in 37.8%

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

   11. Simplified 37.8%

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

   12. Taylor expanded in x around 0 37.8%

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

       1. mul-1-neg 37.8%

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

       2. *-commutative 37.8%

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

       3. sub-neg 37.8%

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

   14. Simplified 37.8%

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

   if 5.6000000000000003e24 < a

   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. Step-by-step derivation

      1. fma-define 93.6%

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

      2. sub-neg 93.6%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 48.4%

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

   6. Taylor expanded in t around inf 31.3%

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

      1. mul-1-neg 31.3%

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

      2. distribute-lft-neg-out 31.3%

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

      3. *-commutative 31.3%

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

   8. Simplified 31.3%

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

   9. Taylor expanded in y around 0 14.1%

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

       1. mul-1-neg 14.1%

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

       2. *-commutative 14.1%

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

       3. distribute-lft-neg-in 14.1%

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

   11. Simplified 14.1%

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

   12. Taylor expanded in y around inf 26.7%

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

       1. mul-1-neg 26.7%

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

       2. *-commutative 26.7%

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

       3. associate-*r* 34.1%

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

       4. distribute-rgt-neg-in 34.1%

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

       5. distribute-rgt-neg-in 34.1%

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

   14. Simplified 34.1%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+155}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.0% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+155}:\\ \;\;\;\;x - z \cdot \left(x \cdot a\right)\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -8e+155)
   (- x (* z (* x a)))
   (if (<= a 2.8e+31) (* x (- 1.0 (* y t))) (* x (* t (- y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -8e+155) {
		tmp = x - (z * (x * a));
	} else if (a <= 2.8e+31) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * (t * -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 (a <= (-8d+155)) then
        tmp = x - (z * (x * a))
    else if (a <= 2.8d+31) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = x * (t * -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 (a <= -8e+155) {
		tmp = x - (z * (x * a));
	} else if (a <= 2.8e+31) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * (t * -y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -8e+155:
		tmp = x - (z * (x * a))
	elif a <= 2.8e+31:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = x * (t * -y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -8e+155)
		tmp = Float64(x - Float64(z * Float64(x * a)));
	elseif (a <= 2.8e+31)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = Float64(x * Float64(t * Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -8e+155)
		tmp = x - (z * (x * a));
	elseif (a <= 2.8e+31)
		tmp = x * (1.0 - (y * t));
	else
		tmp = x * (t * -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -8e+155], N[(x - N[(z * N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.8e+31], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.00000000000000006e155

