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

Percentage Accurate: 96.5% → 99.7%

Time: 17.9s

Alternatives: 15

Speedup: 1.5×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

   1. fma-def 98.0%

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

   2. sub-neg 98.0%

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

   3. log1p-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 85.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right)\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{-86}:\\ \;\;\;\;x \cdot e^{t_1 - z \cdot a}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-92}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* y (- (log z) t))))
   (if (<= y -1.1e-86)
     (* x (exp (- t_1 (* z a))))
     (if (<= y 1.1e-92)
       (* x (exp (* a (- (log1p (- z)) b))))
       (* x (exp t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y * (log(z) - t);
	double tmp;
	if (y <= -1.1e-86) {
		tmp = x * exp((t_1 - (z * a)));
	} else if (y <= 1.1e-92) {
		tmp = x * exp((a * (log1p(-z) - b)));
	} else {
		tmp = x * exp(t_1);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y * (Math.log(z) - t);
	double tmp;
	if (y <= -1.1e-86) {
		tmp = x * Math.exp((t_1 - (z * a)));
	} else if (y <= 1.1e-92) {
		tmp = x * Math.exp((a * (Math.log1p(-z) - b)));
	} else {
		tmp = x * Math.exp(t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y * (math.log(z) - t)
	tmp = 0
	if y <= -1.1e-86:
		tmp = x * math.exp((t_1 - (z * a)))
	elif y <= 1.1e-92:
		tmp = x * math.exp((a * (math.log1p(-z) - b)))
	else:
		tmp = x * math.exp(t_1)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y * Float64(log(z) - t))
	tmp = 0.0
	if (y <= -1.1e-86)
		tmp = Float64(x * exp(Float64(t_1 - Float64(z * a))));
	elseif (y <= 1.1e-92)
		tmp = Float64(x * exp(Float64(a * Float64(log1p(Float64(-z)) - b))));
	else
		tmp = Float64(x * exp(t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.1e-86], N[(x * N[Exp[N[(t$95$1 - N[(z * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e-92], N[(x * N[Exp[N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1000000000000001e-86

   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. Taylor expanded in z around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. associate-*r* 98.7%

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

      3. associate-*r* 98.7%

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

      4. distribute-lft-out 98.7%

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

      5. neg-mul-1 98.7%

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

   4. Simplified 98.7%

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

   5. Taylor expanded in b around 0 92.4%

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

      1. +-commutative 92.4%

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

      2. mul-1-neg 92.4%

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

      3. sub-neg 92.4%

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

   7. Simplified 92.4%

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

   if -1.1000000000000001e-86 < y < 1.09999999999999994e-92

   1. Initial program 96.7%

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

   2. Taylor expanded in y around 0 89.1%

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

      1. sub-neg 89.1%

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

      2. sub-neg 89.1%

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

      3. neg-mul-1 89.1%

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

      4. log1p-def 92.4%

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

      5. neg-mul-1 92.4%

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

      6. sub-neg 92.4%

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

   4. Simplified 92.4%

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

   if 1.09999999999999994e-92 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 85.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-86}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right) - z \cdot a}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-92}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

2. Taylor expanded in z around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. associate-*r* 99.6%

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

   4. distribute-lft-out 99.6%

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

   5. neg-mul-1 99.6%

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

4. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 83.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+38} \lor \neg \left(a \leq 6.4 \cdot 10^{+81}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -2.9e+38) (not (<= a 6.4e+81)))
   (* x (exp (* a (- (log1p (- z)) b))))
   (* x (exp (* y (- (log z) t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -2.9e+38) || !(a <= 6.4e+81)) {
		tmp = x * exp((a * (log1p(-z) - b)));
	} else {
		tmp = x * exp((y * (log(z) - t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -2.9e+38) || !(a <= 6.4e+81)) {
		tmp = x * Math.exp((a * (Math.log1p(-z) - b)));
	} else {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -2.9e+38) or not (a <= 6.4e+81):
		tmp = x * math.exp((a * (math.log1p(-z) - b)))
	else:
		tmp = x * math.exp((y * (math.log(z) - t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -2.9e+38) || !(a <= 6.4e+81))
		tmp = Float64(x * exp(Float64(a * Float64(log1p(Float64(-z)) - b))));
	else
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -2.9e+38], N[Not[LessEqual[a, 6.4e+81]], $MachinePrecision]], N[(x * N[Exp[N[(a * N[(N[Log[1 + (-z)], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.90000000000000007e38 or 6.4e81 < a

