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

Percentage Accurate: 96.4% → 99.7%

Time: 16.0s

Alternatives: 16

Speedup: 1.5×

• 41.8% 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: 96.4% 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: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

   1. fma-define 95.6%

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

   2. sub-neg 95.6%

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

   3. log1p-define 99.2%

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

3. Simplified 99.2%

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

Alternative 2: 91.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{+30} \lor \neg \left(t \leq 3.45 \cdot 10^{+205}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \log z - a \cdot \left(z + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -2.25e+30) (not (<= t 3.45e+205)))
   (* x (exp (* t (- y))))
   (* x (exp (- (* y (log z)) (* a (+ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -2.25e+30) || !(t <= 3.45e+205)) {
		tmp = x * exp((t * -y));
	} else {
		tmp = x * exp(((y * log(z)) - (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 <= (-2.25d+30)) .or. (.not. (t <= 3.45d+205))) then
        tmp = x * exp((t * -y))
    else
        tmp = x * exp(((y * log(z)) - (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 <= -2.25e+30) || !(t <= 3.45e+205)) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = x * Math.exp(((y * Math.log(z)) - (a * (z + b))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -2.25e+30) or not (t <= 3.45e+205):
		tmp = x * math.exp((t * -y))
	else:
		tmp = x * math.exp(((y * math.log(z)) - (a * (z + b))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -2.25e+30) || !(t <= 3.45e+205))
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	else
		tmp = Float64(x * exp(Float64(Float64(y * log(z)) - Float64(a * Float64(z + b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -2.25e+30) || ~((t <= 3.45e+205)))
		tmp = x * exp((t * -y));
	else
		tmp = x * exp(((y * log(z)) - (a * (z + b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -2.25e+30], N[Not[LessEqual[t, 3.45e+205]], $MachinePrecision]], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.24999999999999997e30 or 3.4499999999999999e205 < t

   1. Initial program 98.6%

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

   3. Taylor expanded in t around inf 86.1%

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

      1. mul-1-neg 86.1%

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

      2. distribute-lft-neg-out 86.1%

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

      3. *-commutative 86.1%

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

   5. Simplified 86.1%

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

   if -2.24999999999999997e30 < t < 3.4499999999999999e205

   1. Initial program 94.0%

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

   3. Taylor expanded in z around 0 98.9%

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

      1. mul-1-neg 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in t around 0 98.1%

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

      1. +-commutative 98.1%

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

      2. mul-1-neg 98.1%

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

      3. unsub-neg 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{+30} \lor \neg \left(t \leq 3.45 \cdot 10^{+205}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \log z - a \cdot \left(z + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-42} \lor \neg \left(y \leq 1.6 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -5.8e-42) (not (<= y 1.6e+19)))
   (* x (pow (/ z (exp t)) y))
   (* x (exp (* a (- (log1p (- z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.8e-42) || !(y <= 1.6e+19)) {
		tmp = x * pow((z / exp(t)), y);
	} else {
		tmp = x * exp((a * (log1p(-z) - b)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.8e-42) || !(y <= 1.6e+19)) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else {
		tmp = x * Math.exp((a * (Math.log1p(-z) - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -5.8e-42) or not (y <= 1.6e+19):
		tmp = x * math.pow((z / math.exp(t)), y)
	else:
		tmp = x * math.exp((a * (math.log1p(-z) - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -5.8e-42) || !(y <= 1.6e+19))
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(log1p(Float64(-z)) - b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -5.8e-42], N[Not[LessEqual[y, 1.6e+19]], $MachinePrecision]], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.8000000000000006e-42 or 1.6e19 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in a around 0 90.6%

