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

Percentage Accurate: 96.6% → 99.8%

Time: 26.1s

Alternatives: 16

Speedup: 1.0×

• 20.5% 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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. +-commutative 96.5%

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

   2. fma-def 96.9%

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

   3. sub-neg 96.9%

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

   4. log1p-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 96.5%

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

Alternative 3: 87.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-18} \lor \neg \left(y \leq 6 \cdot 10^{-10}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -5.1e-18) (not (<= y 6e-10)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (* a (- (log1p (- z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.1e-18) || !(y <= 6e-10)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((a * (log1p(-z) - b)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.1e-18) || !(y <= 6e-10)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((a * (Math.log1p(-z) - b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -5.1e-18) or not (y <= 6e-10):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((a * (math.log1p(-z) - b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -5.1e-18) || !(y <= 6e-10))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(a * Float64(log1p(Float64(-z)) - b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -5.1e-18], N[Not[LessEqual[y, 6e-10]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.09999999999999983e-18 or 6e-10 < y

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 89.5%

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

   if -5.09999999999999983e-18 < y < 6e-10

   1. Initial program 93.7%

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

   2. Taylor expanded in y around 0 84.5%

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

      1. sub-neg 84.5%

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

      2. +-commutative 84.5%

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

      3. sub-neg 84.5%

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

      4. neg-mul-1 84.5%

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

      5. log1p-def 90.6%

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

      6. neg-mul-1 90.6%

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

      7. +-commutative 90.6%

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

      8. sub-neg 90.6%

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

   4. Simplified 90.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-18} \lor \neg \left(y \leq 6 \cdot 10^{-10}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \end{array} \]

Alternative 4: 71.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{b \cdot \left(-a\right)}\\ t_2 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -0.0021:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-112}:\\ \;\;\;\;x \cdot t_1\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-162}:\\ \;\;\;\;x \cdot {\left(e^{-t}\right)}^{y}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \left(\left(1 + t_1\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (exp (* b (- a)))) (t_2 (* x (pow z y))))
   (if (<= y -0.0021)
     t_2
     (if (<= y -9e-112)
       (* x t_1)
       (if (<= y -1.2e-162)
         (* x (pow (exp (- t)) y))
         (if (<= y 4.6e+27) (* x (+ (+ 1.0 t_1) -1.0)) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((b * -a));
	double t_2 = x * pow(z, y);
	double tmp;
	if (y <= -0.0021) {
		tmp = t_2;
	} else if (y <= -9e-112) {
		tmp = x * t_1;
	} else if (y <= -1.2e-162) {
		tmp = x * pow(exp(-t), y);
	} else if (y <= 4.6e+27) {
		tmp = x * ((1.0 + t_1) + -1.0);
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = exp((b * -a))
    t_2 = x * (z ** y)
    if (y <= (-0.0021d0)) then
        tmp = t_2
    else if (y <= (-9d-112)) then
        tmp = x * t_1
    else if (y <= (-1.2d-162)) then
        tmp = x * (exp(-t) ** y)
    else if (y <= 4.6d+27) then
        tmp = x * ((1.0d0 + t_1) + (-1.0d0))
    else
        tmp = t_2
    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 = Math.exp((b * -a));
	double t_2 = x * Math.pow(z, y);
	double tmp;
	if (y <= -0.0021) {
		tmp = t_2;
	} else if (y <= -9e-112) {
		tmp = x * t_1;
	} else if (y <= -1.2e-162) {
		tmp = x * Math.pow(Math.exp(-t), y);
	} else if (y <= 4.6e+27) {
		tmp = x * ((1.0 + t_1) + -1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((b * -a))
	t_2 = x * math.pow(z, y)
	tmp = 0
	if y <= -0.0021:
		tmp = t_2
	elif y <= -9e-112:
		tmp = x * t_1
	elif y <= -1.2e-162:
		tmp = x * math.pow(math.exp(-t), y)
	elif y <= 4.6e+27:
		tmp = x * ((1.0 + t_1) + -1.0)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = exp(Float64(b * Float64(-a)))
	t_2 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -0.0021)
		tmp = t_2;
	elseif (y <= -9e-112)
		tmp = Float64(x * t_1);
	elseif (y <= -1.2e-162)
		tmp = Float64(x * (exp(Float64(-t)) ^ y));
	elseif (y <= 4.6e+27)
		tmp = Float64(x * Float64(Float64(1.0 + t_1) + -1.0));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((b * -a));
	t_2 = x * (z ^ y);
	tmp = 0.0;
	if (y <= -0.0021)
		tmp = t_2;
	elseif (y <= -9e-112)
		tmp = x * t_1;
	elseif (y <= -1.2e-162)
		tmp = x * (exp(-t) ^ y);
	elseif (y <= 4.6e+27)
		tmp = x * ((1.0 + t_1) + -1.0);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Exp[N[(b * (-a)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0021], t$95$2, If[LessEqual[y, -9e-112], N[(x * t$95$1), $MachinePrecision], If[LessEqual[y, -1.2e-162], N[(x * N[Power[N[Exp[(-t)], $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+27], N[(x * N[(N[(1.0 + t$95$1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -0.00209999999999999987 or 4.6000000000000001e27 < y