   1. Initial program 94.3%

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

   2. Taylor expanded in y around 0 80.2%

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

      1. sub-neg 80.2%

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

      2. sub-neg 80.2%

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

      3. neg-mul-1 80.2%

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

      4. log1p-def 87.7%

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

      5. neg-mul-1 87.7%

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

      6. sub-neg 87.7%

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

   4. Simplified 87.7%

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

   if -2.90000000000000007e38 < a < 6.4e81

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 90.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+38} \lor \neg \left(a \leq 6.4 \cdot 10^{+81}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]

Alternative 5: 82.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.62 \cdot 10^{-91} \lor \neg \left(y \leq 2.75 \cdot 10^{-94}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.62e-91) (not (<= y 2.75e-94)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (* a (- b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.62e-91) || !(y <= 2.75e-94)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((a * -b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.62d-91)) .or. (.not. (y <= 2.75d-94))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp((a * -b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.62e-91) || !(y <= 2.75e-94)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((a * -b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.62e-91) or not (y <= 2.75e-94):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((a * -b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.62e-91) || !(y <= 2.75e-94))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.62e-91) || ~((y <= 2.75e-94)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp((a * -b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.62e-91], N[Not[LessEqual[y, 2.75e-94]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6199999999999999e-91 or 2.74999999999999995e-94 < y

   1. Initial program 98.2%

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

   2. Taylor expanded in y around inf 87.0%

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

   if -1.6199999999999999e-91 < y < 2.74999999999999995e-94

   1. Initial program 96.7%

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

   2. Taylor expanded in b around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. distribute-rgt-neg-out 86.9%

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

   4. Simplified 86.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.62 \cdot 10^{-91} \lor \neg \left(y \leq 2.75 \cdot 10^{-94}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \end{array} \]

Alternative 6: 71.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -390 \lor \neg \left(t \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -390.0) (not (<= t 5e-7)))
   (* x (exp (* t (- y))))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -390.0) || !(t <= 5e-7)) {
		tmp = x * exp((t * -y));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-390.0d0)) .or. (.not. (t <= 5d-7))) then
        tmp = x * exp((t * -y))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -390.0) || !(t <= 5e-7)) {
		tmp = x * Math.exp((t * -y));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -390.0) or not (t <= 5e-7):
		tmp = x * math.exp((t * -y))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -390.0) || !(t <= 5e-7))
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -390.0) || ~((t <= 5e-7)))
		tmp = x * exp((t * -y));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -390.0], N[Not[LessEqual[t, 5e-7]], $MachinePrecision]], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -390 or 4.99999999999999977e-7 < t

   1. Initial program 96.6%

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

   2. Taylor expanded in t around inf 79.9%

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

      1. mul-1-neg 79.9%

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

      2. *-commutative 79.9%

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

   4. Simplified 79.9%

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

   if -390 < t < 4.99999999999999977e-7

   1. Initial program 99.0%

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

   2. Taylor expanded in y around inf 69.1%

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

   3. Taylor expanded in t around 0 69.1%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -390 \lor \neg \left(t \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]

Alternative 7: 69.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2950000 \lor \neg \left(a \leq 2.7 \cdot 10^{+81}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -2950000.0) (not (<= a 2.7e+81)))
   (* x (exp (* a (- b))))
   (* x (exp (* t (- y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -2950000.0) || !(a <= 2.7e+81)) {
		tmp = x * exp((a * -b));
	} else {
		tmp = x * exp((t * -y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-2950000.0d0)) .or. (.not. (a <= 2.7d+81))) then
        tmp = x * exp((a * -b))
    else
        tmp = x * exp((t * -y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -2950000.0) || !(a <= 2.7e+81)) {
		tmp = x * Math.exp((a * -b));
	} else {
		tmp = x * Math.exp((t * -y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -2950000.0) or not (a <= 2.7e+81):
		tmp = x * math.exp((a * -b))
	else:
		tmp = x * math.exp((t * -y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -2950000.0) || !(a <= 2.7e+81))
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	else
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -2950000.0) || ~((a <= 2.7e+81)))
		tmp = x * exp((a * -b));
	else
		tmp = x * exp((t * -y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -2950000.0], N[Not[LessEqual[a, 2.7e+81]], $MachinePrecision]], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.95e6 or 2.6999999999999999e81 < a