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

      1. *-commutative 90.6%

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

      2. exp-prod 90.6%

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

      3. exp-diff 90.6%

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

      4. rem-exp-log 90.6%

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

   5. Simplified 90.6%

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

   if -5.8000000000000006e-42 < y < 1.6e19

   1. Initial program 92.4%

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

   3. Taylor expanded in y around 0 81.7%

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

      1. sub-neg 81.7%

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

      2. log1p-define 89.1%

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

   5. Simplified 89.1%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-42} \lor \neg \left(y \leq 1.6 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-46} \lor \neg \left(y \leq 4.7 \cdot 10^{-54}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {e}^{\left(a \cdot \left(-b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1e-46) (not (<= y 4.7e-54)))
   (* x (pow (/ z (exp t)) y))
   (* x (pow E (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1e-46) || !(y <= 4.7e-54)) {
		tmp = x * pow((z / exp(t)), y);
	} else {
		tmp = x * pow(((double) M_E), (a * -b));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1e-46) || !(y <= 4.7e-54)) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else {
		tmp = x * Math.pow(Math.E, (a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1e-46) or not (y <= 4.7e-54):
		tmp = x * math.pow((z / math.exp(t)), y)
	else:
		tmp = x * math.pow(math.e, (a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1e-46) || !(y <= 4.7e-54))
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	else
		tmp = Float64(x * (exp(1) ^ Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1e-46) || ~((y <= 4.7e-54)))
		tmp = x * ((z / exp(t)) ^ y);
	else
		tmp = x * (2.71828182845904523536 ^ (a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1e-46], N[Not[LessEqual[y, 4.7e-54]], $MachinePrecision]], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[E, N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000002e-46 or 4.7e-54 < y

   1. Initial program 96.0%

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

   3. Taylor expanded in a around 0 87.8%

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

      1. *-commutative 87.8%

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

      2. exp-prod 87.8%

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

      3. exp-diff 87.8%

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

      4. rem-exp-log 87.8%

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

   5. Simplified 87.8%

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

   if -1.00000000000000002e-46 < y < 4.7e-54

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0 86.0%

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

      1. sub-neg 86.0%

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

      2. log1p-define 91.9%

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

   5. Simplified 91.9%

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

      1. *-un-lft-identity 91.9%

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

      2. exp-prod 91.9%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 83.0%

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

      5. sqr-neg 83.0%

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

      6. sqrt-unprod 83.0%

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

      7. add-sqr-sqrt 83.0%

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

   7. Applied egg-rr 83.0%

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

      1. exp-1-e 83.0%

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

   9. Simplified 83.0%

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

   10. Taylor expanded in z around 0 83.0%

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

       1. +-commutative 83.0%

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

       2. mul-1-neg 83.0%

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

       3. distribute-rgt-neg-out 83.0%

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

       4. distribute-lft-out 83.0%

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

   12. Simplified 83.0%

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

   13. Taylor expanded in z around 0 85.0%

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

       1. associate-*r* 85.0%

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

       2. neg-mul-1 85.0%

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

   15. Simplified 85.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-46} \lor \neg \left(y \leq 4.7 \cdot 10^{-54}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {e}^{\left(a \cdot \left(-b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in z around 0 98.8%

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

   1. mul-1-neg 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 6: 73.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+14} \lor \neg \left(y \leq 1.6 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {e}^{\left(a \cdot \left(-b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -9.5e+14) (not (<= y 1.6e+19)))
   (* x (pow z y))
   (* x (pow E (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -9.5e+14) || !(y <= 1.6e+19)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * pow(((double) M_E), (a * -b));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -9.5e+14) || !(y <= 1.6e+19)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.pow(Math.E, (a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -9.5e+14) or not (y <= 1.6e+19):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.pow(math.e, (a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -9.5e+14) || !(y <= 1.6e+19))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * (exp(1) ^ Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -9.5e+14) || ~((y <= 1.6e+19)))
		tmp = x * (z ^ y);
	else
		tmp = x * (2.71828182845904523536 ^ (a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -9.5e+14], N[Not[LessEqual[y, 1.6e+19]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[E, N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5e14 or 1.6e19 < y

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. exp-prod 92.0%

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

      3. exp-diff 92.0%

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

      4. rem-exp-log 92.0%

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

   5. Simplified 92.0%

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

   6. Taylor expanded in t around 0 74.4%

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

   if -9.5e14 < y < 1.6e19

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0 78.7%

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

      1. sub-neg 78.7%

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

      2. log1p-define 85.4%

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

   5. Simplified 85.4%

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

      1. *-un-lft-identity 85.4%

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

      2. exp-prod 85.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 76.5%

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

      5. sqr-neg 76.5%

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

      6. sqrt-unprod 76.5%

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

      7. add-sqr-sqrt 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. exp-1-e 76.5%

