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

Percentage Accurate: 96.5% → 99.2%

Time: 18.3s

Alternatives: 15

Speedup: 1.5×

• 41.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% 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 98.1%

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

3. Taylor expanded in 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. +-commutative 100.0%

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

   2. associate-*r* 100.0%

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

   3. associate-*r* 100.0%

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

   4. distribute-lft-out 100.0%

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

   5. mul-1-neg 100.0%

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

5. Simplified 100.0%

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

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: 82.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-189} \lor \neg \left(y \leq 1.5 \cdot 10^{-133}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7.8e-189) (not (<= y 1.5e-133)))
   (* 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 <= -7.8e-189) || !(y <= 1.5e-133)) {
		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 <= (-7.8d-189)) .or. (.not. (y <= 1.5d-133))) 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 <= -7.8e-189) || !(y <= 1.5e-133)) {
		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 <= -7.8e-189) or not (y <= 1.5e-133):
		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 <= -7.8e-189) || !(y <= 1.5e-133))
		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 <= -7.8e-189) || ~((y <= 1.5e-133)))
		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, -7.8e-189], N[Not[LessEqual[y, 1.5e-133]], $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 < -7.8000000000000005e-189 or 1.5000000000000001e-133 < y

   1. Initial program 97.9%

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

      1. fma-define 97.9%

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

      2. sub-neg 97.9%

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

      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 86.9%

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

   if -7.8000000000000005e-189 < y < 1.5000000000000001e-133

   1. Initial program 98.5%

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

   3. Taylor expanded in 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. +-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 95.2%

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

      1. associate-*r* 95.2%

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

      2. mul-1-neg 95.2%

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

      3. +-commutative 95.2%

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

   8. Simplified 95.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-189} \lor \neg \left(y \leq 1.5 \cdot 10^{-133}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -1600:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-292}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-78}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- t))))))
   (if (<= t -1600.0)
     t_1
     (if (<= t -9e-292)
       (* x (exp (* a (- b))))
       (if (<= t 5e-78) (* x (pow z y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * -t));
	double tmp;
	if (t <= -1600.0) {
		tmp = t_1;
	} else if (t <= -9e-292) {
		tmp = x * exp((a * -b));
	} else if (t <= 5e-78) {
		tmp = x * pow(z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((y * -t))
    if (t <= (-1600.0d0)) then
        tmp = t_1
    else if (t <= (-9d-292)) then
        tmp = x * exp((a * -b))
    else if (t <= 5d-78) then
        tmp = x * (z ** y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * -t));
	double tmp;
	if (t <= -1600.0) {
		tmp = t_1;
	} else if (t <= -9e-292) {
		tmp = x * Math.exp((a * -b));
	} else if (t <= 5e-78) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * -t))
	tmp = 0
	if t <= -1600.0:
		tmp = t_1
	elif t <= -9e-292:
		tmp = x * math.exp((a * -b))
	elif t <= 5e-78:
		tmp = x * math.pow(z, y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(-t))))
	tmp = 0.0
	if (t <= -1600.0)
		tmp = t_1;
	elseif (t <= -9e-292)
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	elseif (t <= 5e-78)
		tmp = Float64(x * (z ^ y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * -t));
	tmp = 0.0;
	if (t <= -1600.0)
		tmp = t_1;
	elseif (t <= -9e-292)
		tmp = x * exp((a * -b));
	elseif (t <= 5e-78)
		tmp = x * (z ^ y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1600.0], t$95$1, If[LessEqual[t, -9e-292], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e-78], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1600 or 4.9999999999999996e-78 < t

