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

Percentage Accurate: 96.4% → 99.6%

Time: 13.8s

Alternatives: 13

Speedup: 1.0×

• 42.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% 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 95.7%

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

   1. fma-define 96.2%

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

   2. sub-neg 96.2%

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

   3. log1p-define 98.8%

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

3. Simplified 98.8%

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

Alternative 2: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

Alternative 3: 86.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+27} \lor \neg \left(y \leq 7.8 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.2e+27) (not (<= y 7.8e-11)))
   (* x (pow (/ z (exp t)) y))
   (* x (exp (* a (- (log1p (- z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.2e+27) || !(y <= 7.8e-11)) {
		tmp = x * pow((z / exp(t)), y);
	} else {
		tmp = x * exp((a * (log1p(-z) - b)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.2e+27) || !(y <= 7.8e-11)) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else {
		tmp = x * Math.exp((a * (Math.log1p(-z) - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.2e+27) or not (y <= 7.8e-11):
		tmp = x * math.pow((z / math.exp(t)), y)
	else:
		tmp = x * math.exp((a * (math.log1p(-z) - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.2e+27) || !(y <= 7.8e-11))
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(log1p(Float64(-z)) - b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.2e+27], N[Not[LessEqual[y, 7.8e-11]], $MachinePrecision]], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999999e27 or 7.80000000000000021e-11 < y

   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 a around 0 89.4%

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

      1. *-commutative 89.4%

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

      2. exp-prod 89.4%

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

      3. exp-diff 89.4%

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

      4. rem-exp-log 89.4%

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

   5. Simplified 89.4%

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

   if -1.19999999999999999e27 < y < 7.80000000000000021e-11

   1. Initial program 95.8%

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

   3. Taylor expanded in y around 0 81.6%

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

      1. sub-neg 81.6%

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

      2. log1p-define 85.8%

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

   5. Simplified 85.8%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+27} \lor \neg \left(y \leq 7.8 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-25} \lor \neg \left(y \leq 2.5 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -8.5e-25) (not (<= y 2.5e-11)))
   (* 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 <= -8.5e-25) || !(y <= 2.5e-11)) {
		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 <= (-8.5d-25)) .or. (.not. (y <= 2.5d-11))) 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 <= -8.5e-25) || !(y <= 2.5e-11)) {
		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 <= -8.5e-25) or not (y <= 2.5e-11):
		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 <= -8.5e-25) || !(y <= 2.5e-11))
		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 <= -8.5e-25) || ~((y <= 2.5e-11)))
		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, -8.5e-25], N[Not[LessEqual[y, 2.5e-11]], $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 < -8.49999999999999981e-25 or 2.50000000000000009e-11 < 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 a around 0 88.0%

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

      1. *-commutative 88.0%

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

      2. exp-prod 88.0%

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

      3. exp-diff 88.0%

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

      4. rem-exp-log 88.0%

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

   5. Simplified 88.0%

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

   if -8.49999999999999981e-25 < y < 2.50000000000000009e-11

   1. Initial program 96.4%

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

   3. Taylor expanded in b around inf 83.0%

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

      1. associate-*r* 83.0%

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

      2. mul-1-neg 83.0%

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

   5. Simplified 83.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-25} \lor \neg \left(y \leq 2.5 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+229}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{+15} \lor \neg \left(y \leq 1.9 \cdot 10^{-10}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -5.8e+229)
   (* x (exp (* t (- y))))
   (if (or (<= y -2.8e+15) (not (<= y 1.9e-10)))
     (* x (pow z y))
     (* x (exp (* a (- b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5.8e+229) {
		tmp = x * exp((t * -y));
	} else if ((y <= -2.8e+15) || !(y <= 1.9e-10)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-5.8d+229)) then
        tmp = x * exp((t * -y))
    else if ((y <= (-2.8d+15)) .or. (.not. (y <= 1.9d-10))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5.8e+229) {
		tmp = x * Math.exp((t * -y));
	} else if ((y <= -2.8e+15) || !(y <= 1.9e-10)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -5.8e+229:
		tmp = x * math.exp((t * -y))
	elif (y <= -2.8e+15) or not (y <= 1.9e-10):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -5.8e+229)
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	elseif ((y <= -2.8e+15) || !(y <= 1.9e-10))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -5.8e+229)
		tmp = x * exp((t * -y));
	elseif ((y <= -2.8e+15) || ~((y <= 1.9e-10)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -5.8e+229], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -2.8e+15], N[Not[LessEqual[y, 1.9e-10]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.79999999999999963e229

