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

Percentage Accurate: 96.5% → 99.6%

Time: 17.8s

Alternatives: 19

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

   1. fma-define 96.4%

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

   2. sub-neg 96.4%

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

   3. log1p-define 99.5%

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

3. Simplified 99.5%

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

Alternative 2: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

Alternative 3: 87.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0064:\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{e^{y \cdot \left(\log z - t\right) + 1}}{e}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -0.0064)
   (* x (pow (/ z (exp t)) y))
   (if (<= y 2.7e-5)
     (* x (exp (* a (- (log1p (- z)) b))))
     (* x (/ (exp (+ (* y (- (log z) t)) 1.0)) E)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -0.0064) {
		tmp = x * pow((z / exp(t)), y);
	} else if (y <= 2.7e-5) {
		tmp = x * exp((a * (log1p(-z) - b)));
	} else {
		tmp = x * (exp(((y * (log(z) - t)) + 1.0)) / ((double) M_E));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -0.0064) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else if (y <= 2.7e-5) {
		tmp = x * Math.exp((a * (Math.log1p(-z) - b)));
	} else {
		tmp = x * (Math.exp(((y * (Math.log(z) - t)) + 1.0)) / Math.E);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -0.0064:
		tmp = x * math.pow((z / math.exp(t)), y)
	elif y <= 2.7e-5:
		tmp = x * math.exp((a * (math.log1p(-z) - b)))
	else:
		tmp = x * (math.exp(((y * (math.log(z) - t)) + 1.0)) / math.e)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -0.0064)
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	elseif (y <= 2.7e-5)
		tmp = Float64(x * exp(Float64(a * Float64(log1p(Float64(-z)) - b))));
	else
		tmp = Float64(x * Float64(exp(Float64(Float64(y * Float64(log(z) - t)) + 1.0)) / exp(1)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -0.0064], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e-5], N[(x * N[Exp[N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00640000000000000031

   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 inf 89.1%

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

      1. pow-exp 89.1%

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

      2. *-un-lft-identity 89.1%

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

      3. pow-exp 89.1%

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

      4. *-commutative 89.1%

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

      5. exp-prod 89.1%

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

      6. exp-diff 89.1%

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

      7. add-exp-log 89.2%

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

   5. Applied egg-rr 89.2%

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

      1. *-lft-identity 89.2%

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

   7. Simplified 89.2%

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

   if -0.00640000000000000031 < y < 2.6999999999999999e-5

   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 y around 0 80.1%

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

      1. sub-neg 80.1%

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

      2. mul-1-neg 80.1%

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

      3. log1p-define 86.9%

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

      4. mul-1-neg 86.9%

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

   5. Simplified 86.9%

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

   if 2.6999999999999999e-5 < y

   1. Initial program 96.7%

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

   3. Taylor expanded in y around inf 86.8%

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

      1. expm1-log1p-u 20.1%

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(\log z - t\right)\right)\right)}} \]

      2. expm1-define 20.1%

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

      3. exp-diff 20.1%

         \[\leadsto x \cdot \color{blue}{\frac{e^{e^{\mathsf{log1p}\left(y \cdot \left(\log z - t\right)\right)}}}{e^{1}}} \]

      4. log1p-undefine 20.1%

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

      5. rem-exp-log 86.9%

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

      6. exp-1-e 86.9%

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

   5. Applied egg-rr 86.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0064:\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{e^{y \cdot \left(\log z - t\right) + 1}}{e}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0038:\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-5}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -0.0038)
   (* x (pow (/ z (exp t)) y))
   (if (<= y 2.75e-5)
     (* x (exp (* a (- (log1p (- z)) b))))
     (* x (exp (* y (- (log z) t)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -0.0038) {
		tmp = x * pow((z / exp(t)), y);
	} else if (y <= 2.75e-5) {
		tmp = x * exp((a * (log1p(-z) - b)));
	} else {
		tmp = x * exp((y * (log(z) - t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -0.0038) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else if (y <= 2.75e-5) {
		tmp = x * Math.exp((a * (Math.log1p(-z) - b)));
	} else {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -0.0038:
		tmp = x * math.pow((z / math.exp(t)), y)
	elif y <= 2.75e-5:
		tmp = x * math.exp((a * (math.log1p(-z) - b)))
	else:
		tmp = x * math.exp((y * (math.log(z) - t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -0.0038)
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	elseif (y <= 2.75e-5)
		tmp = Float64(x * exp(Float64(a * Float64(log1p(Float64(-z)) - b))));
	else
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -0.0038], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.75e-5], N[(x * N[Exp[N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00379999999999999999