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 91.2%

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

   3. Taylor expanded in t around 0 75.2%

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

   if -0.00209999999999999987 < y < -9.00000000000000024e-112

   1. Initial program 94.8%

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

   2. Taylor expanded in b around inf 84.0%

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

      1. associate-*r* 84.0%

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

      2. neg-mul-1 84.0%

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

   4. Simplified 84.0%

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

   if -9.00000000000000024e-112 < y < -1.2000000000000001e-162

   1. Initial program 75.7%

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

   2. Taylor expanded in t around inf 51.5%

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

      1. mul-1-neg 51.5%

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

      2. distribute-lft-neg-out 51.5%

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

      3. *-commutative 51.5%

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

   4. Simplified 51.5%

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

   5. Taylor expanded in y around inf 51.5%

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

      1. associate-*r* 51.5%

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

      2. mul-1-neg 51.5%

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

      3. exp-prod 83.7%

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

   7. Simplified 83.7%

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

   if -1.2000000000000001e-162 < y < 4.6000000000000001e27

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. sub-neg 96.1%

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

      3. log1p-udef 99.9%

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

      4. fma-udef 99.9%

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

      5. expm1-log1p-u 99.9%

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

      6. fma-udef 99.9%

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

      7. log1p-udef 96.1%

         \[\leadsto x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{a \cdot \left(\color{blue}{\log \left(1 + \left(-z\right)\right)} - b\right) + y \cdot \left(\log z - t\right)}\right)\right) \]

      8. sub-neg 96.1%

         \[\leadsto x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{a \cdot \left(\log \color{blue}{\left(1 - z\right)} - b\right) + y \cdot \left(\log z - t\right)}\right)\right) \]

      9. +-commutative 96.1%

         \[\leadsto x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}}\right)\right) \]

      10. fma-def 96.1%

          \[\leadsto x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\log \left(1 - z\right) - b\right)\right)}}\right)\right) \]

   3. Applied egg-rr 95.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(z\right) - b\right)\right)}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-udef 95.1%

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

      2. log1p-udef 95.1%

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

      3. add-exp-log 95.1%

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

      4. sub-neg 95.1%

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

      5. add-sqr-sqrt 44.7%

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

      6. sqrt-unprod 77.9%

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

      7. sqr-neg 77.9%

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

      8. sqrt-unprod 43.7%

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

      9. add-sqr-sqrt 84.7%

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

   5. Applied egg-rr 84.7%

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

   6. Taylor expanded in b around inf 84.4%

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

      1. mul-1-neg 84.4%

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

      2. distribute-rgt-neg-in 84.4%

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

   8. Simplified 84.4%

      \[\leadsto x \cdot \left(\left(1 + e^{\color{blue}{a \cdot \left(-b\right)}}\right) - 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0021:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-112}:\\ \;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-162}:\\ \;\;\;\;x \cdot {\left(e^{-t}\right)}^{y}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \left(\left(1 + e^{b \cdot \left(-a\right)}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]

Alternative 5: 80.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-8} \lor \neg \left(a \leq 3.4 \cdot 10^{+103}\right):\\ \;\;\;\;x \cdot {\left(e^{-a}\right)}^{b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -4.5e-8) (not (<= a 3.4e+103)))
   (* x (pow (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 ((a <= -4.5e-8) || !(a <= 3.4e+103)) {
		tmp = x * pow(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 ((a <= (-4.5d-8)) .or. (.not. (a <= 3.4d+103))) 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 ((a <= -4.5e-8) || !(a <= 3.4e+103)) {
		tmp = x * Math.pow(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 (a <= -4.5e-8) or not (a <= 3.4e+103):
		tmp = x * math.pow(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 ((a <= -4.5e-8) || !(a <= 3.4e+103))
		tmp = Float64(x * (exp(Float64(-a)) ^ 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 ((a <= -4.5e-8) || ~((a <= 3.4e+103)))
		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[a, -4.5e-8], N[Not[LessEqual[a, 3.4e+103]], $MachinePrecision]], N[(x * N[Power[N[Exp[(-a)], $MachinePrecision], b], $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 a < -4.49999999999999993e-8 or 3.3999999999999998e103 < a