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

   9. Simplified 76.5%

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

   10. Taylor expanded in z around 0 76.5%

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

       1. +-commutative 76.5%

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

       2. mul-1-neg 76.5%

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

       3. distribute-rgt-neg-out 76.5%

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

       4. distribute-lft-out 76.5%

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

   12. Simplified 76.5%

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

   13. Taylor expanded in z around 0 78.0%

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

       1. associate-*r* 78.0%

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

       2. neg-mul-1 78.0%

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

   15. Simplified 78.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+14} \lor \neg \left(y \leq 1.6 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {e}^{\left(a \cdot \left(-b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+14} \lor \neg \left(y \leq 2.8 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -8.2e+14) (not (<= y 2.8e+19)))
   (* x (pow z y))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -8.2e+14) || !(y <= 2.8e+19)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-8.2d+14)) .or. (.not. (y <= 2.8d+19))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -8.2e+14) || !(y <= 2.8e+19)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -8.2e+14) or not (y <= 2.8e+19):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -8.2e+14) || !(y <= 2.8e+19))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -8.2e+14) || ~((y <= 2.8e+19)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -8.2e+14], N[Not[LessEqual[y, 2.8e+19]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.2e14 or 2.8e19 < y

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. exp-prod 92.0%

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

      3. exp-diff 92.0%

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

      4. rem-exp-log 92.0%

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

   5. Simplified 92.0%

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

   6. Taylor expanded in t around 0 74.4%

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

   if -8.2e14 < y < 2.8e19

   1. Initial program 93.1%

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

   3. Taylor expanded in b around inf 78.0%

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

      1. mul-1-neg 78.0%

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

      2. distribute-rgt-neg-out 78.0%

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

   5. Simplified 78.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+14} \lor \neg \left(y \leq 2.8 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-53} \lor \neg \left(y \leq 8.3 \cdot 10^{-47}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(z - b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.6e-53) (not (<= y 8.3e-47)))
   (* x (pow z y))
   (* x (+ 1.0 (* a (- z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.6e-53) || !(y <= 8.3e-47)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * (1.0 + (a * (z - b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-2.6d-53)) .or. (.not. (y <= 8.3d-47))) then
        tmp = x * (z ** y)
    else
        tmp = x * (1.0d0 + (a * (z - b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.6e-53) || !(y <= 8.3e-47)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * (1.0 + (a * (z - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.6e-53) or not (y <= 8.3e-47):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * (1.0 + (a * (z - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.6e-53) || !(y <= 8.3e-47))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * Float64(1.0 + Float64(a * Float64(z - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.6e-53) || ~((y <= 8.3e-47)))
		tmp = x * (z ^ y);
	else
		tmp = x * (1.0 + (a * (z - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.6e-53], N[Not[LessEqual[y, 8.3e-47]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(a * N[(z - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.59999999999999996e-53 or 8.2999999999999997e-47 < y