   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 inf 89.1%

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

      1. pow-exp 89.1%

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

      2. *-un-lft-identity 89.1%

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

      3. pow-exp 89.1%

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

      4. *-commutative 89.1%

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

      5. exp-prod 89.1%

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

      6. exp-diff 89.1%

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

      7. add-exp-log 89.2%

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

   5. Applied egg-rr 89.2%

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

      1. *-lft-identity 89.2%

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

   7. Simplified 89.2%

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

   if -0.00379999999999999999 < y < 2.7500000000000001e-5

   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 y around 0 80.1%

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

      1. sub-neg 80.1%

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

      2. mul-1-neg 80.1%

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

      3. log1p-define 86.9%

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

      4. mul-1-neg 86.9%

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

   5. Simplified 86.9%

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

   if 2.7500000000000001e-5 < y

   1. Initial program 96.7%

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

   3. Taylor expanded in y around inf 86.8%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-14} \lor \neg \left(b \leq 2.8 \cdot 10^{+40}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -9e-14) (not (<= b 2.8e+40)))
   (* x (exp (* a (- b))))
   (* x (exp (* y (- (log z) t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -9e-14) || !(b <= 2.8e+40)) {
		tmp = x * exp((a * -b));
	} else {
		tmp = x * exp((y * (log(z) - t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-9d-14)) .or. (.not. (b <= 2.8d+40))) then
        tmp = x * exp((a * -b))
    else
        tmp = x * exp((y * (log(z) - t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -9e-14) || !(b <= 2.8e+40)) {
		tmp = x * Math.exp((a * -b));
	} else {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -9e-14) or not (b <= 2.8e+40):
		tmp = x * math.exp((a * -b))
	else:
		tmp = x * math.exp((y * (math.log(z) - t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -9e-14) || !(b <= 2.8e+40))
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	else
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -9e-14) || ~((b <= 2.8e+40)))
		tmp = x * exp((a * -b));
	else
		tmp = x * exp((y * (log(z) - t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -9e-14], N[Not[LessEqual[b, 2.8e+40]], $MachinePrecision]], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.9999999999999995e-14 or 2.8000000000000001e40 < b

   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 b around inf 81.7%

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

      1. mul-1-neg 81.7%

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

      2. distribute-rgt-neg-out 81.7%

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

   5. Simplified 81.7%

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

   if -8.9999999999999995e-14 < b < 2.8000000000000001e40

   1. Initial program 93.6%

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

   3. Taylor expanded in y around inf 88.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-14} \lor \neg \left(b \leq 2.8 \cdot 10^{+40}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-71} \lor \neg \left(y \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -5.4e-71) (not (<= y 2e-5)))
   (* x (pow (/ z (exp t)) y))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.4e-71) || !(y <= 2e-5)) {
		tmp = x * pow((z / exp(t)), y);
	} else {
		tmp = x * exp((a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-5.4d-71)) .or. (.not. (y <= 2d-5))) then
        tmp = x * ((z / exp(t)) ** y)
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.4e-71) || !(y <= 2e-5)) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -5.4e-71) or not (y <= 2e-5):
		tmp = x * math.pow((z / math.exp(t)), y)
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -5.4e-71) || !(y <= 2e-5))
		tmp = Float64(x * (Float64(z / exp(t)) ^ 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 <= -5.4e-71) || ~((y <= 2e-5)))
		tmp = x * ((z / exp(t)) ^ y);
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -5.4e-71], N[Not[LessEqual[y, 2e-5]], $MachinePrecision]], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4000000000000003e-71 or 2.00000000000000016e-5 < y

   1. Initial program 95.3%

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

   3. Taylor expanded in y around inf 86.1%

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

      1. pow-exp 82.0%

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

      2. *-un-lft-identity 82.0%

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

      3. pow-exp 86.1%

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

      4. *-commutative 86.1%

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

      5. exp-prod 85.8%

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

      6. exp-diff 85.8%

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

      7. add-exp-log 85.9%

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

   5. Applied egg-rr 85.9%

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

      1. *-lft-identity 85.9%

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

   7. Simplified 85.9%

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

   if -5.4000000000000003e-71 < y < 2.00000000000000016e-5

   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 b around inf 82.6%

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

      1. mul-1-neg 82.6%

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

      2. distribute-rgt-neg-out 82.6%

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

   5. Simplified 82.6%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-71} \lor \neg \left(y \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-13}:\\ \;\;\;\;x \cdot {\left(e^{y}\right)}^{\left(-t\right)}\\ \mathbf{elif}\;y \leq 0.00045:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+186}:\\ \;\;\;\;x \cdot {\left(\frac{z}{t + 1}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3e-13)
   (* x (pow (exp y) (- t)))
   (if (<= y 0.00045)
     (* x (exp (* a (- b))))
     (if (<= y 1.45e+186)
       (* x (pow (/ z (+ t 1.0)) y))
       (* x (exp (* y (- t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3e-13) {
		tmp = x * pow(exp(y), -t);
	} else if (y <= 0.00045) {
		tmp = x * exp((a * -b));
	} else if (y <= 1.45e+186) {
		tmp = x * pow((z / (t + 1.0)), y);
	} else {
		tmp = x * exp((y * -t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-3d-13)) then
        tmp = x * (exp(y) ** -t)
    else if (y <= 0.00045d0) then
        tmp = x * exp((a * -b))
    else if (y <= 1.45d+186) then
        tmp = x * ((z / (t + 1.0d0)) ** y)
    else
        tmp = x * exp((y * -t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3e-13) {
		tmp = x * Math.pow(Math.exp(y), -t);
	} else if (y <= 0.00045) {
		tmp = x * Math.exp((a * -b));
	} else if (y <= 1.45e+186) {
		tmp = x * Math.pow((z / (t + 1.0)), y);
	} else {
		tmp = x * Math.exp((y * -t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3e-13:
		tmp = x * math.pow(math.exp(y), -t)
	elif y <= 0.00045:
		tmp = x * math.exp((a * -b))
	elif y <= 1.45e+186:
		tmp = x * math.pow((z / (t + 1.0)), y)
	else:
		tmp = x * math.exp((y * -t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3e-13)
		tmp = Float64(x * (exp(y) ^ Float64(-t)));
	elseif (y <= 0.00045)
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	elseif (y <= 1.45e+186)
		tmp = Float64(x * (Float64(z / Float64(t + 1.0)) ^ y));
	else
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3e-13)
		tmp = x * (exp(y) ^ -t);
	elseif (y <= 0.00045)
		tmp = x * exp((a * -b));
	elseif (y <= 1.45e+186)
		tmp = x * ((z / (t + 1.0)) ^ y);
	else
		tmp = x * exp((y * -t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3e-13], N[(x * N[Power[N[Exp[y], $MachinePrecision], (-t)], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00045], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e+186], N[(x * N[Power[N[(z / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.99999999999999984e-13