   1. Initial program 93.4%

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

   2. Taylor expanded in b around inf 70.5%

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

      1. associate-*r* 70.5%

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

      2. neg-mul-1 70.5%

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

   4. Simplified 70.5%

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

   5. Taylor expanded in x around 0 70.5%

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

      1. associate-*r* 70.5%

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

      2. exp-prod 79.8%

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

      3. mul-1-neg 79.8%

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

   7. Simplified 79.8%

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

   if -4.49999999999999993e-8 < a < 3.3999999999999998e103

   1. Initial program 98.6%

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

   2. Taylor expanded in y around inf 88.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-8} \lor \neg \left(a \leq 3.4 \cdot 10^{+103}\right):\\ \;\;\;\;x \cdot {\left(e^{-a}\right)}^{b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]

Alternative 6: 72.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.011 \lor \neg \left(y \leq 10^{+31}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(1 + e^{b \cdot \left(-a\right)}\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -0.011) (not (<= y 1e+31)))
   (* x (pow z y))
   (* x (+ (+ 1.0 (exp (* b (- a)))) -1.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -0.011) || !(y <= 1e+31)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * ((1.0 + exp((b * -a))) + -1.0);
	}
	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 <= (-0.011d0)) .or. (.not. (y <= 1d+31))) then
        tmp = x * (z ** y)
    else
        tmp = x * ((1.0d0 + exp((b * -a))) + (-1.0d0))
    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 <= -0.011) || !(y <= 1e+31)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * ((1.0 + Math.exp((b * -a))) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -0.011) or not (y <= 1e+31):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * ((1.0 + math.exp((b * -a))) + -1.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -0.011) || !(y <= 1e+31))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * Float64(Float64(1.0 + exp(Float64(b * Float64(-a)))) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -0.011) || ~((y <= 1e+31)))
		tmp = x * (z ^ y);
	else
		tmp = x * ((1.0 + exp((b * -a))) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -0.011], N[Not[LessEqual[y, 1e+31]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 + N[Exp[N[(b * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.010999999999999999 or 9.9999999999999996e30 < y

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 91.2%

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

   3. Taylor expanded in t around 0 75.2%

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

   if -0.010999999999999999 < y < 9.9999999999999996e30

   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. Step-by-step derivation

      1. +-commutative 94.1%

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

      2. sub-neg 94.1%

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

      3. log1p-udef 99.9%

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

      4. fma-udef 99.9%

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

      5. expm1-log1p-u 99.9%

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

      6. fma-udef 99.9%

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

      7. log1p-udef 94.1%

         \[\leadsto x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{a \cdot \left(\color{blue}{\log \left(1 + \left(-z\right)\right)} - b\right) + y \cdot \left(\log z - t\right)}\right)\right) \]

      8. sub-neg 94.1%

         \[\leadsto x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{a \cdot \left(\log \color{blue}{\left(1 - z\right)} - b\right) + y \cdot \left(\log z - t\right)}\right)\right) \]

      9. +-commutative 94.1%

         \[\leadsto x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}}\right)\right) \]

      10. fma-def 94.1%

          \[\leadsto x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\log \left(1 - z\right) - b\right)\right)}}\right)\right) \]

   3. Applied egg-rr 92.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(z\right) - b\right)\right)}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-udef 92.5%

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

      2. log1p-udef 92.5%

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

      3. add-exp-log 92.5%

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

      4. sub-neg 92.5%

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

      5. add-sqr-sqrt 48.1%

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

      6. sqrt-unprod 78.4%

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

      7. sqr-neg 78.4%

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

      8. sqrt-unprod 39.1%

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

      9. add-sqr-sqrt 80.0%

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

   5. Applied egg-rr 80.0%

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

   6. Taylor expanded in b around inf 80.7%

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

      1. mul-1-neg 80.7%

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

      2. distribute-rgt-neg-in 80.7%

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

   8. Simplified 80.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.011 \lor \neg \left(y \leq 10^{+31}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(1 + e^{b \cdot \left(-a\right)}\right) + -1\right)\\ \end{array} \]

Alternative 7: 71.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+37} \lor \neg \left(t \leq 6.8 \cdot 10^{-89}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -4.1e+37) (not (<= t 6.8e-89)))
   (* x (exp (* y (- t))))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -4.1e+37) || !(t <= 6.8e-89)) {
		tmp = x * exp((y * -t));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-4.1d+37)) .or. (.not. (t <= 6.8d-89))) then
        tmp = x * exp((y * -t))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -4.1e+37) || !(t <= 6.8e-89)) {
		tmp = x * Math.exp((y * -t));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -4.1e+37) or not (t <= 6.8e-89):
		tmp = x * math.exp((y * -t))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -4.1e+37) || !(t <= 6.8e-89))
		tmp = Float64(x * exp(Float64(y * Float64(-t))));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -4.1e+37) || ~((t <= 6.8e-89)))
		tmp = x * exp((y * -t));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -4.1e+37], N[Not[LessEqual[t, 6.8e-89]], $MachinePrecision]], N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.0999999999999998e37 or 6.8000000000000001e-89 < t