   1. Initial program 95.4%

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

   3. Taylor expanded in a around 0 86.7%

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

      1. *-commutative 86.7%

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

      2. exp-prod 86.1%

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

      3. exp-diff 86.1%

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

      4. rem-exp-log 86.1%

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

   5. Simplified 86.1%

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

   6. Taylor expanded in t around 0 66.5%

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

   if -2.59999999999999996e-53 < y < 8.2999999999999997e-47

   1. Initial program 95.0%

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

   3. Taylor expanded in y around 0 85.9%

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

      1. sub-neg 85.9%

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

      2. log1p-define 90.9%

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

   5. Simplified 90.9%

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

      1. *-un-lft-identity 90.9%

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

      2. exp-prod 90.9%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 83.0%

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

      5. sqr-neg 83.0%

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

      6. sqrt-unprod 83.0%

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

      7. add-sqr-sqrt 83.0%

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

   7. Applied egg-rr 83.0%

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

      1. exp-1-e 83.0%

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

   9. Simplified 83.0%

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

   10. Taylor expanded in z around 0 83.0%

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

       1. +-commutative 83.0%

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

       2. mul-1-neg 83.0%

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

       3. distribute-rgt-neg-out 83.0%

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

       4. distribute-lft-out 83.0%

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

   12. Simplified 83.0%

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

   13. Taylor expanded in a around 0 48.3%

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

       1. sub-neg 48.3%

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

       2. mul-1-neg 48.3%

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

       3. log-E 48.3%

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

       4. mul-1-neg 48.3%

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

       5. sub-neg 48.3%

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

   15. Simplified 48.3%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-53} \lor \neg \left(y \leq 8.3 \cdot 10^{-47}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(z - b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.9% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-64}:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot z\right)\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(z - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -7e-64)
   (* a (- (/ x a) (* x z)))
   (if (<= y 7.4e+22) (* x (+ 1.0 (* a (- z b)))) (* x (* z (- a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7e-64) {
		tmp = a * ((x / a) - (x * z));
	} else if (y <= 7.4e+22) {
		tmp = x * (1.0 + (a * (z - b)));
	} 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 <= (-7d-64)) then
        tmp = a * ((x / a) - (x * z))
    else if (y <= 7.4d+22) then
        tmp = x * (1.0d0 + (a * (z - b)))
    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 <= -7e-64) {
		tmp = a * ((x / a) - (x * z));
	} else if (y <= 7.4e+22) {
		tmp = x * (1.0 + (a * (z - b)));
	} else {
		tmp = x * (z * -a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -7e-64:
		tmp = a * ((x / a) - (x * z))
	elif y <= 7.4e+22:
		tmp = x * (1.0 + (a * (z - b)))
	else:
		tmp = x * (z * -a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -7e-64)
		tmp = Float64(a * Float64(Float64(x / a) - Float64(x * z)));
	elseif (y <= 7.4e+22)
		tmp = Float64(x * Float64(1.0 + Float64(a * Float64(z - b))));
	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 <= -7e-64)
		tmp = a * ((x / a) - (x * z));
	elseif (y <= 7.4e+22)
		tmp = x * (1.0 + (a * (z - b)));
	else
		tmp = x * (z * -a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -7e-64], N[(a * N[(N[(x / a), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.4e+22], N[(x * N[(1.0 + N[(a * N[(z - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(z * (-a)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.0000000000000006e-64

   1. Initial program 94.2%

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

   3. Taylor expanded in z around 0 96.5%

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

      1. mul-1-neg 96.5%

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

   5. Simplified 96.5%

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

   6. Taylor expanded in z around inf 21.6%

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

      1. associate-*r* 21.6%

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

      2. mul-1-neg 21.6%

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

   8. Simplified 21.6%

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

   9. Taylor expanded in a around 0 10.3%

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

       1. mul-1-neg 10.3%

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

       2. unsub-neg 10.3%

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

       3. *-commutative 10.3%

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

   11. Simplified 10.3%

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

   12. Taylor expanded in a around inf 20.2%

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

       1. neg-mul-1 20.2%

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

       2. +-commutative 20.2%

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

       3. sub-neg 20.2%

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

   14. Simplified 20.2%

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

   if -7.0000000000000006e-64 < y < 7.3999999999999996e22

   1. Initial program 93.7%

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

   3. Taylor expanded in y around 0 82.7%

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

      1. sub-neg 82.7%

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

      2. log1p-define 88.7%

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

   5. Simplified 88.7%

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

      1. *-un-lft-identity 88.7%

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

      2. exp-prod 88.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 80.1%

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

      5. sqr-neg 80.1%

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

      6. sqrt-unprod 80.1%

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

      7. add-sqr-sqrt 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. exp-1-e 80.1%

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

   9. Simplified 80.1%

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

   10. Taylor expanded in z around 0 80.1%

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

       1. +-commutative 80.1%

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

       2. mul-1-neg 80.1%

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

       3. distribute-rgt-neg-out 80.1%

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

       4. distribute-lft-out 80.1%

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

   12. Simplified 80.1%

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

   13. Taylor expanded in a around 0 47.4%

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

       1. sub-neg 47.4%

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

       2. mul-1-neg 47.4%

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

       3. log-E 47.4%

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

       4. mul-1-neg 47.4%

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

       5. sub-neg 47.4%

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

   15. Simplified 47.4%

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

   if 7.3999999999999996e22 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 16.0%