   1. Initial program 94.6%

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

   2. Taylor expanded in b around inf 76.5%

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

      1. mul-1-neg 76.5%

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

      2. distribute-rgt-neg-out 76.5%

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

   4. Simplified 76.5%

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

   if -2.95e6 < a < 2.6999999999999999e81

   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 82.9%

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

      1. mul-1-neg 82.9%

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

      2. *-commutative 82.9%

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

   4. Simplified 82.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2950000 \lor \neg \left(a \leq 2.7 \cdot 10^{+81}\right):\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \]

Alternative 8: 54.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -320000000000:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -320000000000.0) (* x (- 1.0 (* y t))) (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -320000000000.0) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-320000000000.0d0)) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -320000000000.0) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -320000000000.0:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -320000000000.0)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -320000000000.0)
		tmp = x * (1.0 - (y * t));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -320000000000.0], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.2e11

   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 t around inf 77.1%

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

      1. mul-1-neg 77.1%

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

      2. *-commutative 77.1%

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

   4. Simplified 77.1%

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

   5. Taylor expanded in y around 0 37.9%

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

      1. neg-mul-1 37.9%

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

      2. distribute-rgt-neg-in 37.9%

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

   7. Simplified 37.9%

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

   8. Taylor expanded in x around 0 37.9%

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

      1. mul-1-neg 37.9%

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

      2. *-commutative 37.9%

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

      3. unsub-neg 37.9%

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

   10. Simplified 37.9%

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

   if -3.2e11 < t

   1. Initial program 97.3%

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

   2. Taylor expanded in y around inf 74.5%

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

   3. Taylor expanded in t around 0 65.0%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -320000000000:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]

Alternative 9: 27.5% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{if}\;y \leq -33000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-116}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+161}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* y (* t (- x)))))
   (if (<= y -33000000000000.0)
     t_1
     (if (<= y 2e-116) x (if (<= y 4.5e+161) (* a (* x (- b))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y * (t * -x);
	double tmp;
	if (y <= -33000000000000.0) {
		tmp = t_1;
	} else if (y <= 2e-116) {
		tmp = x;
	} else if (y <= 4.5e+161) {
		tmp = a * (x * -b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (t * -x)
    if (y <= (-33000000000000.0d0)) then
        tmp = t_1
    else if (y <= 2d-116) then
        tmp = x
    else if (y <= 4.5d+161) then
        tmp = a * (x * -b)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y * (t * -x);
	double tmp;
	if (y <= -33000000000000.0) {
		tmp = t_1;
	} else if (y <= 2e-116) {
		tmp = x;
	} else if (y <= 4.5e+161) {
		tmp = a * (x * -b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y * (t * -x)
	tmp = 0
	if y <= -33000000000000.0:
		tmp = t_1
	elif y <= 2e-116:
		tmp = x
	elif y <= 4.5e+161:
		tmp = a * (x * -b)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y * Float64(t * Float64(-x)))
	tmp = 0.0
	if (y <= -33000000000000.0)
		tmp = t_1;
	elseif (y <= 2e-116)
		tmp = x;
	elseif (y <= 4.5e+161)
		tmp = Float64(a * Float64(x * Float64(-b)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y * (t * -x);
	tmp = 0.0;
	if (y <= -33000000000000.0)
		tmp = t_1;
	elseif (y <= 2e-116)
		tmp = x;
	elseif (y <= 4.5e+161)
		tmp = a * (x * -b);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(t * (-x)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -33000000000000.0], t$95$1, If[LessEqual[y, 2e-116], x, If[LessEqual[y, 4.5e+161], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.3e13 or 4.49999999999999992e161 < y