   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. Taylor expanded in t around inf 73.8%

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

      1. mul-1-neg 73.8%

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

      2. distribute-lft-neg-out 73.8%

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

      3. *-commutative 73.8%

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

   4. Simplified 73.8%

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

   if -4.0999999999999998e37 < t < 6.8000000000000001e-89

   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. Taylor expanded in y around inf 70.4%

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

   3. Taylor expanded in t around 0 70.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+37} \lor \neg \left(t \leq 6.8 \cdot 10^{-89}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]

Alternative 8: 72.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.011 \lor \neg \left(y \leq 8.8 \cdot 10^{+27}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -0.011) (not (<= y 8.8e+27)))
   (* x (pow z y))
   (* x (exp (* b (- a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -0.011) || !(y <= 8.8e+27)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((b * -a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-0.011d0)) .or. (.not. (y <= 8.8d+27))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((b * -a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -0.011) || !(y <= 8.8e+27)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((b * -a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -0.011) or not (y <= 8.8e+27):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((b * -a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -0.011) || !(y <= 8.8e+27))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(b * Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -0.011) || ~((y <= 8.8e+27)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((b * -a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -0.011], N[Not[LessEqual[y, 8.8e+27]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(b * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.010999999999999999 or 8.7999999999999995e27 < y

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 91.2%

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

   3. Taylor expanded in t around 0 75.2%

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

   if -0.010999999999999999 < y < 8.7999999999999995e27

   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. Taylor expanded in b around inf 80.6%

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

      1. associate-*r* 80.6%

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

      2. neg-mul-1 80.6%

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

   4. Simplified 80.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.011 \lor \neg \left(y \leq 8.8 \cdot 10^{+27}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\ \end{array} \]

Alternative 9: 55.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-33} \lor \neg \left(y \leq 4.8 \cdot 10^{-67}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.55e-33) (not (<= y 4.8e-67)))
   (* x (pow z y))
   (- x (* x (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.55e-33) || !(y <= 4.8e-67)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x - (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 <= (-1.55d-33)) .or. (.not. (y <= 4.8d-67))) then
        tmp = x * (z ** y)
    else
        tmp = x - (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 <= -1.55e-33) || !(y <= 4.8e-67)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x - (x * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.55e-33) or not (y <= 4.8e-67):
		tmp = x * math.pow(z, y)
	else:
		tmp = x - (x * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.55e-33) || !(y <= 4.8e-67))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x - Float64(x * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.55e-33) || ~((y <= 4.8e-67)))
		tmp = x * (z ^ y);
	else
		tmp = x - (x * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.55e-33], N[Not[LessEqual[y, 4.8e-67]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.54999999999999998e-33 or 4.8e-67 < y

   1. Initial program 98.0%

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

   2. Taylor expanded in y around inf 86.3%

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

   3. Taylor expanded in t around 0 68.1%

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

   if -1.54999999999999998e-33 < y < 4.8e-67

   1. Initial program 94.4%

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

   2. Taylor expanded in b around inf 84.4%

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

      1. associate-*r* 84.4%

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

      2. neg-mul-1 84.4%

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

   4. Simplified 84.4%

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

   5. Taylor expanded in a around 0 50.3%

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

      1. mul-1-neg 50.3%

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

      2. unsub-neg 50.3%

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

      3. associate-*r* 53.9%

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

      4. *-commutative 53.9%

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

   7. Simplified 53.9%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-33} \lor \neg \left(y \leq 4.8 \cdot 10^{-67}\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \end{array} \]

Alternative 10: 39.6% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+73} \lor \neg \left(y \leq 5800\right):\\ \;\;\;\;\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4.6e+73) (not (<= y 5800.0)))
   (* (* 0.5 (* t t)) (* x (* y y)))
   (- x (* x (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.6e+73) || !(y <= 5800.0)) {
		tmp = (0.5 * (t * t)) * (x * (y * y));
	} else {
		tmp = x - (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 <= (-4.6d+73)) .or. (.not. (y <= 5800.0d0))) then
        tmp = (0.5d0 * (t * t)) * (x * (y * y))
    else
        tmp = x - (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 <= -4.6e+73) || !(y <= 5800.0)) {
		tmp = (0.5 * (t * t)) * (x * (y * y));
	} else {
		tmp = x - (x * (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -4.6e+73) or not (y <= 5800.0):
		tmp = (0.5 * (t * t)) * (x * (y * y))
	else:
		tmp = x - (x * (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -4.6e+73) || !(y <= 5800.0))
		tmp = Float64(Float64(0.5 * Float64(t * t)) * Float64(x * Float64(y * y)));
	else
		tmp = Float64(x - Float64(x * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -4.6e+73) || ~((y <= 5800.0)))
		tmp = (0.5 * (t * t)) * (x * (y * y));
	else
		tmp = x - (x * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -4.6e+73], N[Not[LessEqual[y, 5800.0]], $MachinePrecision]], N[(N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6e73 or 5800 < y