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

      1. associate-*r* 16.0%

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

      2. mul-1-neg 16.0%

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

   8. Simplified 16.0%

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

   9. Taylor expanded in a around 0 3.9%

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

       1. mul-1-neg 3.9%

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

       2. unsub-neg 3.9%

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

       3. *-commutative 3.9%

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

   11. Simplified 3.9%

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

   12. Taylor expanded in z around inf 32.3%

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

       1. mul-1-neg 32.3%

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

       2. *-commutative 32.3%

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

       3. distribute-rgt-neg-in 32.3%

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

       4. associate-*r* 33.2%

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

       5. *-commutative 33.2%

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

       6. distribute-lft-neg-in 33.2%

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

       7. distribute-rgt-neg-in 33.2%

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

   14. Simplified 33.2%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-64}:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot z\right)\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \left(1 + a \cdot \left(z - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-237}:\\ \;\;\;\;z \cdot \left(\frac{x}{z} - x \cdot a\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x - z \cdot \left(x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -7.2e-237)
   (* z (- (/ x z) (* x a)))
   (if (<= y 5e-8) (- x (* z (* x a))) (* a (* z (- x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7.2e-237) {
		tmp = z * ((x / z) - (x * a));
	} else if (y <= 5e-8) {
		tmp = x - (z * (x * a));
	} else {
		tmp = a * (z * -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 <= (-7.2d-237)) then
        tmp = z * ((x / z) - (x * a))
    else if (y <= 5d-8) then
        tmp = x - (z * (x * a))
    else
        tmp = a * (z * -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 <= -7.2e-237) {
		tmp = z * ((x / z) - (x * a));
	} else if (y <= 5e-8) {
		tmp = x - (z * (x * a));
	} else {
		tmp = a * (z * -x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -7.2e-237:
		tmp = z * ((x / z) - (x * a))
	elif y <= 5e-8:
		tmp = x - (z * (x * a))
	else:
		tmp = a * (z * -x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -7.2e-237)
		tmp = Float64(z * Float64(Float64(x / z) - Float64(x * a)));
	elseif (y <= 5e-8)
		tmp = Float64(x - Float64(z * Float64(x * a)));
	else
		tmp = Float64(a * Float64(z * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -7.2e-237)
		tmp = z * ((x / z) - (x * a));
	elseif (y <= 5e-8)
		tmp = x - (z * (x * a));
	else
		tmp = a * (z * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -7.2e-237], N[(z * N[(N[(x / z), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-8], N[(x - N[(z * N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(z * (-x)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999993e-237

   1. Initial program 95.0%

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

   3. Taylor expanded in z around 0 97.5%

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

      1. mul-1-neg 97.5%

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

   5. Simplified 97.5%

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

   6. Taylor expanded in z around inf 28.9%

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

      1. associate-*r* 28.9%

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

      2. mul-1-neg 28.9%

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

   8. Simplified 28.9%

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

   9. Taylor expanded in a around 0 16.0%

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

       1. mul-1-neg 16.0%

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

       2. unsub-neg 16.0%

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

       3. *-commutative 16.0%

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

   11. Simplified 16.0%

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

   12. Taylor expanded in z around inf 24.3%

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

       1. +-commutative 24.3%

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

       2. mul-1-neg 24.3%

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

       3. sub-neg 24.3%

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

       4. *-commutative 24.3%

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

   14. Simplified 24.3%

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

   if -7.19999999999999993e-237 < y < 4.9999999999999998e-8

   1. Initial program 92.1%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 60.2%

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

      1. associate-*r* 60.2%

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

      2. mul-1-neg 60.2%

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

   8. Simplified 60.2%

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

   9. Taylor expanded in a around 0 40.3%

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

       1. mul-1-neg 40.3%

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

       2. unsub-neg 40.3%

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

       3. *-commutative 40.3%

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

   11. Simplified 40.3%

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

       1. distribute-rgt-out-- 40.3%

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

       2. *-un-lft-identity 40.3%

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

       3. associate-*l* 41.5%

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

   13. Applied egg-rr 41.5%

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

   if 4.9999999999999998e-8 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 17.3%