   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. Step-by-step derivation

      1. *-un-lft-identity 94.0%

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

      2. exp-prod 94.0%

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

      3. fma-define 95.5%

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

      4. sub-neg 95.5%

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

      5. sub-neg 95.5%

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

      6. sub-neg 95.5%

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

      7. log1p-undefine 98.5%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 95.5%

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

      10. sqr-neg 95.5%

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

      11. sqrt-unprod 95.5%

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

      12. add-sqr-sqrt 95.5%

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

   4. Applied egg-rr 95.5%

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

   5. Taylor expanded in t around inf 62.4%

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

      1. associate-*r* 62.4%

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

      2. neg-mul-1 62.4%

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

   7. Simplified 62.4%

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

   8. Taylor expanded in t around inf 62.4%

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

      1. mul-1-neg 62.4%

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

      2. *-commutative 62.4%

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

      3. distribute-rgt-neg-in 62.4%

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

      4. exp-prod 78.1%

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

   10. Simplified 78.1%

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

   if -2.99999999999999984e-13 < y < 4.4999999999999999e-4

   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 b around inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. distribute-rgt-neg-out 81.2%

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

   5. Simplified 81.2%

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

   if 4.4999999999999999e-4 < y < 1.45e186

   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 inf 85.9%

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

      1. pow-exp 85.8%

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

      2. *-un-lft-identity 85.8%

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

      3. pow-exp 85.9%

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

      4. *-commutative 85.9%

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

      5. exp-prod 85.9%

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

      6. exp-diff 85.9%

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

      7. add-exp-log 85.9%

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

   5. Applied egg-rr 85.9%

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

      1. *-lft-identity 85.9%

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

   7. Simplified 85.9%

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

   8. Taylor expanded in t around 0 81.2%

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

   if 1.45e186 < 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 t around inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. distribute-lft-neg-out 94.5%

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

      3. *-commutative 94.5%

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

   5. Simplified 94.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-13}:\\ \;\;\;\;x \cdot {\left(e^{y}\right)}^{\left(-t\right)}\\ \mathbf{elif}\;y \leq 0.00045:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+186}:\\ \;\;\;\;x \cdot {\left(\frac{z}{t + 1}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+15}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 0.00033:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+187}:\\ \;\;\;\;x \cdot {\left(\frac{z}{t + 1}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -5.2e+15)
   (* x (pow z y))
   (if (<= y 0.00033)
     (* x (exp (* a (- b))))
     (if (<= y 6.5e+187)
       (* x (pow (/ z (+ t 1.0)) y))
       (* x (exp (* y (- t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5.2e+15) {
		tmp = x * pow(z, y);
	} else if (y <= 0.00033) {
		tmp = x * exp((a * -b));
	} else if (y <= 6.5e+187) {
		tmp = x * pow((z / (t + 1.0)), y);
	} else {
		tmp = x * exp((y * -t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-5.2d+15)) then
        tmp = x * (z ** y)
    else if (y <= 0.00033d0) then
        tmp = x * exp((a * -b))
    else if (y <= 6.5d+187) then
        tmp = x * ((z / (t + 1.0d0)) ** y)
    else
        tmp = x * exp((y * -t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5.2e+15) {
		tmp = x * Math.pow(z, y);
	} else if (y <= 0.00033) {
		tmp = x * Math.exp((a * -b));
	} else if (y <= 6.5e+187) {
		tmp = x * Math.pow((z / (t + 1.0)), y);
	} else {
		tmp = x * Math.exp((y * -t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -5.2e+15:
		tmp = x * math.pow(z, y)
	elif y <= 0.00033:
		tmp = x * math.exp((a * -b))
	elif y <= 6.5e+187:
		tmp = x * math.pow((z / (t + 1.0)), y)
	else:
		tmp = x * math.exp((y * -t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -5.2e+15)
		tmp = Float64(x * (z ^ y));
	elseif (y <= 0.00033)
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	elseif (y <= 6.5e+187)
		tmp = Float64(x * (Float64(z / Float64(t + 1.0)) ^ y));
	else
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -5.2e+15)
		tmp = x * (z ^ y);
	elseif (y <= 0.00033)
		tmp = x * exp((a * -b));
	elseif (y <= 6.5e+187)
		tmp = x * ((z / (t + 1.0)) ^ y);
	else
		tmp = x * exp((y * -t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -5.2e+15], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00033], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+187], N[(x * N[Power[N[(z / N[(t + 1.0), $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.2e15

   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 inf 88.5%

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

   4. Taylor expanded in t around 0 65.5%

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

   if -5.2e15 < y < 3.3e-4

   1. Initial program 95.5%

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

   3. Taylor expanded in b around inf 79.3%

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

      1. mul-1-neg 79.3%

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

      2. distribute-rgt-neg-out 79.3%

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

   5. Simplified 79.3%

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

   if 3.3e-4 < y < 6.49999999999999969e187

   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 inf 85.9%

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

      1. pow-exp 85.8%

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

      2. *-un-lft-identity 85.8%

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

      3. pow-exp 85.9%

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

      4. *-commutative 85.9%

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

      5. exp-prod 85.9%

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

      6. exp-diff 85.9%

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

      7. add-exp-log 85.9%

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

   5. Applied egg-rr 85.9%

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

      1. *-lft-identity 85.9%

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

   7. Simplified 85.9%

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

   8. Taylor expanded in t around 0 81.2%

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

   if 6.49999999999999969e187 < 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 t around inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. distribute-lft-neg-out 94.5%