   1. Initial program 99.1%

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

   2. Taylor expanded in t around inf 60.4%

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

      1. mul-1-neg 60.4%

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

      2. distribute-lft-neg-out 60.4%

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

      3. *-commutative 60.4%

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

   4. Simplified 60.4%

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

   5. Taylor expanded in y around 0 28.6%

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

      1. +-commutative 28.6%

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

      2. mul-1-neg 28.6%

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

      3. unsub-neg 28.6%

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

      4. *-commutative 28.6%

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

      5. unpow2 28.6%

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

      6. unpow2 28.6%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) - t \cdot y\right)\right) \]

      7. unswap-sqr 25.6%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \color{blue}{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)} - t \cdot y\right)\right) \]

      8. *-commutative 25.6%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - \color{blue}{y \cdot t}\right)\right) \]

   7. Simplified 25.6%

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - y \cdot t\right)\right)} \]

   8. Taylor expanded in y around inf 33.1%

      \[\leadsto \color{blue}{0.5 \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 33.1%

         \[\leadsto \color{blue}{\left(0.5 \cdot {t}^{2}\right) \cdot \left(x \cdot {y}^{2}\right)} \]

      2. unpow2 33.1%

         \[\leadsto \left(0.5 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(x \cdot {y}^{2}\right) \]

      3. unpow2 33.1%

         \[\leadsto \left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   10. Simplified 33.1%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)} \]

   if -4.6e73 < y < 5800

   1. Initial program 94.5%

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

   2. Taylor expanded in b around inf 78.1%

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

      1. associate-*r* 78.1%

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

      2. neg-mul-1 78.1%

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

   4. Simplified 78.1%

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

   5. Taylor expanded in a around 0 43.9%

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

      1. mul-1-neg 43.9%

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

      2. unsub-neg 43.9%

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

      3. associate-*r* 46.6%

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

      4. *-commutative 46.6%

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

   7. Simplified 46.6%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+73} \lor \neg \left(y \leq 5800\right):\\ \;\;\;\;\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \end{array} \]

Alternative 11: 39.5% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(1 + t_1 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 5800:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 0.5 (* t t))))
   (if (<= y -1.9e+36)
     (* x (+ 1.0 (* t_1 (* y y))))
     (if (<= y 5800.0) (- x (* x (* a b))) (* t_1 (* x (* y y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.5 * (t * t);
	double tmp;
	if (y <= -1.9e+36) {
		tmp = x * (1.0 + (t_1 * (y * y)));
	} else if (y <= 5800.0) {
		tmp = x - (x * (a * b));
	} else {
		tmp = t_1 * (x * (y * 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) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 * (t * t)
    if (y <= (-1.9d+36)) then
        tmp = x * (1.0d0 + (t_1 * (y * y)))
    else if (y <= 5800.0d0) then
        tmp = x - (x * (a * b))
    else
        tmp = t_1 * (x * (y * 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 t_1 = 0.5 * (t * t);
	double tmp;
	if (y <= -1.9e+36) {
		tmp = x * (1.0 + (t_1 * (y * y)));
	} else if (y <= 5800.0) {
		tmp = x - (x * (a * b));
	} else {
		tmp = t_1 * (x * (y * y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 0.5 * (t * t)
	tmp = 0
	if y <= -1.9e+36:
		tmp = x * (1.0 + (t_1 * (y * y)))
	elif y <= 5800.0:
		tmp = x - (x * (a * b))
	else:
		tmp = t_1 * (x * (y * y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.5 * Float64(t * t))
	tmp = 0.0
	if (y <= -1.9e+36)
		tmp = Float64(x * Float64(1.0 + Float64(t_1 * Float64(y * y))));
	elseif (y <= 5800.0)
		tmp = Float64(x - Float64(x * Float64(a * b)));
	else
		tmp = Float64(t_1 * Float64(x * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 0.5 * (t * t);
	tmp = 0.0;
	if (y <= -1.9e+36)
		tmp = x * (1.0 + (t_1 * (y * y)));
	elseif (y <= 5800.0)
		tmp = x - (x * (a * b));
	else
		tmp = t_1 * (x * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.9e+36], N[(x * N[(1.0 + N[(t$95$1 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5800.0], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.90000000000000012e36

   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. Taylor expanded in t around inf 56.9%

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

      1. mul-1-neg 56.9%

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

      2. distribute-lft-neg-out 56.9%

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

      3. *-commutative 56.9%

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

   4. Simplified 56.9%

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

   5. Taylor expanded in y around 0 41.2%

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

      1. +-commutative 41.2%

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

      2. mul-1-neg 41.2%

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

      3. unsub-neg 41.2%

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

      4. *-commutative 41.2%

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

      5. unpow2 41.2%

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

      6. unpow2 41.2%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) - t \cdot y\right)\right) \]