   1. Initial program 98.1%

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

      1. fma-define 98.1%

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

      2. sub-neg 98.1%

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

      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 82.0%

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

   6. Taylor expanded in t around inf 82.0%

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

      1. mul-1-neg 82.0%

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

      2. distribute-lft-neg-out 82.0%

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

      3. *-commutative 82.0%

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

   8. Simplified 82.0%

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

   if -1600 < t < -8.99999999999999913e-292

   1. Initial program 96.4%

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

      1. fma-define 96.4%

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

      2. sub-neg 96.4%

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

      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 69.8%

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

   6. Taylor expanded in z around 0 69.8%

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

      1. associate-*r* 69.8%

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

      2. mul-1-neg 69.8%

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

   8. Simplified 69.8%

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

   if -8.99999999999999913e-292 < t < 4.9999999999999996e-78

   1. Initial program 99.9%

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

      1. fma-define 99.9%

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

      2. sub-neg 99.9%

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

      3. log1p-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 81.4%

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

   6. Taylor expanded in t around 0 81.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1600:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-292}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-78}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+70} \lor \neg \left(t \leq 1.5 \cdot 10^{+50}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -9e+70) (not (<= t 1.5e+50)))
   (* x (exp (* y (- t))))
   (* x (exp (* a (- (- z) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -9e+70) || !(t <= 1.5e+50)) {
		tmp = x * exp((y * -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 ((t <= (-9d+70)) .or. (.not. (t <= 1.5d+50))) then
        tmp = x * exp((y * -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 ((t <= -9e+70) || !(t <= 1.5e+50)) {
		tmp = x * Math.exp((y * -t));
	} else {
		tmp = x * Math.exp((a * (-z - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -9e+70) or not (t <= 1.5e+50):
		tmp = x * math.exp((y * -t))
	else:
		tmp = x * math.exp((a * (-z - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -9e+70) || !(t <= 1.5e+50))
		tmp = Float64(x * exp(Float64(y * Float64(-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 ((t <= -9e+70) || ~((t <= 1.5e+50)))
		tmp = x * exp((y * -t));
	else
		tmp = x * exp((a * (-z - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -9e+70], N[Not[LessEqual[t, 1.5e+50]], $MachinePrecision]], N[(x * N[Exp[N[(y * (-t)), $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 < -8.9999999999999999e70 or 1.4999999999999999e50 < t

   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 86.9%

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

   6. Taylor expanded in t around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. distribute-lft-neg-out 86.9%

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

      3. *-commutative 86.9%

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

   8. Simplified 86.9%

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

   if -8.9999999999999999e70 < t < 1.4999999999999999e50

   1. Initial program 96.6%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 74.2%

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

      1. associate-*r* 74.2%

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

      2. mul-1-neg 74.2%

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

      3. +-commutative 74.2%

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

   8. Simplified 74.2%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+70} \lor \neg \left(t \leq 1.5 \cdot 10^{+50}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -400 \lor \neg \left(t \leq 1.3 \cdot 10^{-76}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -400.0) (not (<= t 1.3e-76)))
   (* x (exp (* 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 <= -400.0) || !(t <= 1.3e-76)) {
		tmp = x * exp((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 <= (-400.0d0)) .or. (.not. (t <= 1.3d-76))) then
        tmp = x * exp((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 <= -400.0) || !(t <= 1.3e-76)) {
		tmp = x * Math.exp((y * -t));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -400.0) or not (t <= 1.3e-76):
		tmp = x * math.exp((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 <= -400.0) || !(t <= 1.3e-76))
		tmp = Float64(x * exp(Float64(y * Float64(-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 <= -400.0) || ~((t <= 1.3e-76)))
		tmp = x * exp((y * -t));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -400.0], N[Not[LessEqual[t, 1.3e-76]], $MachinePrecision]], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -400 or 1.3e-76 < t