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

      3. *-commutative 94.5%

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

   5. Simplified 94.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+15}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 0.00033:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+187}:\\ \;\;\;\;x \cdot {\left(\frac{z}{t + 1}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -2.55 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00045:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+185}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -2.55e+16)
     t_1
     (if (<= y 0.00045)
       (* x (exp (* a (- b))))
       (if (<= y 2.7e+185) t_1 (* x (exp (* y (- t)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double tmp;
	if (y <= -2.55e+16) {
		tmp = t_1;
	} else if (y <= 0.00045) {
		tmp = x * exp((a * -b));
	} else if (y <= 2.7e+185) {
		tmp = t_1;
	} else {
		tmp = x * exp((y * -t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (z ** y)
    if (y <= (-2.55d+16)) then
        tmp = t_1
    else if (y <= 0.00045d0) then
        tmp = x * exp((a * -b))
    else if (y <= 2.7d+185) then
        tmp = t_1
    else
        tmp = x * exp((y * -t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.pow(z, y);
	double tmp;
	if (y <= -2.55e+16) {
		tmp = t_1;
	} else if (y <= 0.00045) {
		tmp = x * Math.exp((a * -b));
	} else if (y <= 2.7e+185) {
		tmp = t_1;
	} else {
		tmp = x * Math.exp((y * -t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	tmp = 0
	if y <= -2.55e+16:
		tmp = t_1
	elif y <= 0.00045:
		tmp = x * math.exp((a * -b))
	elif y <= 2.7e+185:
		tmp = t_1
	else:
		tmp = x * math.exp((y * -t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -2.55e+16)
		tmp = t_1;
	elseif (y <= 0.00045)
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	elseif (y <= 2.7e+185)
		tmp = t_1;
	else
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	tmp = 0.0;
	if (y <= -2.55e+16)
		tmp = t_1;
	elseif (y <= 0.00045)
		tmp = x * exp((a * -b));
	elseif (y <= 2.7e+185)
		tmp = t_1;
	else
		tmp = x * exp((y * -t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.55e+16], t$95$1, If[LessEqual[y, 0.00045], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+185], t$95$1, N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.55e16 or 4.4999999999999999e-4 < y < 2.70000000000000007e185

   1. Initial program 95.1%

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

   3. Taylor expanded in y around inf 87.4%

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

   4. Taylor expanded in t around 0 71.8%

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

   if -2.55e16 < y < 4.4999999999999999e-4

   1. Initial program 95.5%

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

   3. Taylor expanded in b around inf 79.3%

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

      1. mul-1-neg 79.3%

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

      2. distribute-rgt-neg-out 79.3%

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

   5. Simplified 79.3%

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

   if 2.70000000000000007e185 < 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 t around inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. distribute-lft-neg-out 94.5%

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

      3. *-commutative 94.5%

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

   5. Simplified 94.5%

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

Alternative 10: 68.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-13} \lor \neg \left(b \leq 1.22 \cdot 10^{-98}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -1.25e-13) (not (<= b 1.22e-98)))
   (* x (exp (* a (- b))))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -1.25e-13) || !(b <= 1.22e-98)) {
		tmp = x * exp((a * -b));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.25d-13)) .or. (.not. (b <= 1.22d-98))) then
        tmp = x * exp((a * -b))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -1.25e-13) || !(b <= 1.22e-98)) {
		tmp = x * Math.exp((a * -b));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -1.25e-13) or not (b <= 1.22e-98):
		tmp = x * math.exp((a * -b))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -1.25e-13) || !(b <= 1.22e-98))
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -1.25e-13) || ~((b <= 1.22e-98)))
		tmp = x * exp((a * -b));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -1.25e-13], N[Not[LessEqual[b, 1.22e-98]], $MachinePrecision]], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.24999999999999997e-13 or 1.2200000000000001e-98 < b

   1. Initial program 97.9%

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

   3. Taylor expanded in b around inf 78.8%

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

      1. mul-1-neg 78.8%

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

      2. distribute-rgt-neg-out 78.8%

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

   5. Simplified 78.8%

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

   if -1.24999999999999997e-13 < b < 1.2200000000000001e-98

   1. Initial program 92.6%

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

   3. Taylor expanded in y around inf 90.1%

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

   4. Taylor expanded in t around 0 69.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-13} \lor \neg \left(b \leq 1.22 \cdot 10^{-98}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-18} \lor \neg \left(y \leq 1.65 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x + b \cdot \left(a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot b\right)\right) - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.7e-18) (not (<= y 1.65e-36)))
   (* x (pow z y))
   (+ x (* b (* a (- (* 0.5 (* a (* x b))) x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.7e-18) || !(y <= 1.65e-36)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - 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 <= (-1.7d-18)) .or. (.not. (y <= 1.65d-36))) then
        tmp = x * (z ** y)
    else
        tmp = x + (b * (a * ((0.5d0 * (a * (x * b))) - 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 <= -1.7e-18) || !(y <= 1.65e-36)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.7e-18) or not (y <= 1.65e-36):
		tmp = x * math.pow(z, y)
	else:
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.7e-18) || !(y <= 1.65e-36))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x + Float64(b * Float64(a * Float64(Float64(0.5 * Float64(a * Float64(x * b))) - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.7e-18) || ~((y <= 1.65e-36)))
		tmp = x * (z ^ y);
	else
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.7e-18], N[Not[LessEqual[y, 1.65e-36]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x + N[(b * N[(a * N[(N[(0.5 * N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.70000000000000001e-18 or 1.64999999999999995e-36 < y