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

      1. associate-*r* 17.3%

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

      2. mul-1-neg 17.3%

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

   8. Simplified 17.3%

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

   9. Taylor expanded in a around 0 4.0%

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

       1. mul-1-neg 4.0%

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

       2. unsub-neg 4.0%

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

       3. *-commutative 4.0%

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

   11. Simplified 4.0%

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

   12. Taylor expanded in z around inf 33.0%

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

       1. mul-1-neg 33.0%

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

       2. distribute-rgt-neg-out 33.0%

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

       3. distribute-rgt-neg-in 33.0%

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

   14. Simplified 33.0%

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

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-237}:\\ \;\;\;\;z \cdot \left(\frac{x}{z} - x \cdot a\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x - z \cdot \left(x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.5% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-26}:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot z\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;x - z \cdot \left(x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -9.2e-26)
   (* a (- (/ x a) (* x z)))
   (if (<= y 4.8e-8) (- x (* z (* x a))) (* a (* z (- x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9.2e-26) {
		tmp = a * ((x / a) - (x * z));
	} else if (y <= 4.8e-8) {
		tmp = x - (z * (x * a));
	} else {
		tmp = a * (z * -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 <= (-9.2d-26)) then
        tmp = a * ((x / a) - (x * z))
    else if (y <= 4.8d-8) then
        tmp = x - (z * (x * a))
    else
        tmp = a * (z * -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 <= -9.2e-26) {
		tmp = a * ((x / a) - (x * z));
	} else if (y <= 4.8e-8) {
		tmp = x - (z * (x * a));
	} else {
		tmp = a * (z * -x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -9.2e-26:
		tmp = a * ((x / a) - (x * z))
	elif y <= 4.8e-8:
		tmp = x - (z * (x * a))
	else:
		tmp = a * (z * -x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -9.2e-26)
		tmp = Float64(a * Float64(Float64(x / a) - Float64(x * z)));
	elseif (y <= 4.8e-8)
		tmp = Float64(x - Float64(z * Float64(x * a)));
	else
		tmp = Float64(a * Float64(z * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -9.2e-26)
		tmp = a * ((x / a) - (x * z));
	elseif (y <= 4.8e-8)
		tmp = x - (z * (x * a));
	else
		tmp = a * (z * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -9.2e-26], N[(a * N[(N[(x / a), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e-8], N[(x - N[(z * N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(z * (-x)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.20000000000000035e-26

   1. Initial program 96.2%

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

   3. Taylor expanded in z around 0 96.2%

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

      1. mul-1-neg 96.2%

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

   5. Simplified 96.2%

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

   6. Taylor expanded in z around inf 16.0%

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

      1. associate-*r* 16.0%

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

      2. mul-1-neg 16.0%

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

   8. Simplified 16.0%

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

   9. Taylor expanded in a around 0 6.1%

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

       1. mul-1-neg 6.1%

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

       2. unsub-neg 6.1%

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

       3. *-commutative 6.1%

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

   11. Simplified 6.1%

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

   12. Taylor expanded in a around inf 17.0%

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

       1. neg-mul-1 17.0%

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

       2. +-commutative 17.0%

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

       3. sub-neg 17.0%

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

   14. Simplified 17.0%

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

   if -9.20000000000000035e-26 < y < 4.79999999999999997e-8

   1. Initial program 92.4%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf 57.3%

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

      1. associate-*r* 57.3%

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

      2. mul-1-neg 57.3%

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

   8. Simplified 57.3%

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

   9. Taylor expanded in a around 0 38.0%

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

       1. mul-1-neg 38.0%

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

       2. unsub-neg 38.0%

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

       3. *-commutative 38.0%

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

   11. Simplified 38.0%

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

       1. distribute-rgt-out-- 38.0%

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

       2. *-un-lft-identity 38.0%

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

       3. associate-*l* 39.5%

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

   13. Applied egg-rr 39.5%

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

   if 4.79999999999999997e-8 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 17.3%