   1. Initial program 93.3%

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

   3. Taylor expanded in t around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. distribute-lft-neg-out 86.9%

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

      3. *-commutative 86.9%

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

   5. Simplified 86.9%

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

   if -5.79999999999999963e229 < y < -2.8e15 or 1.8999999999999999e-10 < 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 a around 0 86.9%

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

      1. *-commutative 86.9%

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

      2. exp-prod 86.9%

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

      3. exp-diff 86.9%

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

      4. rem-exp-log 86.9%

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

   5. Simplified 86.9%

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

   6. Taylor expanded in t around 0 71.6%

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

   if -2.8e15 < y < 1.8999999999999999e-10

   1. Initial program 96.5%

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

   3. Taylor expanded in b around inf 81.9%

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

      1. associate-*r* 81.9%

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

      2. mul-1-neg 81.9%

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

   5. Simplified 81.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+229}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{+15} \lor \neg \left(y \leq 1.9 \cdot 10^{-10}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -560 \lor \neg \left(t \leq 5 \cdot 10^{-22}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -560.0) (not (<= t 5e-22)))
   (* x (exp (* t (- y))))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -560.0) || !(t <= 5e-22)) {
		tmp = x * exp((t * -y));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-560.0d0)) .or. (.not. (t <= 5d-22))) then
        tmp = x * exp((t * -y))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -560.0) || !(t <= 5e-22)) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -560.0) or not (t <= 5e-22):
		tmp = x * math.exp((t * -y))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -560.0) || !(t <= 5e-22))
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -560.0) || ~((t <= 5e-22)))
		tmp = x * exp((t * -y));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -560.0], N[Not[LessEqual[t, 5e-22]], $MachinePrecision]], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -560 or 4.99999999999999954e-22 < t

   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 t around inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. distribute-lft-neg-out 81.8%

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

      3. *-commutative 81.8%

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

   5. Simplified 81.8%

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

   if -560 < t < 4.99999999999999954e-22

   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 a around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. exp-prod 67.8%

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

      3. exp-diff 67.8%

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

      4. rem-exp-log 67.8%

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

   5. Simplified 67.8%

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

   6. Taylor expanded in t around 0 67.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -560 \lor \neg \left(t \leq 5 \cdot 10^{-22}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+15} \lor \neg \left(y \leq 7.8 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7.8e+15) (not (<= y 7.8e-11)))
   (* x (pow z y))
   (* x (- 1.0 (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7.8e+15) || !(y <= 7.8e-11)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * (1.0 - (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 <= (-7.8d+15)) .or. (.not. (y <= 7.8d-11))) then
        tmp = x * (z ** y)
    else
        tmp = x * (1.0d0 - (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 <= -7.8e+15) || !(y <= 7.8e-11)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -7.8e+15) or not (y <= 7.8e-11):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * (1.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -7.8e+15) || !(y <= 7.8e-11))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -7.8e+15) || ~((y <= 7.8e-11)))
		tmp = x * (z ^ y);
	else
		tmp = x * (1.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -7.8e+15], N[Not[LessEqual[y, 7.8e-11]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8e15 or 7.80000000000000021e-11 < y