   1. Initial program 98.1%

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

      1. fma-define 98.1%

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

      2. sub-neg 98.1%

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

      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 82.0%

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

   6. Taylor expanded in t around inf 82.0%

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

      1. mul-1-neg 82.0%

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

      2. distribute-lft-neg-out 82.0%

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

      3. *-commutative 82.0%

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

   8. Simplified 82.0%

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

   if -400 < t < 1.3e-76

   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 70.3%

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

   6. Taylor expanded in t around 0 70.3%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -400 \lor \neg \left(t \leq 1.3 \cdot 10^{-76}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+25} \lor \neg \left(a \leq 5.7 \cdot 10^{+73}\right):\\ \;\;\;\;x \cdot {\left(-z\right)}^{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -9e+25) (not (<= a 5.7e+73)))
   (* 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 <= -9e+25) || !(a <= 5.7e+73)) {
		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 <= (-9d+25)) .or. (.not. (a <= 5.7d+73))) 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 <= -9e+25) || !(a <= 5.7e+73)) {
		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 <= -9e+25) or not (a <= 5.7e+73):
		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 <= -9e+25) || !(a <= 5.7e+73))
		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 <= -9e+25) || ~((a <= 5.7e+73)))
		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, -9e+25], N[Not[LessEqual[a, 5.7e+73]], $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 < -9.0000000000000006e25 or 5.70000000000000001e73 < a

   1. Initial program 95.9%

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

      1. fma-define 95.9%

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

      2. sub-neg 95.9%

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

      3. log1p-define 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 73.1%

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

   6. Taylor expanded in b around 0 8.6%

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

   7. Taylor expanded in z around inf 58.4%

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

      1. neg-mul-1 58.4%

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

   9. Simplified 58.4%

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

   if -9.0000000000000006e25 < a < 5.70000000000000001e73

   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 87.3%

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

   6. Taylor expanded in t around 0 57.4%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+25} \lor \neg \left(a \leq 5.7 \cdot 10^{+73}\right):\\ \;\;\;\;x \cdot {\left(-z\right)}^{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.85e+56) (* 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 <= -1.85e+56) {
		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 <= (-1.85d+56)) 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 <= -1.85e+56) {
		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 <= -1.85e+56:
		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 <= -1.85e+56)
		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 <= -1.85e+56)
		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, -1.85e+56], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.84999999999999998e56

   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 86.4%

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

   6. Taylor expanded in t around inf 86.4%

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

      1. mul-1-neg 86.4%

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

      2. distribute-lft-neg-out 86.4%

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

      3. *-commutative 86.4%

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

   8. Simplified 86.4%

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

   9. Taylor expanded in y around 0 36.5%

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

       1. mul-1-neg 36.5%

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

       2. *-commutative 36.5%

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

       3. distribute-rgt-neg-in 36.5%

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

   11. Simplified 36.5%

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

       1. distribute-rgt-neg-out 36.5%

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

       2. unsub-neg 36.5%

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

   13. Applied egg-rr 36.5%

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

   if -1.84999999999999998e56 < t

   1. Initial program 97.4%

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

      1. fma-define 97.4%

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

      2. sub-neg 97.4%

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

      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 a around 0 74.3%

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

   6. Taylor expanded in t around 0 62.2%

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

Alternative 8: 31.2% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+46}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-143}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2e+46)
   (- x (* t (* x y)))
   (if (<= y 1.9e-143) (- x (* x (* a b))) (* a (* x (- z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2e+46) {
		tmp = x - (t * (x * y));
	} else if (y <= 1.9e-143) {
		tmp = x - (x * (a * b));
	} 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 <= (-2d+46)) then
        tmp = x - (t * (x * y))
    else if (y <= 1.9d-143) then
        tmp = x - (x * (a * b))
    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 <= -2e+46) {
		tmp = x - (t * (x * y));
	} else if (y <= 1.9e-143) {
		tmp = x - (x * (a * b));
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2e+46:
		tmp = x - (t * (x * y))
	elif y <= 1.9e-143:
		tmp = x - (x * (a * b))
	else:
		tmp = a * (x * -z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2e+46)
		tmp = Float64(x - Float64(t * Float64(x * y)));
	elseif (y <= 1.9e-143)
		tmp = Float64(x - Float64(x * Float64(a * b)));
	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 <= -2e+46)
		tmp = x - (t * (x * y));
	elseif (y <= 1.9e-143)
		tmp = x - (x * (a * b));
	else
		tmp = a * (x * -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2e+46], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e-143], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e46