   1. Initial program 91.9%

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

      1. fma-define 91.9%

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

      2. sub-neg 91.9%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 67.5%

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

   6. Taylor expanded in b around 0 14.2%

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

   7. Taylor expanded in z around 0 14.6%

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

      1. mul-1-neg 14.6%

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

      2. unsub-neg 14.6%

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

      3. associate-*r* 27.4%

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

      4. *-commutative 27.4%

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

   9. Simplified 27.4%

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

   if -8.00000000000000006e155 < a < 2.80000000000000017e31

   1. Initial program 99.3%

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

      1. fma-define 99.3%

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

      2. sub-neg 99.3%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 87.2%

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

   6. Taylor expanded in t around inf 71.2%

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

      1. mul-1-neg 71.2%

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

      2. distribute-lft-neg-out 71.2%

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

      3. *-commutative 71.2%

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

   8. Simplified 71.2%

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

   9. Taylor expanded in y around 0 37.8%

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

       1. mul-1-neg 37.8%

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

       2. *-commutative 37.8%

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

       3. distribute-lft-neg-in 37.8%

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

   11. Simplified 37.8%

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

   12. Taylor expanded in x around 0 37.8%

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

       1. mul-1-neg 37.8%

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

       2. *-commutative 37.8%

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

       3. sub-neg 37.8%

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

   14. Simplified 37.8%

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

   if 2.80000000000000017e31 < a

   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. Step-by-step derivation

      1. fma-define 93.6%

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

      2. sub-neg 93.6%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 48.4%

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

   6. Taylor expanded in t around inf 31.3%

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

      1. mul-1-neg 31.3%

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

      2. distribute-lft-neg-out 31.3%

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

      3. *-commutative 31.3%

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

   8. Simplified 31.3%

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

   9. Taylor expanded in y around 0 14.1%

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

       1. mul-1-neg 14.1%

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

       2. *-commutative 14.1%

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

       3. distribute-lft-neg-in 14.1%

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

   11. Simplified 14.1%

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

   12. Taylor expanded in y around inf 26.7%

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

       1. mul-1-neg 26.7%

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

       2. *-commutative 26.7%

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

       3. associate-*r* 34.1%

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

       4. distribute-rgt-neg-in 34.1%

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

       5. distribute-rgt-neg-in 34.1%

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

   14. Simplified 34.1%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+155}:\\ \;\;\;\;x - z \cdot \left(x \cdot a\right)\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.0% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-87}:\\ \;\;\;\;x \cdot \left(y \cdot t + 1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4e-21)
   (* x (* t (- y)))
   (if (<= y 5.8e-87) (* x (+ (* y t) 1.0)) (* a (* x (- z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4e-21) {
		tmp = x * (t * -y);
	} else if (y <= 5.8e-87) {
		tmp = x * ((y * t) + 1.0);
	} 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 <= (-4d-21)) then
        tmp = x * (t * -y)
    else if (y <= 5.8d-87) then
        tmp = x * ((y * t) + 1.0d0)
    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 <= -4e-21) {
		tmp = x * (t * -y);
	} else if (y <= 5.8e-87) {
		tmp = x * ((y * t) + 1.0);
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4e-21:
		tmp = x * (t * -y)
	elif y <= 5.8e-87:
		tmp = x * ((y * t) + 1.0)
	else:
		tmp = a * (x * -z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4e-21)
		tmp = Float64(x * Float64(t * Float64(-y)));
	elseif (y <= 5.8e-87)
		tmp = Float64(x * Float64(Float64(y * t) + 1.0));
	else
		tmp = Float64(a * Float64(x * Float64(-z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4e-21)
		tmp = x * (t * -y);
	elseif (y <= 5.8e-87)
		tmp = x * ((y * t) + 1.0);
	else
		tmp = a * (x * -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4e-21], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e-87], N[(x * N[(N[(y * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.99999999999999963e-21

   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. Step-by-step derivation

      1. fma-define 100.0%

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

      2. sub-neg 100.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 90.4%

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

   6. Taylor expanded in t around inf 64.8%

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

      1. mul-1-neg 64.8%

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

      2. distribute-lft-neg-out 64.8%

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

      3. *-commutative 64.8%

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

   8. Simplified 64.8%

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

   9. Taylor expanded in y around 0 24.7%

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

       1. mul-1-neg 24.7%

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

       2. *-commutative 24.7%

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

       3. distribute-lft-neg-in 24.7%

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

   11. Simplified 24.7%

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

   12. Taylor expanded in y around inf 19.9%

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

       1. mul-1-neg 19.9%

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

       2. *-commutative 19.9%

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

       3. associate-*r* 24.5%

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

       4. distribute-rgt-neg-in 24.5%

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

       5. distribute-rgt-neg-in 24.5%

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

   14. Simplified 24.5%

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

   if -3.99999999999999963e-21 < y < 5.7999999999999998e-87

   1. Initial program 93.0%

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

      1. fma-define 93.0%

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

      2. sub-neg 93.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 48.8%

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

   6. Taylor expanded in t around inf 48.8%

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

      1. mul-1-neg 48.8%

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

      2. distribute-lft-neg-out 48.8%

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

      3. *-commutative 48.8%

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

   8. Simplified 48.8%

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

   9. Taylor expanded in y around 0 36.4%

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

       1. mul-1-neg 36.4%

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

       2. *-commutative 36.4%

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

       3. distribute-lft-neg-in 36.4%

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

   11. Simplified 36.4%

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

       1. neg-sub0 36.4%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(0 - y\right)} \cdot t\right) \]

       2. sub-neg 36.4%

          \[\leadsto x \cdot \left(1 + \color{blue}{\left(0 + \left(-y\right)\right)} \cdot t\right) \]