   1. Initial program 95.6%

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

   3. Taylor expanded in y around inf 84.3%

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

   4. Taylor expanded in t around 0 64.4%

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

   if -1.70000000000000001e-18 < y < 1.64999999999999995e-36

   1. Initial program 95.7%

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

   3. Taylor expanded in b around inf 81.0%

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

      1. mul-1-neg 81.0%

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

      2. distribute-rgt-neg-out 81.0%

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

   5. Simplified 81.0%

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

   6. Taylor expanded in b around 0 51.2%

      \[\leadsto \color{blue}{x + b \cdot \left(-1 \cdot \left(a \cdot x\right) + 0.5 \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 51.2%

         \[\leadsto x + b \cdot \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) + -1 \cdot \left(a \cdot x\right)\right)} \]

      2. mul-1-neg 51.2%

         \[\leadsto x + b \cdot \left(0.5 \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) + \color{blue}{\left(-a \cdot x\right)}\right) \]

      3. unsub-neg 51.2%

         \[\leadsto x + b \cdot \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) - a \cdot x\right)} \]

      4. associate-*r* 54.5%

         \[\leadsto x + b \cdot \left(0.5 \cdot \color{blue}{\left(\left({a}^{2} \cdot b\right) \cdot x\right)} - a \cdot x\right) \]

      5. *-commutative 54.5%

         \[\leadsto x + b \cdot \left(0.5 \cdot \left(\color{blue}{\left(b \cdot {a}^{2}\right)} \cdot x\right) - a \cdot x\right) \]

   8. Simplified 54.5%

      \[\leadsto \color{blue}{x + b \cdot \left(0.5 \cdot \left(\left(b \cdot {a}^{2}\right) \cdot x\right) - a \cdot x\right)} \]

   9. Taylor expanded in a around 0 52.3%

      \[\leadsto x + b \cdot \color{blue}{\left(a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot x\right)\right) - x\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 52.3%

          \[\leadsto x + b \cdot \left(a \cdot \left(0.5 \cdot \left(a \cdot \color{blue}{\left(x \cdot b\right)}\right) - x\right)\right) \]

   11. Simplified 52.3%

       \[\leadsto x + b \cdot \color{blue}{\left(a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot b\right)\right) - x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-18} \lor \neg \left(y \leq 1.65 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x + b \cdot \left(a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot b\right)\right) - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.0% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-47}:\\ \;\;\;\;b \cdot \left(\frac{x}{b} - x \cdot a\right)\\ \mathbf{elif}\;y \leq 0.036:\\ \;\;\;\;x + b \cdot \left(a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot b\right)\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.6e-47)
   (* b (- (/ x b) (* x a)))
   (if (<= y 0.036)
     (+ x (* b (* a (- (* 0.5 (* a (* x b))) x))))
     (* x (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.6e-47) {
		tmp = b * ((x / b) - (x * a));
	} else if (y <= 0.036) {
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.6d-47)) then
        tmp = b * ((x / b) - (x * a))
    else if (y <= 0.036d0) then
        tmp = x + (b * (a * ((0.5d0 * (a * (x * b))) - x)))
    else
        tmp = x * (a * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.6e-47) {
		tmp = b * ((x / b) - (x * a));
	} else if (y <= 0.036) {
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.6e-47:
		tmp = b * ((x / b) - (x * a))
	elif y <= 0.036:
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)))
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.6e-47)
		tmp = Float64(b * Float64(Float64(x / b) - Float64(x * a)));
	elseif (y <= 0.036)
		tmp = Float64(x + Float64(b * Float64(a * Float64(Float64(0.5 * Float64(a * Float64(x * b))) - x))));
	else
		tmp = Float64(x * Float64(a * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.6e-47)
		tmp = b * ((x / b) - (x * a));
	elseif (y <= 0.036)
		tmp = x + (b * (a * ((0.5 * (a * (x * b))) - x)));
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.6e-47], N[(b * N[(N[(x / b), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.036], N[(x + N[(b * N[(a * N[(N[(0.5 * N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6e-47

   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 b around inf 44.5%

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

      1. mul-1-neg 44.5%

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

      2. distribute-rgt-neg-out 44.5%

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

   5. Simplified 44.5%

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

   6. Taylor expanded in a around 0 15.1%

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

      1. neg-mul-1 15.1%

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

      2. unsub-neg 15.1%

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

   8. Simplified 15.1%

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

   9. Taylor expanded in b around inf 24.9%

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

       1. +-commutative 24.9%

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

       2. mul-1-neg 24.9%

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

       3. *-commutative 24.9%

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

       4. unsub-neg 24.9%

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

   11. Simplified 24.9%

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

   if -2.6e-47 < y < 0.0359999999999999973

   1. Initial program 96.3%

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

   3. Taylor expanded in b around inf 80.6%

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

      1. mul-1-neg 80.6%

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

      2. distribute-rgt-neg-out 80.6%

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

   5. Simplified 80.6%

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

   6. Taylor expanded in b around 0 49.0%

      \[\leadsto \color{blue}{x + b \cdot \left(-1 \cdot \left(a \cdot x\right) + 0.5 \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 49.0%

         \[\leadsto x + b \cdot \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) + -1 \cdot \left(a \cdot x\right)\right)} \]

      2. mul-1-neg 49.0%

         \[\leadsto x + b \cdot \left(0.5 \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) + \color{blue}{\left(-a \cdot x\right)}\right) \]

      3. unsub-neg 49.0%

         \[\leadsto x + b \cdot \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) - a \cdot x\right)} \]