   1. Initial program 97.9%

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

      1. fma-define 97.9%

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

      2. sub-neg 97.9%

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

      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 95.9%

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

   6. Taylor expanded in y around 0 45.5%

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

      1. associate-*r* 43.5%

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

   8. Simplified 43.5%

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

   9. Taylor expanded in t around inf 34.6%

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

       1. neg-mul-1 34.6%

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

   11. Simplified 34.6%

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

   if -2e46 < y < 1.89999999999999991e-143

   1. Initial program 98.2%

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

      1. fma-define 98.2%

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

      2. sub-neg 98.2%

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

      3. log1p-define 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 78.9%

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

   6. Taylor expanded in z around 0 78.9%

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

      1. associate-*r* 78.9%

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

      2. mul-1-neg 78.9%

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

   8. Simplified 78.9%

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

   9. Taylor expanded in a around 0 38.8%

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

       1. mul-1-neg 38.8%

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

       2. unsub-neg 38.8%

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

       3. associate-*r* 41.4%

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

       4. *-commutative 41.4%

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

   11. Simplified 41.4%

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

   if 1.89999999999999991e-143 < y

   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 y around 0 42.2%

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

   6. Taylor expanded in b around 0 12.7%

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

   7. Taylor expanded in z around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. unsub-neg 9.4%

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

      3. *-commutative 9.4%

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

   9. Simplified 9.4%

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

   10. Taylor expanded in z around inf 32.6%

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

       1. associate-*r* 32.6%

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

       2. mul-1-neg 32.6%

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

   12. Simplified 32.6%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+46}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-143}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.3% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+46}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.92 \cdot 10^{-143}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.4e+46)
   (* t (* x (- y)))
   (if (<= y 1.92e-143) (- x (* x (* a b))) (* a (* x (- z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.4e+46) {
		tmp = t * (x * -y);
	} else if (y <= 1.92e-143) {
		tmp = x - (x * (a * b));
	} 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 <= (-2.4d+46)) then
        tmp = t * (x * -y)
    else if (y <= 1.92d-143) then
        tmp = x - (x * (a * b))
    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 <= -2.4e+46) {
		tmp = t * (x * -y);
	} else if (y <= 1.92e-143) {
		tmp = x - (x * (a * b));
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.4e+46:
		tmp = t * (x * -y)
	elif y <= 1.92e-143:
		tmp = x - (x * (a * b))
	else:
		tmp = a * (x * -z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.4e+46)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 1.92e-143)
		tmp = Float64(x - Float64(x * Float64(a * b)));
	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 <= -2.4e+46)
		tmp = t * (x * -y);
	elseif (y <= 1.92e-143)
		tmp = x - (x * (a * b));
	else
		tmp = a * (x * -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.4e+46], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.92e-143], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.40000000000000008e46