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

      1. associate-*r* 17.3%

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

      2. mul-1-neg 17.3%

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

   8. Simplified 17.3%

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

   9. Taylor expanded in a around 0 4.0%

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

       1. mul-1-neg 4.0%

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

       2. unsub-neg 4.0%

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

       3. *-commutative 4.0%

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

   11. Simplified 4.0%

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

   12. Taylor expanded in z around inf 33.0%

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

       1. mul-1-neg 33.0%

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

       2. distribute-rgt-neg-out 33.0%

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

       3. distribute-rgt-neg-in 33.0%

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

   14. Simplified 33.0%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-26}:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot z\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;x - z \cdot \left(x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.9% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-24} \lor \neg \left(y \leq 2.8 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -9.6e-24) (not (<= y 2.8e-7))) (* a (* z (- x))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -9.6e-24) || !(y <= 2.8e-7)) {
		tmp = a * (z * -x);
	} 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 <= (-9.6d-24)) .or. (.not. (y <= 2.8d-7))) then
        tmp = a * (z * -x)
    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 <= -9.6e-24) || !(y <= 2.8e-7)) {
		tmp = a * (z * -x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -9.6e-24) or not (y <= 2.8e-7):
		tmp = a * (z * -x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -9.6e-24) || !(y <= 2.8e-7))
		tmp = Float64(a * Float64(z * Float64(-x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -9.6e-24) || ~((y <= 2.8e-7)))
		tmp = a * (z * -x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -9.6e-24], N[Not[LessEqual[y, 2.8e-7]], $MachinePrecision]], N[(a * N[(z * (-x)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -9.5999999999999993e-24 or 2.80000000000000019e-7 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in z around 0 97.8%

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

      1. mul-1-neg 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in z around inf 16.6%

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

      1. associate-*r* 16.6%

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

      2. mul-1-neg 16.6%

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

   8. Simplified 16.6%

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

   9. Taylor expanded in a around 0 5.2%

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

       1. mul-1-neg 5.2%

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

       2. unsub-neg 5.2%

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

       3. *-commutative 5.2%

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

   11. Simplified 5.2%

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

   12. Taylor expanded in z around inf 19.7%

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

       1. mul-1-neg 19.7%

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

       2. distribute-rgt-neg-out 19.7%

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

       3. distribute-rgt-neg-in 19.7%

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

   14. Simplified 19.7%

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

   if -9.5999999999999993e-24 < y < 2.80000000000000019e-7

   1. Initial program 92.5%

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

   3. Taylor expanded in y around 0 80.5%

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

      1. sub-neg 80.5%

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

      2. log1p-define 87.7%

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

   5. Simplified 87.7%

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

   6. Taylor expanded in a around 0 35.6%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-24} \lor \neg \left(y \leq 2.8 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.1% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;x - z \cdot \left(x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 1.85e-6) (- x (* z (* x a))) (* a (* z (- x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 1.85e-6) {
		tmp = x - (z * (x * a));
	} else {
		tmp = a * (z * -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 <= 1.85d-6) then
        tmp = x - (z * (x * a))
    else
        tmp = a * (z * -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 <= 1.85e-6) {
		tmp = x - (z * (x * a));
	} else {
		tmp = a * (z * -x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 1.85e-6:
		tmp = x - (z * (x * a))
	else:
		tmp = a * (z * -x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 1.85e-6)
		tmp = Float64(x - Float64(z * Float64(x * a)));
	else
		tmp = Float64(a * Float64(z * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 1.85e-6)
		tmp = x - (z * (x * a));
	else
		tmp = a * (z * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 1.85e-6], N[(x - N[(z * N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(z * (-x)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8500000000000001e-6

   1. Initial program 93.9%

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

   3. Taylor expanded in z around 0 98.5%

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

      1. mul-1-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in z around inf 41.1%

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

      1. associate-*r* 41.1%

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

      2. mul-1-neg 41.1%

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

   8. Simplified 41.1%

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

   9. Taylor expanded in a around 0 25.5%

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

       1. mul-1-neg 25.5%

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

       2. unsub-neg 25.5%

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

       3. *-commutative 25.5%

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

   11. Simplified 25.5%

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

       1. distribute-rgt-out-- 25.5%

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

       2. *-un-lft-identity 25.5%

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

       3. associate-*l* 26.8%

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

   13. Applied egg-rr 26.8%

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

   if 1.8500000000000001e-6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 17.3%