      7. unswap-sqr 36.3%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \color{blue}{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)} - t \cdot y\right)\right) \]

      8. *-commutative 36.3%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - \color{blue}{y \cdot t}\right)\right) \]

   7. Simplified 36.3%

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - y \cdot t\right)\right)} \]

   8. Taylor expanded in y around inf 41.4%

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

      1. associate-*r* 41.4%

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot {t}^{2}\right) \cdot {y}^{2}}\right) \]

      2. unpow2 41.4%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {y}^{2}\right) \]

      3. unpow2 41.4%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   10. Simplified 41.4%

       \[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(y \cdot y\right)}\right) \]

   if -1.90000000000000012e36 < y < 5800

   1. Initial program 94.2%

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

   2. Taylor expanded in b around inf 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

   4. Simplified 79.8%

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

   5. Taylor expanded in a around 0 46.0%

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

      1. mul-1-neg 46.0%

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

      2. unsub-neg 46.0%

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

      3. associate-*r* 48.8%

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

      4. *-commutative 48.8%

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

   7. Simplified 48.8%

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

   if 5800 < y

   1. Initial program 98.5%

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

   2. Taylor expanded in t around inf 60.0%

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

      1. mul-1-neg 60.0%

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

      2. distribute-lft-neg-out 60.0%

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

      3. *-commutative 60.0%

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

   4. Simplified 60.0%

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

   5. Taylor expanded in y around 0 16.3%

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

      1. +-commutative 16.3%

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

      2. mul-1-neg 16.3%

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

      3. unsub-neg 16.3%

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

      4. *-commutative 16.3%

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

      5. unpow2 16.3%

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

      6. unpow2 16.3%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) - t \cdot y\right)\right) \]

      7. unswap-sqr 15.1%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \color{blue}{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)} - t \cdot y\right)\right) \]

      8. *-commutative 15.1%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - \color{blue}{y \cdot t}\right)\right) \]

   7. Simplified 15.1%

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - y \cdot t\right)\right)} \]

   8. Taylor expanded in y around inf 25.1%

      \[\leadsto \color{blue}{0.5 \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 25.1%

         \[\leadsto \color{blue}{\left(0.5 \cdot {t}^{2}\right) \cdot \left(x \cdot {y}^{2}\right)} \]

      2. unpow2 25.1%

         \[\leadsto \left(0.5 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(x \cdot {y}^{2}\right) \]

      3. unpow2 25.1%

         \[\leadsto \left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   10. Simplified 25.1%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(1 + \left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 5800:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 12: 29.7% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{+147}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -6.2e-12)
   (* x (* y (- t)))
   (if (<= a 3.7e+147) (* x (- 1.0 (* y t))) (* y (* x (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -6.2e-12) {
		tmp = x * (y * -t);
	} else if (a <= 3.7e+147) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = y * (x * -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 (a <= (-6.2d-12)) then
        tmp = x * (y * -t)
    else if (a <= 3.7d+147) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = y * (x * -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 (a <= -6.2e-12) {
		tmp = x * (y * -t);
	} else if (a <= 3.7e+147) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = y * (x * -t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -6.2e-12:
		tmp = x * (y * -t)
	elif a <= 3.7e+147:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = y * (x * -t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -6.2e-12)
		tmp = Float64(x * Float64(y * Float64(-t)));
	elseif (a <= 3.7e+147)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = Float64(y * Float64(x * Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -6.2e-12)
		tmp = x * (y * -t);
	elseif (a <= 3.7e+147)
		tmp = x * (1.0 - (y * t));
	else
		tmp = y * (x * -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -6.2e-12], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.7e+147], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.2000000000000002e-12

   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. Taylor expanded in t around inf 36.0%