       3. add-sqr-sqrt 20.3%

          \[\leadsto x \cdot \left(1 + \left(0 + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right) \cdot t\right) \]

       4. sqrt-unprod 36.4%

          \[\leadsto x \cdot \left(1 + \left(0 + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right) \cdot t\right) \]

       5. sqr-neg 36.4%

          \[\leadsto x \cdot \left(1 + \left(0 + \sqrt{\color{blue}{y \cdot y}}\right) \cdot t\right) \]

       6. sqrt-unprod 16.0%

          \[\leadsto x \cdot \left(1 + \left(0 + \color{blue}{\sqrt{y} \cdot \sqrt{y}}\right) \cdot t\right) \]

       7. add-sqr-sqrt 37.0%

          \[\leadsto x \cdot \left(1 + \left(0 + \color{blue}{y}\right) \cdot t\right) \]

   13. Applied egg-rr 37.0%

       \[\leadsto x \cdot \left(1 + \color{blue}{\left(0 + y\right)} \cdot t\right) \]
   14. Step-by-step derivation

       1. +-lft-identity 37.0%

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

   15. Simplified 37.0%

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

   if 5.7999999999999998e-87 < y

   1. Initial program 99.0%

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

      1. fma-define 99.0%

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

      2. sub-neg 99.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 36.6%

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

   6. Taylor expanded in b around 0 12.6%

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

   7. Taylor expanded in z around 0 8.5%

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

      1. mul-1-neg 8.5%

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

      2. unsub-neg 8.5%

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

      3. *-commutative 8.5%

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

   9. Simplified 8.5%

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

   10. Taylor expanded in z around inf 30.5%

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

       1. associate-*r* 30.5%

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

       2. mul-1-neg 30.5%

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

   12. Simplified 30.5%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-87}:\\ \;\;\;\;x \cdot \left(y \cdot t + 1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 28.1% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-46} \lor \neg \left(y \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.1e-46) (not (<= y 5e-6))) (* x (* t (- y))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.1e-46) || !(y <= 5e-6)) {
		tmp = x * (t * -y);
	} 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 <= (-2.1d-46)) .or. (.not. (y <= 5d-6))) then
        tmp = x * (t * -y)
    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 <= -2.1e-46) || !(y <= 5e-6)) {
		tmp = x * (t * -y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.1e-46) or not (y <= 5e-6):
		tmp = x * (t * -y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.1e-46) || !(y <= 5e-6))
		tmp = Float64(x * Float64(t * Float64(-y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.1e-46) || ~((y <= 5e-6)))
		tmp = x * (t * -y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.1e-46], N[Not[LessEqual[y, 5e-6]], $MachinePrecision]], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.09999999999999987e-46 or 5.00000000000000041e-6 < 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. Step-by-step derivation

      1. fma-define 100.0%

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

      2. sub-neg 100.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 84.5%

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

   6. Taylor expanded in t around inf 59.8%

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

      1. mul-1-neg 59.8%

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

      2. distribute-lft-neg-out 59.8%

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

      3. *-commutative 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in y around 0 22.2%