   1. Initial program 95.8%

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

   3. Taylor expanded in a around 0 88.9%

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

      1. *-commutative 88.9%

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

      2. exp-prod 88.9%

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

      3. exp-diff 88.9%

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

      4. rem-exp-log 88.9%

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

   5. Simplified 88.9%

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

   6. Taylor expanded in t around 0 69.5%

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

   if -7.8e15 < y < 7.80000000000000021e-11

   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 81.2%

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

      1. associate-*r* 81.2%

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

      2. mul-1-neg 81.2%

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

   5. Simplified 81.2%

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

   6. Taylor expanded in a around 0 51.4%

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

      1. mul-1-neg 51.4%

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

      2. unsub-neg 51.4%

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

   8. Simplified 51.4%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+15} \lor \neg \left(y \leq 7.8 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 27.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-68}:\\ \;\;\;\;a \cdot \frac{x}{a}\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-130} \lor \neg \left(y \leq 2.5 \cdot 10^{-55}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (* a (- b)))))
   (if (<= y -1.55e+22)
     t_1
     (if (<= y -8.2e-68)
       (* a (/ x a))
       (if (or (<= y -7.8e-130) (not (<= y 2.5e-55))) t_1 x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (a * -b);
	double tmp;
	if (y <= -1.55e+22) {
		tmp = t_1;
	} else if (y <= -8.2e-68) {
		tmp = a * (x / a);
	} else if ((y <= -7.8e-130) || !(y <= 2.5e-55)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = x * (a * -b)
    if (y <= (-1.55d+22)) then
        tmp = t_1
    else if (y <= (-8.2d-68)) then
        tmp = a * (x / a)
    else if ((y <= (-7.8d-130)) .or. (.not. (y <= 2.5d-55))) then
        tmp = t_1
    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 t_1 = x * (a * -b);
	double tmp;
	if (y <= -1.55e+22) {
		tmp = t_1;
	} else if (y <= -8.2e-68) {
		tmp = a * (x / a);
	} else if ((y <= -7.8e-130) || !(y <= 2.5e-55)) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * (a * -b)
	tmp = 0
	if y <= -1.55e+22:
		tmp = t_1
	elif y <= -8.2e-68:
		tmp = a * (x / a)
	elif (y <= -7.8e-130) or not (y <= 2.5e-55):
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(a * Float64(-b)))
	tmp = 0.0
	if (y <= -1.55e+22)
		tmp = t_1;
	elseif (y <= -8.2e-68)
		tmp = Float64(a * Float64(x / a));
	elseif ((y <= -7.8e-130) || !(y <= 2.5e-55))
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (a * -b);
	tmp = 0.0;
	if (y <= -1.55e+22)
		tmp = t_1;
	elseif (y <= -8.2e-68)
		tmp = a * (x / a);
	elseif ((y <= -7.8e-130) || ~((y <= 2.5e-55)))
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.55e+22], t$95$1, If[LessEqual[y, -8.2e-68], N[(a * N[(x / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -7.8e-130], N[Not[LessEqual[y, 2.5e-55]], $MachinePrecision]], t$95$1, x]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5500000000000001e22 or -8.20000000000000042e-68 < y < -7.8000000000000002e-130 or 2.5000000000000001e-55 < y

   1. Initial program 96.2%

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

   3. Taylor expanded in b around inf 42.7%

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

      1. associate-*r* 42.7%

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

      2. mul-1-neg 42.7%

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

   5. Simplified 42.7%

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

   6. Taylor expanded in a around 0 14.7%

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

      1. mul-1-neg 14.7%

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

      2. unsub-neg 14.7%

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

   8. Simplified 14.7%

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

   9. Taylor expanded in a around inf 21.5%

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

       1. associate-*r* 25.4%

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

       2. associate-*r* 25.4%

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

       3. *-commutative 25.4%

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

       4. associate-*r* 25.4%

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

       5. mul-1-neg 25.4%

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

   11. Simplified 25.4%

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

   if -1.5500000000000001e22 < y < -8.20000000000000042e-68

   1. Initial program 93.4%

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

   3. Taylor expanded in b around inf 67.4%

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

      1. associate-*r* 67.4%

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

      2. mul-1-neg 67.4%

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

   5. Simplified 67.4%

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

   6. Taylor expanded in a around 0 34.9%

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

      1. mul-1-neg 34.9%

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

      2. unsub-neg 34.9%

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

   8. Simplified 34.9%

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

   9. Taylor expanded in a around inf 34.8%

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

       1. +-commutative 34.8%

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

       2. mul-1-neg 34.8%

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

       3. sub-neg 34.8%

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

       4. *-commutative 34.8%

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

   11. Simplified 34.8%

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

   12. Taylor expanded in a around 0 48.2%

       \[\leadsto a \cdot \color{blue}{\frac{x}{a}} \]