      4. associate-*r* 51.5%

         \[\leadsto x + b \cdot \left(0.5 \cdot \color{blue}{\left(\left({a}^{2} \cdot b\right) \cdot x\right)} - a \cdot x\right) \]

      5. *-commutative 51.5%

         \[\leadsto x + b \cdot \left(0.5 \cdot \left(\color{blue}{\left(b \cdot {a}^{2}\right)} \cdot x\right) - a \cdot x\right) \]

   8. Simplified 51.5%

      \[\leadsto \color{blue}{x + b \cdot \left(0.5 \cdot \left(\left(b \cdot {a}^{2}\right) \cdot x\right) - a \cdot x\right)} \]

   9. Taylor expanded in a around 0 50.1%

      \[\leadsto x + b \cdot \color{blue}{\left(a \cdot \left(0.5 \cdot \left(a \cdot \left(b \cdot x\right)\right) - x\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 50.1%

          \[\leadsto x + b \cdot \left(a \cdot \left(0.5 \cdot \left(a \cdot \color{blue}{\left(x \cdot b\right)}\right) - x\right)\right) \]

   11. Simplified 50.1%

       \[\leadsto x + b \cdot \color{blue}{\left(a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot b\right)\right) - x\right)\right)} \]

   if 0.0359999999999999973 < y

   1. Initial program 96.6%

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

   3. Taylor expanded in b around inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. distribute-rgt-neg-out 37.7%

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

   5. Simplified 37.7%

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

   6. Taylor expanded in a around 0 12.8%

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

      1. neg-mul-1 12.8%

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

      2. unsub-neg 12.8%

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

   8. Simplified 12.8%

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

   9. Taylor expanded in a around inf 29.0%

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

       1. mul-1-neg 29.0%

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

       2. *-commutative 29.0%

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

       3. *-commutative 29.0%

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

       4. distribute-rgt-neg-in 29.0%

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

       5. associate-*r* 32.3%

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

   11. Simplified 32.3%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-47}:\\ \;\;\;\;b \cdot \left(\frac{x}{b} - x \cdot a\right)\\ \mathbf{elif}\;y \leq 0.036:\\ \;\;\;\;x + b \cdot \left(a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot b\right)\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 32.6% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-201}:\\ \;\;\;\;b \cdot \left(\frac{x}{b} - x \cdot a\right)\\ \mathbf{elif}\;y \leq 0.0032:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4e-201)
   (* b (- (/ x b) (* x a)))
   (if (<= y 0.0032) (- x (* x (* a b))) (* x (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4e-201) {
		tmp = b * ((x / b) - (x * a));
	} else if (y <= 0.0032) {
		tmp = x - (x * (a * b));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-4d-201)) then
        tmp = b * ((x / b) - (x * a))
    else if (y <= 0.0032d0) then
        tmp = x - (x * (a * b))
    else
        tmp = x * (a * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4e-201) {
		tmp = b * ((x / b) - (x * a));
	} else if (y <= 0.0032) {
		tmp = x - (x * (a * b));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4e-201:
		tmp = b * ((x / b) - (x * a))
	elif y <= 0.0032:
		tmp = x - (x * (a * b))
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4e-201)
		tmp = Float64(b * Float64(Float64(x / b) - Float64(x * a)));
	elseif (y <= 0.0032)
		tmp = Float64(x - Float64(x * Float64(a * b)));
	else
		tmp = Float64(x * Float64(a * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4e-201)
		tmp = b * ((x / b) - (x * a));
	elseif (y <= 0.0032)
		tmp = x - (x * (a * b));
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4e-201], N[(b * N[(N[(x / b), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0032], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.99999999999999978e-201