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

      1. associate-*r* 17.3%

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

      2. mul-1-neg 17.3%

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

   8. Simplified 17.3%

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

   9. Taylor expanded in a around 0 4.0%

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

       1. mul-1-neg 4.0%

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

       2. unsub-neg 4.0%

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

       3. *-commutative 4.0%

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

   11. Simplified 4.0%

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

   12. Taylor expanded in z around inf 33.0%

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

       1. mul-1-neg 33.0%

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

       2. distribute-rgt-neg-out 33.0%

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

       3. distribute-rgt-neg-in 33.0%

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

   14. Simplified 33.0%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;x - z \cdot \left(x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.2% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;x - a \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 5.5e-8) (- x (* a (* x z))) (* a (* z (- x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 5.5e-8) {
		tmp = x - (a * (x * z));
	} else {
		tmp = a * (z * -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 <= 5.5d-8) then
        tmp = x - (a * (x * z))
    else
        tmp = a * (z * -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 <= 5.5e-8) {
		tmp = x - (a * (x * z));
	} else {
		tmp = a * (z * -x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 5.5e-8:
		tmp = x - (a * (x * z))
	else:
		tmp = a * (z * -x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 5.5e-8)
		tmp = Float64(x - Float64(a * Float64(x * z)));
	else
		tmp = Float64(a * Float64(z * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 5.5e-8)
		tmp = x - (a * (x * z));
	else
		tmp = a * (z * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 5.5e-8], N[(x - N[(a * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(z * (-x)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5000000000000003e-8

   1. Initial program 93.9%

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

   3. Taylor expanded in z around 0 98.5%

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

      1. mul-1-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in z around inf 41.1%

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

      1. associate-*r* 41.1%

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

      2. mul-1-neg 41.1%

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

   8. Simplified 41.1%

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

   9. Taylor expanded in a around 0 25.5%

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

       1. mul-1-neg 25.5%

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

       2. unsub-neg 25.5%

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

       3. *-commutative 25.5%

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

   11. Simplified 25.5%

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

   if 5.5000000000000003e-8 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 17.3%

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

      1. associate-*r* 17.3%

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

      2. mul-1-neg 17.3%

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

   8. Simplified 17.3%

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

   9. Taylor expanded in a around 0 4.0%

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

       1. mul-1-neg 4.0%

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

       2. unsub-neg 4.0%

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

       3. *-commutative 4.0%

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

   11. Simplified 4.0%

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

   12. Taylor expanded in z around inf 33.0%

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

       1. mul-1-neg 33.0%

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

       2. distribute-rgt-neg-out 33.0%

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

       3. distribute-rgt-neg-in 33.0%

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

   14. Simplified 33.0%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;x - a \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.2% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(1 - z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 1.55e-7) (* x (- 1.0 (* z a))) (* a (* z (- x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 1.55e-7) {
		tmp = x * (1.0 - (z * a));
	} else {
		tmp = a * (z * -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 <= 1.55d-7) then
        tmp = x * (1.0d0 - (z * a))
    else
        tmp = a * (z * -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 <= 1.55e-7) {
		tmp = x * (1.0 - (z * a));
	} else {
		tmp = a * (z * -x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 1.55e-7:
		tmp = x * (1.0 - (z * a))
	else:
		tmp = a * (z * -x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 1.55e-7)
		tmp = Float64(x * Float64(1.0 - Float64(z * a)));
	else
		tmp = Float64(a * Float64(z * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 1.55e-7)
		tmp = x * (1.0 - (z * a));
	else
		tmp = a * (z * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 1.55e-7], N[(x * N[(1.0 - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(z * (-x)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55e-7

   1. Initial program 93.9%

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

   3. Taylor expanded in z around 0 98.5%

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

      1. mul-1-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in z around inf 41.1%

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

      1. associate-*r* 41.1%

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

      2. mul-1-neg 41.1%

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

   8. Simplified 41.1%

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

   9. Taylor expanded in a around 0 25.5%

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

       1. mul-1-neg 25.5%

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

       2. unsub-neg 25.5%

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

       3. *-commutative 25.5%

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

   11. Simplified 25.5%

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

   if 1.55e-7 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 17.3%

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

      1. associate-*r* 17.3%

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

      2. mul-1-neg 17.3%

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

   8. Simplified 17.3%

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

   9. Taylor expanded in a around 0 4.0%

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

       1. mul-1-neg 4.0%

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

       2. unsub-neg 4.0%

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

       3. *-commutative 4.0%

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

   11. Simplified 4.0%

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

   12. Taylor expanded in z around inf 33.0%

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

       1. mul-1-neg 33.0%

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

       2. distribute-rgt-neg-out 33.0%

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

       3. distribute-rgt-neg-in 33.0%

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

   14. Simplified 33.0%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(1 - z \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 19.5% accurate, 315.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in y around 0 56.4%

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

   1. sub-neg 56.4%

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

   2. log1p-define 60.6%

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

5. Simplified 60.6%

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

6. Taylor expanded in a around 0 19.0%

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

Reproduce

herbie shell --seed 2024088 
(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))))))