   if -7.8000000000000002e-130 < y < 2.5000000000000001e-55

   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 b around inf 82.7%

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

      1. associate-*r* 82.7%

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

      2. mul-1-neg 82.7%

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

   5. Simplified 82.7%

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

   6. Taylor expanded in a around 0 46.1%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-68}:\\ \;\;\;\;a \cdot \frac{x}{a}\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-130} \lor \neg \left(y \leq 2.5 \cdot 10^{-55}\right):\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.1% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.06 \cdot 10^{-63}:\\ \;\;\;\;a \cdot \frac{x}{a}\\ \mathbf{elif}\;y \leq 0.00038:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (* a (- b)))))
   (if (<= y -1.5e+22)
     t_1
     (if (<= y -1.06e-63)
       (* a (/ x a))
       (if (<= y 0.00038) (* x (- 1.0 (* a b))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (a * -b);
	double tmp;
	if (y <= -1.5e+22) {
		tmp = t_1;
	} else if (y <= -1.06e-63) {
		tmp = a * (x / a);
	} else if (y <= 0.00038) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (a * -b)
    if (y <= (-1.5d+22)) then
        tmp = t_1
    else if (y <= (-1.06d-63)) then
        tmp = a * (x / a)
    else if (y <= 0.00038d0) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (a * -b);
	double tmp;
	if (y <= -1.5e+22) {
		tmp = t_1;
	} else if (y <= -1.06e-63) {
		tmp = a * (x / a);
	} else if (y <= 0.00038) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * (a * -b)
	tmp = 0
	if y <= -1.5e+22:
		tmp = t_1
	elif y <= -1.06e-63:
		tmp = a * (x / a)
	elif y <= 0.00038:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(a * Float64(-b)))
	tmp = 0.0
	if (y <= -1.5e+22)
		tmp = t_1;
	elseif (y <= -1.06e-63)
		tmp = Float64(a * Float64(x / a));
	elseif (y <= 0.00038)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (a * -b);
	tmp = 0.0;
	if (y <= -1.5e+22)
		tmp = t_1;
	elseif (y <= -1.06e-63)
		tmp = a * (x / a);
	elseif (y <= 0.00038)
		tmp = x * (1.0 - (a * b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.5e+22], t$95$1, If[LessEqual[y, -1.06e-63], N[(a * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00038], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e22 or 3.8000000000000002e-4 < y

   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 39.0%

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

      1. associate-*r* 39.0%

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

      2. mul-1-neg 39.0%

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

   5. Simplified 39.0%

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

   6. Taylor expanded in a around 0 11.5%

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

      1. mul-1-neg 11.5%

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

      2. unsub-neg 11.5%

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

   8. Simplified 11.5%

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

   9. Taylor expanded in a around inf 19.6%

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

       1. associate-*r* 23.3%

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

       2. associate-*r* 23.3%

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

       3. *-commutative 23.3%

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

       4. associate-*r* 23.3%

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

       5. mul-1-neg 23.3%

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

   11. Simplified 23.3%

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

   if -1.5e22 < y < -1.06000000000000004e-63

   1. Initial program 93.0%

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

   3. Taylor expanded in b around inf 65.1%

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

      1. associate-*r* 65.1%

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

      2. mul-1-neg 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in a around 0 30.2%

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

      1. mul-1-neg 30.2%

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

      2. unsub-neg 30.2%

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

   8. Simplified 30.2%

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

   9. Taylor expanded in a around inf 30.1%

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

       1. +-commutative 30.1%

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

       2. mul-1-neg 30.1%

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

       3. sub-neg 30.1%

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

       4. *-commutative 30.1%

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

   11. Simplified 30.1%

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

   12. Taylor expanded in a around 0 44.5%

       \[\leadsto a \cdot \color{blue}{\frac{x}{a}} \]