   1. Initial program 97.9%

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

      1. fma-define 97.9%

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

      2. sub-neg 97.9%

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

      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 95.9%

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

   6. Taylor expanded in t around inf 59.4%

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

      1. mul-1-neg 59.4%

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

      2. distribute-lft-neg-out 59.4%

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

      3. *-commutative 59.4%

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

   8. Simplified 59.4%

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

   9. Taylor expanded in y around 0 28.9%

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

       1. mul-1-neg 28.9%

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

       2. *-commutative 28.9%

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

       3. distribute-rgt-neg-in 28.9%

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

   11. Simplified 28.9%

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

   12. Taylor expanded in y around inf 34.4%

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

       1. associate-*r* 34.4%

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

       2. mul-1-neg 34.4%

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

   14. Simplified 34.4%

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

   if -2.40000000000000008e46 < y < 1.91999999999999995e-143

   1. Initial program 98.2%

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

      1. fma-define 98.2%

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

      2. sub-neg 98.2%

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

      3. log1p-define 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 78.9%

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

   6. Taylor expanded in z around 0 78.9%

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

      1. associate-*r* 78.9%

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

      2. mul-1-neg 78.9%

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

   8. Simplified 78.9%

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

   9. Taylor expanded in a around 0 38.8%

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

       1. mul-1-neg 38.8%

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

       2. unsub-neg 38.8%

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

       3. associate-*r* 41.4%

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

       4. *-commutative 41.4%

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

   11. Simplified 41.4%

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

   if 1.91999999999999995e-143 < y

   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 y around 0 42.2%

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

   6. Taylor expanded in b around 0 12.7%

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

   7. Taylor expanded in z around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. unsub-neg 9.4%

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

      3. *-commutative 9.4%

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

   9. Simplified 9.4%

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

   10. Taylor expanded in z around inf 32.6%

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

       1. associate-*r* 32.6%

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

       2. mul-1-neg 32.6%

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

   12. Simplified 32.6%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+46}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.92 \cdot 10^{-143}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 25.5% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{-71} \lor \neg \left(y \leq 1.92 \cdot 10^{-143}\right):\\ \;\;\;\;x \cdot \left(z \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.26e-71) (not (<= y 1.92e-143))) (* x (* z (- a))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.26e-71) || !(y <= 1.92e-143)) {
		tmp = x * (z * -a);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.26d-71)) .or. (.not. (y <= 1.92d-143))) then
        tmp = x * (z * -a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.26e-71) || !(y <= 1.92e-143)) {
		tmp = x * (z * -a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.26e-71) or not (y <= 1.92e-143):
		tmp = x * (z * -a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.26e-71) || !(y <= 1.92e-143))
		tmp = Float64(x * Float64(z * Float64(-a)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.26e-71) || ~((y <= 1.92e-143)))
		tmp = x * (z * -a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.26e-71], N[Not[LessEqual[y, 1.92e-143]], $MachinePrecision]], N[(x * N[(z * (-a)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.2600000000000001e-71 or 1.91999999999999995e-143 < y

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. 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 y around 0 42.2%

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

   6. Taylor expanded in b around 0 10.2%

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

   7. Taylor expanded in z around 0 9.2%

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

      1. mul-1-neg 9.2%

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

      2. unsub-neg 9.2%

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

      3. *-commutative 9.2%

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

   9. Simplified 9.2%

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

   10. Taylor expanded in z around inf 23.2%

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

       1. mul-1-neg 23.2%

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

       2. *-commutative 23.2%

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

       3. distribute-rgt-neg-in 23.2%

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

   12. Simplified 23.2%

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

   if -1.2600000000000001e-71 < y < 1.91999999999999995e-143

   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 60.8%

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

   6. Taylor expanded in y around 0 37.8%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{-71} \lor \neg \left(y \leq 1.92 \cdot 10^{-143}\right):\\ \;\;\;\;x \cdot \left(z \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.2% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-60}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-143}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -9e-60)
   (* t (* x (- y)))
   (if (<= y 1.35e-143) x (* a (* x (- z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9e-60) {
		tmp = t * (x * -y);
	} else if (y <= 1.35e-143) {
		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 <= (-9d-60)) then
        tmp = t * (x * -y)
    else if (y <= 1.35d-143) 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 <= -9e-60) {
		tmp = t * (x * -y);
	} else if (y <= 1.35e-143) {
		tmp = x;
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -9e-60:
		tmp = t * (x * -y)
	elif y <= 1.35e-143:
		tmp = x
	else:
		tmp = a * (x * -z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -9e-60)
		tmp = Float64(t * Float64(x * Float64(-y)));
	elseif (y <= 1.35e-143)
		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 <= -9e-60)
		tmp = t * (x * -y);
	elseif (y <= 1.35e-143)
		tmp = x;
	else
		tmp = a * (x * -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -9e-60], N[(t * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e-143], x, N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.00000000000000001e-60