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

      1. mul-1-neg 36.0%

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

      2. distribute-lft-neg-out 36.0%

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

      3. *-commutative 36.0%

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

   4. Simplified 36.0%

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

   5. Taylor expanded in y around 0 16.4%

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

      1. mul-1-neg 16.4%

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

      2. unsub-neg 16.4%

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

      3. *-commutative 16.4%

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

   7. Simplified 16.4%

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

   8. Taylor expanded in y around inf 28.0%

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

      1. mul-1-neg 28.0%

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

      2. *-commutative 28.0%

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

   10. Simplified 28.0%

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

   if -6.2000000000000002e-12 < a < 3.7e147

   1. Initial program 98.1%

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

   2. Taylor expanded in t around inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. distribute-lft-neg-out 69.3%

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

      3. *-commutative 69.3%

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

   4. Simplified 69.3%

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

   5. Taylor expanded in y around 0 36.0%

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

      1. mul-1-neg 36.0%

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

      2. unsub-neg 36.0%

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

      3. *-commutative 36.0%

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

   7. Simplified 36.0%

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

   if 3.7e147 < a

   1. Initial program 93.6%

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

   2. Taylor expanded in t around inf 28.9%

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

      1. mul-1-neg 28.9%

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

      2. distribute-lft-neg-out 28.9%

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

      3. *-commutative 28.9%

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

   4. Simplified 28.9%

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

   5. Taylor expanded in y around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. unsub-neg 9.4%

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

      3. *-commutative 9.4%

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

   7. Simplified 9.4%

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

   8. Taylor expanded in y around inf 15.7%

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

      1. associate-*r* 15.7%

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

      2. neg-mul-1 15.7%

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

      3. associate-*r* 28.3%

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

      4. *-commutative 28.3%

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

      5. *-commutative 28.3%

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

   10. Simplified 28.3%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{+147}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \end{array} \]

Alternative 13: 23.5% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{+150}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.9e-88)
   (* x (* y (- t)))
   (if (<= a 1.46e+150) x (* y (* x (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.9e-88) {
		tmp = x * (y * -t);
	} else if (a <= 1.46e+150) {
		tmp = x;
	} else {
		tmp = y * (x * -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 (a <= (-1.9d-88)) then
        tmp = x * (y * -t)
    else if (a <= 1.46d+150) then
        tmp = x
    else
        tmp = y * (x * -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 (a <= -1.9e-88) {
		tmp = x * (y * -t);
	} else if (a <= 1.46e+150) {
		tmp = x;
	} else {
		tmp = y * (x * -t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.9e-88:
		tmp = x * (y * -t)
	elif a <= 1.46e+150:
		tmp = x
	else:
		tmp = y * (x * -t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.9e-88)
		tmp = Float64(x * Float64(y * Float64(-t)));
	elseif (a <= 1.46e+150)
		tmp = x;
	else
		tmp = Float64(y * Float64(x * Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -1.9e-88)
		tmp = x * (y * -t);
	elseif (a <= 1.46e+150)
		tmp = x;
	else
		tmp = y * (x * -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.9e-88], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.46e+150], x, N[(y * N[(x * (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.90000000000000006e-88

   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. Taylor expanded in t around inf 42.1%

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

      1. mul-1-neg 42.1%

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

      2. distribute-lft-neg-out 42.1%

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

      3. *-commutative 42.1%

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

   4. Simplified 42.1%

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

   5. Taylor expanded in y around 0 19.5%

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

      1. mul-1-neg 19.5%

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

      2. unsub-neg 19.5%

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

      3. *-commutative 19.5%

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

   7. Simplified 19.5%

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

   8. Taylor expanded in y around inf 28.7%

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

      1. mul-1-neg 28.7%

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

      2. *-commutative 28.7%

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

   10. Simplified 28.7%

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

   if -1.90000000000000006e-88 < a < 1.4599999999999999e150

   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. Taylor expanded in t around inf 69.8%

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

      1. mul-1-neg 69.8%

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

      2. distribute-lft-neg-out 69.8%

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

      3. *-commutative 69.8%

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

   4. Simplified 69.8%

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

   5. Taylor expanded in y around 0 37.3%

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

      1. +-commutative 37.3%

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

      2. mul-1-neg 37.3%

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

      3. unsub-neg 37.3%

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

      4. *-commutative 37.3%

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

      5. unpow2 37.3%

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

      6. unpow2 37.3%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) - t \cdot y\right)\right) \]

      7. unswap-sqr 39.8%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \color{blue}{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)} - t \cdot y\right)\right) \]

      8. *-commutative 39.8%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - \color{blue}{y \cdot t}\right)\right) \]

   7. Simplified 39.8%

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - y \cdot t\right)\right)} \]

   8. Taylor expanded in y around 0 32.3%

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

   if 1.4599999999999999e150 < a

   1. Initial program 93.6%

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

   2. Taylor expanded in t around inf 28.9%

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

      1. mul-1-neg 28.9%

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

      2. distribute-lft-neg-out 28.9%

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

      3. *-commutative 28.9%

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

   4. Simplified 28.9%

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

   5. Taylor expanded in y around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. unsub-neg 9.4%

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

      3. *-commutative 9.4%

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

   7. Simplified 9.4%

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

   8. Taylor expanded in y around inf 15.7%

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

      1. associate-*r* 15.7%

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

      2. neg-mul-1 15.7%

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

      3. associate-*r* 28.3%

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

      4. *-commutative 28.3%

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

      5. *-commutative 28.3%

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

   10. Simplified 28.3%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{+150}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-t\right)\right)\\ \end{array} \]