   if -1.06000000000000004e-63 < y < 3.8000000000000002e-4

   1. Initial program 96.2%

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

   3. Taylor expanded in b around inf 80.9%

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

      1. associate-*r* 80.9%

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

      2. mul-1-neg 80.9%

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

   5. Simplified 80.9%

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

   6. Taylor expanded in a around 0 53.0%

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

      1. mul-1-neg 53.0%

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

      2. unsub-neg 53.0%

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

   8. Simplified 53.0%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq -1.06 \cdot 10^{-63}:\\ \;\;\;\;a \cdot \frac{x}{a}\\ \mathbf{elif}\;y \leq 0.00038:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+27}:\\ \;\;\;\;a \cdot \left(b \cdot \left(\frac{x}{a \cdot b} - x\right)\right)\\ \mathbf{elif}\;y \leq 0.0136:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.3e+27)
   (* a (* b (- (/ x (* a b)) x)))
   (if (<= y 0.0136) (* 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 <= -1.3e+27) {
		tmp = a * (b * ((x / (a * b)) - x));
	} else if (y <= 0.0136) {
		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 <= (-1.3d+27)) then
        tmp = a * (b * ((x / (a * b)) - x))
    else if (y <= 0.0136d0) 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 <= -1.3e+27) {
		tmp = a * (b * ((x / (a * b)) - x));
	} else if (y <= 0.0136) {
		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 <= -1.3e+27:
		tmp = a * (b * ((x / (a * b)) - x))
	elif y <= 0.0136:
		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 <= -1.3e+27)
		tmp = Float64(a * Float64(b * Float64(Float64(x / Float64(a * b)) - x)));
	elseif (y <= 0.0136)
		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 <= -1.3e+27)
		tmp = a * (b * ((x / (a * b)) - x));
	elseif (y <= 0.0136)
		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, -1.3e+27], N[(a * N[(b * N[(N[(x / N[(a * b), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0136], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.30000000000000004e27

   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 b around inf 34.3%

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

      1. associate-*r* 34.3%

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

      2. mul-1-neg 34.3%

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

   5. Simplified 34.3%

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

   6. Taylor expanded in a around 0 10.5%

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

      1. mul-1-neg 10.5%

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

      2. unsub-neg 10.5%

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

   8. Simplified 10.5%

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

   9. Taylor expanded in a around inf 11.9%

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

       1. +-commutative 11.9%

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

       2. mul-1-neg 11.9%

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

       3. sub-neg 11.9%

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

       4. *-commutative 11.9%

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

   11. Simplified 11.9%

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

   12. Taylor expanded in b around inf 19.3%

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

   if -1.30000000000000004e27 < y < 0.0135999999999999992

   1. Initial program 95.9%

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

   3. Taylor expanded in b around inf 79.6%

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

      1. associate-*r* 79.6%

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

      2. mul-1-neg 79.6%

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

   5. Simplified 79.6%

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

   6. Taylor expanded in a around 0 50.7%

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

      1. mul-1-neg 50.7%

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

      2. unsub-neg 50.7%

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

   8. Simplified 50.7%

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

   if 0.0135999999999999992 < y

   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 40.6%

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

      1. associate-*r* 40.6%

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

      2. mul-1-neg 40.6%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in a around 0 10.0%

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

      1. mul-1-neg 10.0%

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

      2. unsub-neg 10.0%

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

   8. Simplified 10.0%

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

   9. Taylor expanded in a around inf 24.9%

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

       1. associate-*r* 29.7%

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

       2. associate-*r* 29.7%

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

       3. *-commutative 29.7%

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

       4. associate-*r* 29.7%

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

       5. mul-1-neg 29.7%

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

   11. Simplified 29.7%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+27}:\\ \;\;\;\;a \cdot \left(b \cdot \left(\frac{x}{a \cdot b} - x\right)\right)\\ \mathbf{elif}\;y \leq 0.0136:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.5% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-249}:\\ \;\;\;\;a \cdot \frac{x}{a}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -8.2e-249) (* a (/ x a)) (if (<= y 3.8e-46) x (* a (* x (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8.2e-249) {
		tmp = a * (x / a);
	} else if (y <= 3.8e-46) {
		tmp = x;
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-8.2d-249)) then
        tmp = a * (x / a)
    else if (y <= 3.8d-46) then
        tmp = x
    else
        tmp = a * (x * -b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8.2e-249) {
		tmp = a * (x / a);
	} else if (y <= 3.8e-46) {
		tmp = x;
	} else {
		tmp = a * (x * -b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -8.2e-249:
		tmp = a * (x / a)
	elif y <= 3.8e-46:
		tmp = x
	else:
		tmp = a * (x * -b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -8.2e-249)
		tmp = Float64(a * Float64(x / a));
	elseif (y <= 3.8e-46)
		tmp = x;
	else
		tmp = Float64(a * Float64(x * Float64(-b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -8.2e-249)
		tmp = a * (x / a);
	elseif (y <= 3.8e-46)
		tmp = x;
	else
		tmp = a * (x * -b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -8.2e-249], N[(a * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-46], x, N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.20000000000000007e-249