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

       1. mul-1-neg 22.2%

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

       2. *-commutative 22.2%

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

       3. distribute-lft-neg-in 22.2%

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

   11. Simplified 22.2%

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

   12. Taylor expanded in y around inf 19.6%

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

       1. mul-1-neg 19.6%

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

       2. *-commutative 19.6%

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

       3. associate-*r* 24.8%

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

       4. distribute-rgt-neg-in 24.8%

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

       5. distribute-rgt-neg-in 24.8%

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

   14. Simplified 24.8%

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

   if -2.09999999999999987e-46 < y < 5.00000000000000041e-6

   1. Initial program 92.9%

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

      1. fma-define 92.9%

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

      2. sub-neg 92.9%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 56.5%

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

   6. Taylor expanded in y around 0 34.9%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-46} \lor \neg \left(y \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.0% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-81}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.1e-46)
   (* x (* t (- y)))
   (if (<= y 1.85e-81) x (* a (* x (- z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.1e-46) {
		tmp = x * (t * -y);
	} else if (y <= 1.85e-81) {
		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 <= (-2.1d-46)) then
        tmp = x * (t * -y)
    else if (y <= 1.85d-81) 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 <= -2.1e-46) {
		tmp = x * (t * -y);
	} else if (y <= 1.85e-81) {
		tmp = x;
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.1e-46:
		tmp = x * (t * -y)
	elif y <= 1.85e-81:
		tmp = x
	else:
		tmp = a * (x * -z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.1e-46)
		tmp = Float64(x * Float64(t * Float64(-y)));
	elseif (y <= 1.85e-81)
		tmp = x;
	else
		tmp = Float64(a * Float64(x * Float64(-z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.1e-46)
		tmp = x * (t * -y);
	elseif (y <= 1.85e-81)
		tmp = x;
	else
		tmp = a * (x * -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.1e-46], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e-81], x, N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.09999999999999987e-46

   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. Step-by-step derivation

      1. fma-define 100.0%

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

      2. sub-neg 100.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 79.5%

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

   6. Taylor expanded in t around inf 57.5%

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

      1. mul-1-neg 57.5%

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

      2. distribute-lft-neg-out 57.5%

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

      3. *-commutative 57.5%

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

   8. Simplified 57.5%

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

   9. Taylor expanded in y around 0 21.7%

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

       1. mul-1-neg 21.7%

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

       2. *-commutative 21.7%

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

       3. distribute-lft-neg-in 21.7%

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

   11. Simplified 21.7%

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

   12. Taylor expanded in y around inf 18.9%

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

       1. mul-1-neg 18.9%

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

       2. *-commutative 18.9%

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

       3. associate-*r* 22.9%

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

       4. distribute-rgt-neg-in 22.9%

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

       5. distribute-rgt-neg-in 22.9%

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

   14. Simplified 22.9%

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

   if -2.09999999999999987e-46 < y < 1.84999999999999993e-81

   1. Initial program 92.2%

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

      1. fma-define 92.2%

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

      2. sub-neg 92.2%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 52.8%

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

   6. Taylor expanded in y around 0 39.0%

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

   if 1.84999999999999993e-81 < y

   1. Initial program 99.0%

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

      1. fma-define 99.0%

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

      2. sub-neg 99.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 36.6%

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

   6. Taylor expanded in b around 0 12.6%

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

   7. Taylor expanded in z around 0 8.5%

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

      1. mul-1-neg 8.5%

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

      2. unsub-neg 8.5%

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

      3. *-commutative 8.5%

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

   9. Simplified 8.5%

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

   10. Taylor expanded in z around inf 30.5%

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

       1. associate-*r* 30.5%

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

       2. mul-1-neg 30.5%

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

   12. Simplified 30.5%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-81}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.2% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.1e-46)
   (* x (* t (- y)))
   (if (<= y 1.56e-56) x (* x (* z (- a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.1e-46) {
		tmp = x * (t * -y);
	} else if (y <= 1.56e-56) {
		tmp = x;
	} else {
		tmp = x * (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 (y <= (-2.1d-46)) then
        tmp = x * (t * -y)
    else if (y <= 1.56d-56) then
        tmp = x
    else
        tmp = x * (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 (y <= -2.1e-46) {
		tmp = x * (t * -y);
	} else if (y <= 1.56e-56) {
		tmp = x;
	} else {
		tmp = x * (z * -a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.1e-46:
		tmp = x * (t * -y)
	elif y <= 1.56e-56:
		tmp = x
	else:
		tmp = x * (z * -a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.1e-46)
		tmp = Float64(x * Float64(t * Float64(-y)));
	elseif (y <= 1.56e-56)
		tmp = x;
	else
		tmp = Float64(x * Float64(z * Float64(-a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.1e-46)
		tmp = x * (t * -y);
	elseif (y <= 1.56e-56)
		tmp = x;
	else
		tmp = x * (z * -a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.1e-46], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.56e-56], x, N[(x * N[(z * (-a)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.09999999999999987e-46