   1. Initial program 98.8%

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

   2. Taylor expanded in t around inf 75.5%

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

      1. mul-1-neg 75.5%

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

      2. *-commutative 75.5%

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

   4. Simplified 75.5%

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

   5. Taylor expanded in y around 0 26.9%

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

      1. neg-mul-1 26.9%

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

      2. distribute-rgt-neg-in 26.9%

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

   7. Simplified 26.9%

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

   8. Taylor expanded in t around inf 22.3%

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

      1. mul-1-neg 22.3%

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

      2. associate-*r* 25.7%

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

      3. *-commutative 25.7%

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

      4. *-commutative 25.7%

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

   10. Simplified 25.7%

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

   if -3.3e13 < y < 2e-116

   1. Initial program 95.4%

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

   2. Taylor expanded in t around inf 57.3%

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

      1. mul-1-neg 57.3%

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

      2. *-commutative 57.3%

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

   4. Simplified 57.3%

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

   5. Taylor expanded in y around 0 41.3%

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

   if 2e-116 < y < 4.49999999999999992e161

   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 b around inf 54.2%

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

      1. mul-1-neg 54.2%

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

      2. distribute-rgt-neg-out 54.2%

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

   4. Simplified 54.2%

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

   5. Taylor expanded in a around 0 18.8%

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

      1. mul-1-neg 18.8%

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

      2. unsub-neg 18.8%

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

   7. Simplified 18.8%

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

   8. Taylor expanded in a around inf 28.7%

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

      1. mul-1-neg 28.7%

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

      2. distribute-rgt-neg-in 28.7%

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

      3. *-commutative 28.7%

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

   10. Simplified 28.7%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -33000000000000:\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-116}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+161}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 10: 28.2% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7600000000000:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-116}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+161}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -7600000000000.0)
   (* x (* t (- y)))
   (if (<= y 3.4e-116)
     x
     (if (<= y 1.35e+161) (* a (* x (- b))) (* y (* t (- x)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7600000000000.0) {
		tmp = x * (t * -y);
	} else if (y <= 3.4e-116) {
		tmp = x;
	} else if (y <= 1.35e+161) {
		tmp = a * (x * -b);
	} else {
		tmp = y * (t * -x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-7600000000000.0d0)) then
        tmp = x * (t * -y)
    else if (y <= 3.4d-116) then
        tmp = x
    else if (y <= 1.35d+161) then
        tmp = a * (x * -b)
    else
        tmp = y * (t * -x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7600000000000.0) {
		tmp = x * (t * -y);
	} else if (y <= 3.4e-116) {
		tmp = x;
	} else if (y <= 1.35e+161) {
		tmp = a * (x * -b);
	} else {
		tmp = y * (t * -x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -7600000000000.0:
		tmp = x * (t * -y)
	elif y <= 3.4e-116:
		tmp = x
	elif y <= 1.35e+161:
		tmp = a * (x * -b)
	else:
		tmp = y * (t * -x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -7600000000000.0)
		tmp = Float64(x * Float64(t * Float64(-y)));
	elseif (y <= 3.4e-116)
		tmp = x;
	elseif (y <= 1.35e+161)
		tmp = Float64(a * Float64(x * Float64(-b)));
	else
		tmp = Float64(y * Float64(t * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -7600000000000.0)
		tmp = x * (t * -y);
	elseif (y <= 3.4e-116)
		tmp = x;
	elseif (y <= 1.35e+161)
		tmp = a * (x * -b);
	else
		tmp = y * (t * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -7600000000000.0], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e-116], x, If[LessEqual[y, 1.35e+161], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * (-x)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.6e12