   1. Initial program 94.7%

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

   3. Taylor expanded in b around inf 51.0%

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

      1. associate-*r* 51.0%

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

      2. mul-1-neg 51.0%

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

   5. Simplified 51.0%

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

   6. Taylor expanded in a around 0 25.7%

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

      1. mul-1-neg 25.7%

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

      2. unsub-neg 25.7%

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

   8. Simplified 25.7%

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

   9. Taylor expanded in a around inf 24.8%

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

       1. +-commutative 24.8%

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

       2. mul-1-neg 24.8%

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

       3. sub-neg 24.8%

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

       4. *-commutative 24.8%

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

   11. Simplified 24.8%

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

   12. Taylor expanded in a around 0 20.8%

       \[\leadsto a \cdot \color{blue}{\frac{x}{a}} \]

   if -8.20000000000000007e-249 < y < 3.7999999999999997e-46

   1. Initial program 96.9%

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

   3. Taylor expanded in b around inf 86.4%

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

      1. associate-*r* 86.4%

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

      2. mul-1-neg 86.4%

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

   5. Simplified 86.4%

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

   6. Taylor expanded in a around 0 50.2%

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

   if 3.7999999999999997e-46 < y

   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 42.8%

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

      1. associate-*r* 42.8%

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

      2. mul-1-neg 42.8%

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

   5. Simplified 42.8%

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

   6. Taylor expanded in a around 0 12.4%

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

      1. mul-1-neg 12.4%

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

      2. unsub-neg 12.4%

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

   8. Simplified 12.4%

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

   9. Taylor expanded in a around inf 25.9%

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

       1. associate-*r* 28.9%

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

       2. associate-*r* 28.9%

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

       3. *-commutative 28.9%

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

       4. associate-*r* 28.9%

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

       5. mul-1-neg 28.9%

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

   11. Simplified 28.9%

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

   12. Taylor expanded in x around 0 25.9%

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

       1. *-commutative 25.9%

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

       2. neg-mul-1 25.9%

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

       3. distribute-rgt-neg-in 25.9%

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

       4. distribute-rgt-neg-out 25.9%

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

   14. Simplified 25.9%

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

Alternative 12: 24.5% accurate, 63.0× speedup

Math

\[\begin{array}{l} \\ a \cdot \frac{x}{a} \end{array} \]

FPCore

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

C

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

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 = a * (x / a)
end function

Java

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

Python

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

Julia

function code(x, y, z, t, a, b)
	return Float64(a * Float64(x / a))
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a * N[(x / a), $MachinePrecision]), $MachinePrecision]

Derivation

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 57.4%

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

   1. associate-*r* 57.4%

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

   2. mul-1-neg 57.4%

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

5. Simplified 57.4%

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

6. Taylor expanded in a around 0 29.4%

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

   1. mul-1-neg 29.4%

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

   2. unsub-neg 29.4%

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

8. Simplified 29.4%

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

9. Taylor expanded in a around inf 24.8%

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

    1. +-commutative 24.8%

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

    2. mul-1-neg 24.8%

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

    3. sub-neg 24.8%

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

    4. *-commutative 24.8%

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

11. Simplified 24.8%

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

12. Taylor expanded in a around 0 23.3%

    \[\leadsto a \cdot \color{blue}{\frac{x}{a}} \]
13. Add Preprocessing

Alternative 13: 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 95.7%

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

3. Taylor expanded in b around inf 57.4%

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

   1. associate-*r* 57.4%

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

   2. mul-1-neg 57.4%

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

5. Simplified 57.4%

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

6. Taylor expanded in a around 0 19.6%

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

Reproduce

herbie shell --seed 2024085 
(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))))))