   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 a around 0 85.2%

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

   6. Taylor expanded in t around inf 56.8%

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

      1. mul-1-neg 56.8%

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

      2. distribute-lft-neg-out 56.8%

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

      3. *-commutative 56.8%

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

   8. Simplified 56.8%

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

   9. Taylor expanded in y around 0 22.7%

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

       1. mul-1-neg 22.7%

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

       2. *-commutative 22.7%

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

       3. distribute-rgt-neg-in 22.7%

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

   11. Simplified 22.7%

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

   12. Taylor expanded in y around inf 28.4%

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

       1. associate-*r* 28.4%

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

       2. mul-1-neg 28.4%

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

   14. Simplified 28.4%

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

   if -9.00000000000000001e-60 < y < 1.35000000000000005e-143

   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 61.2%

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

   6. Taylor expanded in y around 0 37.4%

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

   if 1.35000000000000005e-143 < y

   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 y around 0 42.2%

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

   6. Taylor expanded in b around 0 12.7%

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

   7. Taylor expanded in z around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. unsub-neg 9.4%

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

      3. *-commutative 9.4%

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

   9. Simplified 9.4%

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

   10. Taylor expanded in z around inf 32.6%

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

       1. associate-*r* 32.6%

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

       2. mul-1-neg 32.6%

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

   12. Simplified 32.6%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-60}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-143}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.5% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-144}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.6e-59)
   (* x (* y (- t)))
   (if (<= y 8.4e-144) x (* a (* x (- z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.6e-59) {
		tmp = x * (y * -t);
	} else if (y <= 8.4e-144) {
		tmp = x;
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-1.6d-59)) then
        tmp = x * (y * -t)
    else if (y <= 8.4d-144) then
        tmp = x
    else
        tmp = a * (x * -z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.6e-59) {
		tmp = x * (y * -t);
	} else if (y <= 8.4e-144) {
		tmp = x;
	} else {
		tmp = a * (x * -z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.6e-59:
		tmp = x * (y * -t)
	elif y <= 8.4e-144:
		tmp = x
	else:
		tmp = a * (x * -z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.6e-59)
		tmp = Float64(x * Float64(y * Float64(-t)));
	elseif (y <= 8.4e-144)
		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 <= -1.6e-59)
		tmp = x * (y * -t);
	elseif (y <= 8.4e-144)
		tmp = x;
	else
		tmp = a * (x * -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.6e-59], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.4e-144], x, N[(a * N[(x * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6e-59