   1. Initial program 98.2%

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

   2. Taylor expanded in t around inf 77.3%

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

      1. mul-1-neg 77.3%

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

      2. *-commutative 77.3%

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

   4. Simplified 77.3%

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

   5. Taylor expanded in y around 0 26.9%

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

      1. neg-mul-1 26.9%

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

      2. distribute-rgt-neg-in 26.9%

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

   7. Simplified 26.9%

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

   8. Taylor expanded in t around inf 20.0%

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

      1. mul-1-neg 20.0%

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

      2. *-commutative 20.0%

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

      3. associate-*l* 26.7%

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

      4. distribute-rgt-neg-out 26.7%

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

      5. distribute-rgt-neg-in 26.7%

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

   10. Simplified 26.7%

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

   if -7.6e12 < y < 3.39999999999999992e-116

   1. Initial program 95.4%

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

   2. Taylor expanded in t around inf 57.3%

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

      1. mul-1-neg 57.3%

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

      2. *-commutative 57.3%

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

   4. Simplified 57.3%

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

   5. Taylor expanded in y around 0 41.3%

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

   if 3.39999999999999992e-116 < y < 1.3499999999999999e161

   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 b around inf 54.2%

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

      1. mul-1-neg 54.2%

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

      2. distribute-rgt-neg-out 54.2%

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

   4. Simplified 54.2%

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

   5. Taylor expanded in a around 0 18.8%

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

      1. mul-1-neg 18.8%

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

      2. unsub-neg 18.8%

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

   7. Simplified 18.8%

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

   8. Taylor expanded in a around inf 28.7%

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

      1. mul-1-neg 28.7%

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

      2. distribute-rgt-neg-in 28.7%

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

      3. *-commutative 28.7%

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

   10. Simplified 28.7%

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

   if 1.3499999999999999e161 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 71.9%

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

      1. mul-1-neg 71.9%

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

      2. *-commutative 71.9%

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

   4. Simplified 71.9%

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

   5. Taylor expanded in y around 0 27.1%

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

      1. neg-mul-1 27.1%

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

      2. distribute-rgt-neg-in 27.1%

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

   7. Simplified 27.1%

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

   8. Taylor expanded in t around inf 27.0%

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

      1. mul-1-neg 27.0%

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

      2. associate-*r* 37.2%

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

      3. *-commutative 37.2%

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

      4. *-commutative 37.2%

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

   10. Simplified 37.2%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7600000000000:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-116}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+161}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 11: 33.3% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -26000000000000:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -26000000000000.0)
   (* x (* t (- y)))
   (if (<= y 3.2e-10) (* x (- 1.0 (* a b))) (* y (* t (- x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -26000000000000.0) {
		tmp = x * (t * -y);
	} else if (y <= 3.2e-10) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = y * (t * -x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-26000000000000.0d0)) then
        tmp = x * (t * -y)
    else if (y <= 3.2d-10) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = y * (t * -x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -26000000000000.0) {
		tmp = x * (t * -y);
	} else if (y <= 3.2e-10) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = y * (t * -x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -26000000000000.0:
		tmp = x * (t * -y)
	elif y <= 3.2e-10:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = y * (t * -x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -26000000000000.0)
		tmp = Float64(x * Float64(t * Float64(-y)));
	elseif (y <= 3.2e-10)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(y * Float64(t * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -26000000000000.0)
		tmp = x * (t * -y);
	elseif (y <= 3.2e-10)
		tmp = x * (1.0 - (a * b));
	else
		tmp = y * (t * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -26000000000000.0], N[(x * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e-10], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * (-x)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6e13

   1. Initial program 98.2%

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

   2. Taylor expanded in t around inf 77.3%

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

      1. mul-1-neg 77.3%

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

      2. *-commutative 77.3%

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

   4. Simplified 77.3%

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

   5. Taylor expanded in y around 0 26.9%

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

      1. neg-mul-1 26.9%

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

      2. distribute-rgt-neg-in 26.9%

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

   7. Simplified 26.9%

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

   8. Taylor expanded in t around inf 20.0%

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

      1. mul-1-neg 20.0%

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

      2. *-commutative 20.0%

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

      3. associate-*l* 26.7%

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

      4. distribute-rgt-neg-out 26.7%

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

      5. distribute-rgt-neg-in 26.7%

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

   10. Simplified 26.7%

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

   if -2.6e13 < y < 3.19999999999999981e-10

   1. Initial program 96.0%

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

   2. Taylor expanded in b around inf 77.3%

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

      1. mul-1-neg 77.3%

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

      2. distribute-rgt-neg-out 77.3%

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

   4. Simplified 77.3%

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

   5. Taylor expanded in a around 0 48.7%

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

      1. mul-1-neg 48.7%

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

      2. unsub-neg 48.7%

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

   7. Simplified 48.7%

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

   if 3.19999999999999981e-10 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 62.5%