   1. Initial program 94.6%

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

   3. Taylor expanded in b around inf 52.5%

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

      1. mul-1-neg 52.5%

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

      2. distribute-rgt-neg-out 52.5%

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

   5. Simplified 52.5%

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

   6. Taylor expanded in a around 0 21.0%

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

      1. neg-mul-1 21.0%

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

      2. unsub-neg 21.0%

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

   8. Simplified 21.0%

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

   9. Taylor expanded in b around inf 28.1%

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

       1. +-commutative 28.1%

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

       2. mul-1-neg 28.1%

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

       3. *-commutative 28.1%

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

       4. unsub-neg 28.1%

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

   11. Simplified 28.1%

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

   if -3.99999999999999978e-201 < y < 0.00320000000000000015

   1. Initial program 96.1%

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

   3. Taylor expanded in b around inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. distribute-rgt-neg-out 82.9%

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

   5. Simplified 82.9%

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

   6. Taylor expanded in a around 0 48.3%

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

      1. mul-1-neg 48.3%

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

      2. unsub-neg 48.3%

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

      3. associate-*r* 50.2%

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

   8. Simplified 50.2%

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

   if 0.00320000000000000015 < y

   1. Initial program 96.6%

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

   3. Taylor expanded in b around inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. distribute-rgt-neg-out 37.7%

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

   5. Simplified 37.7%

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

   6. Taylor expanded in a around 0 12.8%

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

      1. neg-mul-1 12.8%

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

      2. unsub-neg 12.8%

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

   8. Simplified 12.8%

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

   9. Taylor expanded in a around inf 29.0%

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

       1. mul-1-neg 29.0%

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

       2. *-commutative 29.0%

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

       3. *-commutative 29.0%

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

       4. distribute-rgt-neg-in 29.0%

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

       5. associate-*r* 32.3%

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

   11. Simplified 32.3%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-201}:\\ \;\;\;\;b \cdot \left(\frac{x}{b} - x \cdot a\right)\\ \mathbf{elif}\;y \leq 0.0032:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.5% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-11}:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot b\right)\\ \mathbf{elif}\;y \leq 0.102:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.8e-11)
   (* a (- (/ x a) (* x b)))
   (if (<= y 0.102) (- x (* x (* a b))) (* x (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.8e-11) {
		tmp = a * ((x / a) - (x * b));
	} else if (y <= 0.102) {
		tmp = x - (x * (a * b));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-1.8d-11)) then
        tmp = a * ((x / a) - (x * b))
    else if (y <= 0.102d0) then
        tmp = x - (x * (a * b))
    else
        tmp = x * (a * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.8e-11) {
		tmp = a * ((x / a) - (x * b));
	} else if (y <= 0.102) {
		tmp = x - (x * (a * b));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.8e-11:
		tmp = a * ((x / a) - (x * b))
	elif y <= 0.102:
		tmp = x - (x * (a * b))
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.8e-11)
		tmp = Float64(a * Float64(Float64(x / a) - Float64(x * b)));
	elseif (y <= 0.102)
		tmp = Float64(x - Float64(x * Float64(a * b)));
	else
		tmp = Float64(x * Float64(a * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.8e-11)
		tmp = a * ((x / a) - (x * b));
	elseif (y <= 0.102)
		tmp = x - (x * (a * b));
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.8e-11], N[(a * N[(N[(x / a), $MachinePrecision] - N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.102], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.79999999999999992e-11

   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 b around inf 39.4%

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

      1. mul-1-neg 39.4%

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

      2. distribute-rgt-neg-out 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in a around 0 8.9%

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

      1. neg-mul-1 8.9%

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

      2. unsub-neg 8.9%

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

   8. Simplified 8.9%

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

   9. Taylor expanded in a around inf 17.3%

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

       1. +-commutative 17.3%

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

       2. mul-1-neg 17.3%

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

       3. *-commutative 17.3%

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

       4. unsub-neg 17.3%

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

   11. Simplified 17.3%

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

   if -1.79999999999999992e-11 < y < 0.101999999999999993

   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 b around inf 80.8%

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

      1. mul-1-neg 80.8%

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

      2. distribute-rgt-neg-out 80.8%

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

   5. Simplified 80.8%

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

   6. Taylor expanded in a around 0 45.1%

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

      1. mul-1-neg 45.1%

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

      2. unsub-neg 45.1%

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

      3. associate-*r* 47.9%

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

   8. Simplified 47.9%

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

   if 0.101999999999999993 < y

   1. Initial program 96.6%

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

   3. Taylor expanded in b around inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. distribute-rgt-neg-out 37.7%

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

   5. Simplified 37.7%

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

   6. Taylor expanded in a around 0 12.8%

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

      1. neg-mul-1 12.8%

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

      2. unsub-neg 12.8%

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

   8. Simplified 12.8%

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

   9. Taylor expanded in a around inf 29.0%

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

       1. mul-1-neg 29.0%

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

       2. *-commutative 29.0%

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

       3. *-commutative 29.0%

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

       4. distribute-rgt-neg-in 29.0%

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

       5. associate-*r* 32.3%

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

   11. Simplified 32.3%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-11}:\\ \;\;\;\;a \cdot \left(\frac{x}{a} - x \cdot b\right)\\ \mathbf{elif}\;y \leq 0.102:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 22.5% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{+58} \lor \neg \left(b \leq 5.5 \cdot 10^{-167}\right):\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -1.12e+58) (not (<= b 5.5e-167))) (* x (* a (- b))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -1.12e+58) || !(b <= 5.5e-167)) {
		tmp = x * (a * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.12d+58)) .or. (.not. (b <= 5.5d-167))) then
        tmp = x * (a * -b)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -1.12e+58) || !(b <= 5.5e-167)) {
		tmp = x * (a * -b);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -1.12e+58) or not (b <= 5.5e-167):
		tmp = x * (a * -b)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -1.12e+58) || !(b <= 5.5e-167))
		tmp = Float64(x * Float64(a * Float64(-b)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -1.12e+58) || ~((b <= 5.5e-167)))
		tmp = x * (a * -b);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -1.12e+58], N[Not[LessEqual[b, 5.5e-167]], $MachinePrecision]], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if b < -1.12e58 or 5.5000000000000003e-167 < b

   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 b around inf 72.5%

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

      1. mul-1-neg 72.5%

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

      2. distribute-rgt-neg-out 72.5%

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

   5. Simplified 72.5%

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

   6. Taylor expanded in a around 0 28.3%

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

      1. neg-mul-1 28.3%

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

      2. unsub-neg 28.3%

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

   8. Simplified 28.3%

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

   9. Taylor expanded in a around inf 24.7%

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

       1. mul-1-neg 24.7%

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

       2. *-commutative 24.7%

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

       3. *-commutative 24.7%

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

       4. distribute-rgt-neg-in 24.7%

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

       5. associate-*r* 26.8%

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

   11. Simplified 26.8%

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

   if -1.12e58 < b < 5.5000000000000003e-167

   1. Initial program 93.2%

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

   3. Taylor expanded in b around inf 45.7%

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

      1. mul-1-neg 45.7%

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

      2. distribute-rgt-neg-out 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in a around 0 30.9%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{+58} \lor \neg \left(b \leq 5.5 \cdot 10^{-167}\right):\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 22.6% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;b \leq 3.15 \cdot 10^{-167}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -3.2e+153)
   (* a (* x (- b)))
   (if (<= b 3.15e-167) x (* x (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.2e+153) {
		tmp = a * (x * -b);
	} else if (b <= 3.15e-167) {
		tmp = x;
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-3.2d+153)) then
        tmp = a * (x * -b)
    else if (b <= 3.15d-167) then
        tmp = x
    else
        tmp = x * (a * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.2e+153) {
		tmp = a * (x * -b);
	} else if (b <= 3.15e-167) {
		tmp = x;
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -3.2e+153:
		tmp = a * (x * -b)
	elif b <= 3.15e-167:
		tmp = x
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -3.2e+153)
		tmp = Float64(a * Float64(x * Float64(-b)));
	elseif (b <= 3.15e-167)
		tmp = x;
	else
		tmp = Float64(x * Float64(a * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -3.2e+153)
		tmp = a * (x * -b);
	elseif (b <= 3.15e-167)
		tmp = x;
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -3.2e+153], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.15e-167], x, N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.2000000000000001e153