   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. Step-by-step derivation

      1. fma-define 100.0%

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

      2. sub-neg 100.0%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 79.5%

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

   6. Taylor expanded in t around inf 57.5%

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

      1. mul-1-neg 57.5%

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

      2. distribute-lft-neg-out 57.5%

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

      3. *-commutative 57.5%

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

   8. Simplified 57.5%

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

   9. Taylor expanded in y around 0 21.7%

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

       1. mul-1-neg 21.7%

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

       2. *-commutative 21.7%

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

       3. distribute-lft-neg-in 21.7%

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

   11. Simplified 21.7%

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

   12. Taylor expanded in y around inf 18.9%

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

       1. mul-1-neg 18.9%

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

       2. *-commutative 18.9%

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

       3. associate-*r* 22.9%

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

       4. distribute-rgt-neg-in 22.9%

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

       5. distribute-rgt-neg-in 22.9%

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

   14. Simplified 22.9%

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

   if -2.09999999999999987e-46 < y < 1.56000000000000003e-56

   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. Step-by-step derivation

      1. fma-define 92.7%

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

      2. sub-neg 92.7%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 52.8%

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

   6. Taylor expanded in y around 0 38.9%

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

   if 1.56000000000000003e-56 < y

   1. Initial program 98.9%

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

      1. fma-define 98.9%

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

      2. sub-neg 98.9%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 33.5%

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

   6. Taylor expanded in b around 0 11.0%

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

   7. Taylor expanded in z around 0 6.6%

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

      1. mul-1-neg 6.6%

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

      2. unsub-neg 6.6%

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

      3. *-commutative 6.6%

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

   9. Simplified 6.6%

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

   10. Taylor expanded in z around inf 27.7%

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

       1. mul-1-neg 27.7%

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

       2. *-commutative 27.7%

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

       3. distribute-rgt-neg-in 27.7%

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

   12. Simplified 27.7%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 23.4% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-41}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 6.8e-41) (* a (* x (- z))) (* x (- 1.0 (* y t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 6.8e-41) {
		tmp = a * (x * -z);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= 6.8d-41) then
        tmp = a * (x * -z)
    else
        tmp = x * (1.0d0 - (y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 6.8e-41) {
		tmp = a * (x * -z);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 6.8e-41:
		tmp = a * (x * -z)
	else:
		tmp = x * (1.0 - (y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 6.8e-41)
		tmp = Float64(a * Float64(x * Float64(-z)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 6.8e-41)
		tmp = a * (x * -z);
	else
		tmp = x * (1.0 - (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 6.8e-41], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.7999999999999997e-41

   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 97.3%

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

      2. sub-neg 97.3%

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

      3. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 54.0%

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

   6. Taylor expanded in b around 0 18.7%

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

   7. Taylor expanded in z around 0 17.1%

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

      1. mul-1-neg 17.1%

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

      2. unsub-neg 17.1%

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

      3. *-commutative 17.1%

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

   9. Simplified 17.1%

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

   10. Taylor expanded in z around inf 24.0%

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

       1. associate-*r* 24.0%

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

       2. mul-1-neg 24.0%

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

   12. Simplified 24.0%

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

   if 6.7999999999999997e-41 < x

   1. Initial program 96.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 69.4%

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

   6. Taylor expanded in t around inf 55.0%

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

      1. mul-1-neg 55.0%

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

      2. distribute-lft-neg-out 55.0%

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

      3. *-commutative 55.0%

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

   8. Simplified 55.0%

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

   9. Taylor expanded in y around 0 26.8%

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

       1. mul-1-neg 26.8%

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

       2. *-commutative 26.8%

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

       3. distribute-lft-neg-in 26.8%

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

   11. Simplified 26.8%

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

   12. Taylor expanded in x around 0 26.8%

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

       1. mul-1-neg 26.8%

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

       2. *-commutative 26.8%

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

       3. sub-neg 26.8%

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

   14. Simplified 26.8%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-41}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 20.0% 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 96.9%

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

   1. fma-define 96.9%

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

   2. sub-neg 96.9%

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

   3. log1p-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0 72.4%

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

6. Taylor expanded in y around 0 17.3%

   \[\leadsto \color{blue}{x} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(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))))))