Alternative 14: 30.9% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 51000000000:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 51000000000.0) (- x (* x (* a b))) (* x (- 1.0 (* y t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 51000000000.0) {
		tmp = x - (x * (a * b));
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= 51000000000.0d0) then
        tmp = x - (x * (a * b))
    else
        tmp = x * (1.0d0 - (y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 51000000000.0) {
		tmp = x - (x * (a * b));
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 51000000000.0:
		tmp = x - (x * (a * b))
	else:
		tmp = x * (1.0 - (y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 51000000000.0)
		tmp = Float64(x - Float64(x * Float64(a * b)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 51000000000.0)
		tmp = x - (x * (a * b));
	else
		tmp = x * (1.0 - (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 51000000000.0], N[(x - N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.1e10

   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. Taylor expanded in b around inf 59.4%

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

      1. associate-*r* 59.4%

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

      2. neg-mul-1 59.4%

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

   4. Simplified 59.4%

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

   5. Taylor expanded in a around 0 33.0%

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

      1. mul-1-neg 33.0%

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

      2. unsub-neg 33.0%

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

      3. associate-*r* 35.6%

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

      4. *-commutative 35.6%

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

   7. Simplified 35.6%

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

   if 5.1e10 < t

   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. Taylor expanded in t around inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. distribute-lft-neg-out 75.3%

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

      3. *-commutative 75.3%

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

   4. Simplified 75.3%

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

   5. Taylor expanded in y around 0 30.1%

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

      1. mul-1-neg 30.1%

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

      2. unsub-neg 30.1%

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

      3. *-commutative 30.1%

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

   7. Simplified 30.1%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 51000000000:\\ \;\;\;\;x - x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]

Alternative 15: 21.6% accurate, 39.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -3.4e-88) (* x (* y (- t))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -3.4e-88) {
		tmp = x * (y * -t);
	} 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 (a <= (-3.4d-88)) then
        tmp = x * (y * -t)
    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 (a <= -3.4e-88) {
		tmp = x * (y * -t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -3.4e-88:
		tmp = x * (y * -t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -3.4e-88)
		tmp = Float64(x * Float64(y * Float64(-t)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -3.4e-88)
		tmp = x * (y * -t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -3.4e-88], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if a < -3.39999999999999975e-88

   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. Taylor expanded in t around inf 42.1%

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

      1. mul-1-neg 42.1%

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

      2. distribute-lft-neg-out 42.1%

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

      3. *-commutative 42.1%

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

   4. Simplified 42.1%

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

   5. Taylor expanded in y around 0 19.5%

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

      1. mul-1-neg 19.5%

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

      2. unsub-neg 19.5%

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

      3. *-commutative 19.5%

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

   7. Simplified 19.5%

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

   8. Taylor expanded in y around inf 28.7%

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

      1. mul-1-neg 28.7%

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

      2. *-commutative 28.7%

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

   10. Simplified 28.7%

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

   if -3.39999999999999975e-88 < a

   1. Initial program 97.1%

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

   2. Taylor expanded in t around inf 62.7%

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

      1. mul-1-neg 62.7%

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

      2. distribute-lft-neg-out 62.7%

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

      3. *-commutative 62.7%

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

   4. Simplified 62.7%

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

   5. Taylor expanded in y around 0 34.6%

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

      1. +-commutative 34.6%

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

      2. mul-1-neg 34.6%

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

      3. unsub-neg 34.6%

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

      4. *-commutative 34.6%

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

      5. unpow2 34.6%

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

      6. unpow2 34.6%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) - t \cdot y\right)\right) \]

      7. unswap-sqr 35.7%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \color{blue}{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)} - t \cdot y\right)\right) \]

      8. *-commutative 35.7%

         \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - \color{blue}{y \cdot t}\right)\right) \]

   7. Simplified 35.7%

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - y \cdot t\right)\right)} \]

   8. Taylor expanded in y around 0 27.4%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16: 19.7% accurate, 315.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in t around inf 56.1%

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

   1. mul-1-neg 56.1%

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

   2. distribute-lft-neg-out 56.1%

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

   3. *-commutative 56.1%

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

4. Simplified 56.1%

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

5. Taylor expanded in y around 0 31.2%

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

   1. +-commutative 31.2%

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

   2. mul-1-neg 31.2%

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

   3. unsub-neg 31.2%

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

   4. *-commutative 31.2%

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

   5. unpow2 31.2%

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

   6. unpow2 31.2%

      \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) - t \cdot y\right)\right) \]

   7. unswap-sqr 32.1%

      \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \color{blue}{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)} - t \cdot y\right)\right) \]

   8. *-commutative 32.1%

      \[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - \color{blue}{y \cdot t}\right)\right) \]

7. Simplified 32.1%

   \[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - y \cdot t\right)\right)} \]

8. Taylor expanded in y around 0 20.9%

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

9. Final simplification 20.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023275 
(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))))))