   1. Initial program 96.7%

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

   3. Taylor expanded in b around inf 74.2%

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

      1. mul-1-neg 74.2%

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

      2. distribute-rgt-neg-out 74.2%

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

   5. Simplified 74.2%

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

   6. Taylor expanded in a around 0 32.2%

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

      1. neg-mul-1 32.2%

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

      2. unsub-neg 32.2%

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

   8. Simplified 32.2%

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

   9. Taylor expanded in a around inf 35.1%

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

   if -3.2000000000000001e153 < b < 3.1500000000000001e-167

   1. Initial program 94.1%

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

   3. Taylor expanded in b around inf 50.7%

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

      1. mul-1-neg 50.7%

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

      2. distribute-rgt-neg-out 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in a around 0 28.0%

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

   if 3.1500000000000001e-167 < b

   1. Initial program 97.7%

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

   3. Taylor expanded in b around inf 69.6%

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

      1. mul-1-neg 69.6%

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

      2. distribute-rgt-neg-out 69.6%

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

   5. Simplified 69.6%

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

   6. Taylor expanded in a around 0 29.0%

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

      1. neg-mul-1 29.0%

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

      2. unsub-neg 29.0%

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

   8. Simplified 29.0%

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

   9. Taylor expanded in a around inf 24.6%

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

       1. mul-1-neg 24.6%

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

       2. *-commutative 24.6%

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

       3. *-commutative 24.6%

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

       4. distribute-rgt-neg-in 24.6%

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

       5. associate-*r* 28.9%

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

   11. Simplified 28.9%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;b \leq 3.15 \cdot 10^{-167}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 31.7% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0005:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 0.0005) (- x (* x (* a b))) (* x (* a (- b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 0.0005) {
		tmp = x - (x * (a * b));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= 0.0005d0) then
        tmp = x - (x * (a * b))
    else
        tmp = x * (a * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 0.0005) {
		tmp = x - (x * (a * b));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 0.0005:
		tmp = x - (x * (a * b))
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 0.0005)
		tmp = Float64(x - Float64(x * Float64(a * b)));
	else
		tmp = Float64(x * Float64(a * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 0.0005)
		tmp = x - (x * (a * b));
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 0.0005], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000001e-4

   1. Initial program 95.3%

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

   3. Taylor expanded in b around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. distribute-rgt-neg-out 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in a around 0 33.2%

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

      1. mul-1-neg 33.2%

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

      2. unsub-neg 33.2%

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

      3. associate-*r* 34.6%

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

   8. Simplified 34.6%

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

   if 5.0000000000000001e-4 < y

   1. Initial program 96.6%

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

   3. Taylor expanded in b around inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. distribute-rgt-neg-out 37.7%

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

   5. Simplified 37.7%

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

   6. Taylor expanded in a around 0 12.8%

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

      1. neg-mul-1 12.8%

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

      2. unsub-neg 12.8%

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

   8. Simplified 12.8%

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

   9. Taylor expanded in a around inf 29.0%

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

       1. mul-1-neg 29.0%

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

       2. *-commutative 29.0%

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

       3. *-commutative 29.0%

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

       4. distribute-rgt-neg-in 29.0%

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

       5. associate-*r* 32.3%

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

   11. Simplified 32.3%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0005:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 31.7% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.023:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y 0.023) (* x (- 1.0 (* a b))) (* x (* a (- b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 0.023) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= 0.023d0) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = x * (a * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= 0.023) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = x * (a * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= 0.023:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = x * (a * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= 0.023)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(x * Float64(a * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= 0.023)
		tmp = x * (1.0 - (a * b));
	else
		tmp = x * (a * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, 0.023], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.023

   1. Initial program 95.3%

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

   3. Taylor expanded in b around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. distribute-rgt-neg-out 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in a around 0 34.6%

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

      1. neg-mul-1 34.6%

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

      2. unsub-neg 34.6%

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

   8. Simplified 34.6%

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

   if 0.023 < y

   1. Initial program 96.6%

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

   3. Taylor expanded in b around inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. distribute-rgt-neg-out 37.7%

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

   5. Simplified 37.7%

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

   6. Taylor expanded in a around 0 12.8%

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

      1. neg-mul-1 12.8%

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

      2. unsub-neg 12.8%

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

   8. Simplified 12.8%

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

   9. Taylor expanded in a around inf 29.0%

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

       1. mul-1-neg 29.0%

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

       2. *-commutative 29.0%

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

       3. *-commutative 29.0%

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

       4. distribute-rgt-neg-in 29.0%

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

       5. associate-*r* 32.3%

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

   11. Simplified 32.3%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.023:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 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 95.6%

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

3. Taylor expanded in b around inf 60.0%

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

   1. mul-1-neg 60.0%

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

   2. distribute-rgt-neg-out 60.0%

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

5. Simplified 60.0%

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

6. Taylor expanded in a around 0 19.2%

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

Reproduce

herbie shell --seed 2024137 
(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))))))