   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 a around 0 85.2%

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

   6. Taylor expanded in t around inf 56.8%

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

      1. mul-1-neg 56.8%

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

      2. distribute-lft-neg-out 56.8%

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

      3. *-commutative 56.8%

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

   8. Simplified 56.8%

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

   9. Taylor expanded in y around 0 22.7%

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

       1. mul-1-neg 22.7%

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

       2. *-commutative 22.7%

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

       3. distribute-rgt-neg-in 22.7%

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

   11. Simplified 22.7%

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

   12. Taylor expanded in y around inf 23.1%

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

       1. associate-*r* 23.1%

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

       2. mul-1-neg 23.1%

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

   14. Simplified 23.1%

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

   if -1.6e-59 < y < 8.4000000000000005e-144

   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 61.2%

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

   6. Taylor expanded in y around 0 37.4%

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

   if 8.4000000000000005e-144 < y

   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 y around 0 42.2%

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

   6. Taylor expanded in b around 0 12.7%

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

   7. Taylor expanded in z around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. unsub-neg 9.4%

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

      3. *-commutative 9.4%

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

   9. Simplified 9.4%

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

   10. Taylor expanded in z around inf 32.6%

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

       1. associate-*r* 32.6%

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

       2. mul-1-neg 32.6%

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

   12. Simplified 32.6%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-144}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.4% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \leq 1.92 \cdot 10^{-143}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.8e-60)
   (* x (* y (- t)))
   (if (<= y 1.92e-143) x (* x (* z (- a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.8e-60) {
		tmp = x * (y * -t);
	} else if (y <= 1.92e-143) {
		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 <= (-6.8d-60)) then
        tmp = x * (y * -t)
    else if (y <= 1.92d-143) 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 <= -6.8e-60) {
		tmp = x * (y * -t);
	} else if (y <= 1.92e-143) {
		tmp = x;
	} else {
		tmp = x * (z * -a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.8e-60:
		tmp = x * (y * -t)
	elif y <= 1.92e-143:
		tmp = x
	else:
		tmp = x * (z * -a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.8e-60)
		tmp = Float64(x * Float64(y * Float64(-t)));
	elseif (y <= 1.92e-143)
		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 <= -6.8e-60)
		tmp = x * (y * -t);
	elseif (y <= 1.92e-143)
		tmp = x;
	else
		tmp = x * (z * -a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.8e-60], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.92e-143], x, N[(x * N[(z * (-a)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.80000000000000013e-60

   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 a around 0 85.2%

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

   6. Taylor expanded in t around inf 56.8%

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

      1. mul-1-neg 56.8%

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

      2. distribute-lft-neg-out 56.8%

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

      3. *-commutative 56.8%

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

   8. Simplified 56.8%

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

   9. Taylor expanded in y around 0 22.7%

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

       1. mul-1-neg 22.7%

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

       2. *-commutative 22.7%

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

       3. distribute-rgt-neg-in 22.7%

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

   11. Simplified 22.7%

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

   12. Taylor expanded in y around inf 23.1%

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

       1. associate-*r* 23.1%

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

       2. mul-1-neg 23.1%

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

   14. Simplified 23.1%

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

   if -6.80000000000000013e-60 < y < 1.91999999999999995e-143

   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 61.2%

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

   6. Taylor expanded in y around 0 37.4%

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

   if 1.91999999999999995e-143 < y

   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 y around 0 42.2%

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

   6. Taylor expanded in b around 0 12.7%

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

   7. Taylor expanded in z around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. unsub-neg 9.4%

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

      3. *-commutative 9.4%

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

   9. Simplified 9.4%

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

   10. Taylor expanded in z around inf 28.1%

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

       1. mul-1-neg 28.1%

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

       2. *-commutative 28.1%

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

       3. distribute-rgt-neg-in 28.1%

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

   12. Simplified 28.1%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \leq 1.92 \cdot 10^{-143}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 24.4% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{-63}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 2.25e-63) (* 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 <= 2.25e-63) {
		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 <= 2.25d-63) 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 <= 2.25e-63) {
		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 <= 2.25e-63:
		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 <= 2.25e-63)
		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 <= 2.25e-63)
		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, 2.25e-63], 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 < 2.25e-63

   1. Initial program 98.2%

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

      1. fma-define 98.2%

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

      2. sub-neg 98.2%

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

      3. log1p-define 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 56.8%

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

   6. Taylor expanded in b around 0 19.8%

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

   7. Taylor expanded in z around 0 19.0%

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

      1. mul-1-neg 19.0%

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

      2. unsub-neg 19.0%

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

      3. *-commutative 19.0%

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

   9. Simplified 19.0%

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

   10. Taylor expanded in z around inf 27.7%

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

       1. associate-*r* 27.7%

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

       2. mul-1-neg 27.7%

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

   12. Simplified 27.7%

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

   if 2.25e-63 < x

   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 80.2%

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

   6. Taylor expanded in t around inf 64.6%

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

      1. mul-1-neg 64.6%

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

      2. distribute-lft-neg-out 64.6%

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

      3. *-commutative 64.6%

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

   8. Simplified 64.6%

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

   9. Taylor expanded in y around 0 29.4%

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

       1. mul-1-neg 29.4%

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

       2. *-commutative 29.4%

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

       3. distribute-rgt-neg-in 29.4%

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

   11. Simplified 29.4%

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

       1. distribute-rgt-neg-out 29.4%

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

       2. unsub-neg 29.4%

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

   13. Applied egg-rr 29.4%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{-63}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 19.0% 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 98.1%

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

   1. fma-define 98.1%

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

   2. sub-neg 98.1%

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

   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 77.4%

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

6. Taylor expanded in y around 0 17.6%

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

Reproduce

herbie shell --seed 2024146 
(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))))))