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

      1. mul-1-neg 62.5%

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

      2. *-commutative 62.5%

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

   4. Simplified 62.5%

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

   5. Taylor expanded in y around 0 21.1%

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

      1. neg-mul-1 21.1%

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

      2. distribute-rgt-neg-in 21.1%

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

   7. Simplified 21.1%

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

   8. Taylor expanded in t around inf 24.3%

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

      1. mul-1-neg 24.3%

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

      2. associate-*r* 29.3%

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

      3. *-commutative 29.3%

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

      4. *-commutative 29.3%

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

   10. Simplified 29.3%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -26000000000000:\\ \;\;\;\;x \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 12: 33.2% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -42000000000000:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -42000000000000.0)
   (* x (- 1.0 (* y t)))
   (if (<= y 4.6e-10) (* x (- 1.0 (* a b))) (* y (* t (- x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -42000000000000.0) {
		tmp = x * (1.0 - (y * t));
	} else if (y <= 4.6e-10) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = y * (t * -x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-42000000000000.0d0)) then
        tmp = x * (1.0d0 - (y * t))
    else if (y <= 4.6d-10) then
        tmp = x * (1.0d0 - (a * b))
    else
        tmp = y * (t * -x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -42000000000000.0) {
		tmp = x * (1.0 - (y * t));
	} else if (y <= 4.6e-10) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = y * (t * -x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -42000000000000.0:
		tmp = x * (1.0 - (y * t))
	elif y <= 4.6e-10:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = y * (t * -x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -42000000000000.0)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	elseif (y <= 4.6e-10)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(y * Float64(t * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -42000000000000.0)
		tmp = x * (1.0 - (y * t));
	elseif (y <= 4.6e-10)
		tmp = x * (1.0 - (a * b));
	else
		tmp = y * (t * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -42000000000000.0], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e-10], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * (-x)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.2e13

   1. Initial program 98.2%

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

   2. Taylor expanded in t around inf 77.3%

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

      1. mul-1-neg 77.3%

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

      2. *-commutative 77.3%

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

   4. Simplified 77.3%

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

   5. Taylor expanded in y around 0 26.9%

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

      1. neg-mul-1 26.9%

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

      2. distribute-rgt-neg-in 26.9%

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

   7. Simplified 26.9%

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

   8. Taylor expanded in x around 0 26.9%

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

      1. mul-1-neg 26.9%

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

      2. *-commutative 26.9%

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

      3. unsub-neg 26.9%

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

   10. Simplified 26.9%

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

   if -4.2e13 < y < 4.60000000000000014e-10

   1. Initial program 96.0%

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

   2. Taylor expanded in b around inf 77.3%

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

      1. mul-1-neg 77.3%

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

      2. distribute-rgt-neg-out 77.3%

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

   4. Simplified 77.3%

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

   5. Taylor expanded in a around 0 48.7%

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

      1. mul-1-neg 48.7%

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

      2. unsub-neg 48.7%

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

   7. Simplified 48.7%

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

   if 4.60000000000000014e-10 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 62.5%

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

      1. mul-1-neg 62.5%

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

      2. *-commutative 62.5%

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

   4. Simplified 62.5%

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

   5. Taylor expanded in y around 0 21.1%

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

      1. neg-mul-1 21.1%

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

      2. distribute-rgt-neg-in 21.1%

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

   7. Simplified 21.1%

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

   8. Taylor expanded in t around inf 24.3%

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

      1. mul-1-neg 24.3%

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

      2. associate-*r* 29.3%

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

      3. *-commutative 29.3%

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

      4. *-commutative 29.3%

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

   10. Simplified 29.3%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -42000000000000:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 13: 26.9% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-38} \lor \neg \left(y \leq 3.4 \cdot 10^{-116}\right):\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4.2e-38) (not (<= y 3.4e-116))) (* (* a b) (- x)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.2e-38) || !(y <= 3.4e-116)) {
		tmp = (a * b) * -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-4.2d-38)) .or. (.not. (y <= 3.4d-116))) then
        tmp = (a * b) * -x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.2e-38) || !(y <= 3.4e-116)) {
		tmp = (a * b) * -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -4.2e-38) or not (y <= 3.4e-116):
		tmp = (a * b) * -x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -4.2e-38) || !(y <= 3.4e-116))
		tmp = Float64(Float64(a * b) * Float64(-x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -4.2e-38) || ~((y <= 3.4e-116)))
		tmp = (a * b) * -x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -4.2e-38], N[Not[LessEqual[y, 3.4e-116]], $MachinePrecision]], N[(N[(a * b), $MachinePrecision] * (-x)), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -4.20000000000000026e-38 or 3.39999999999999992e-116 < y

   1. Initial program 98.7%

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

   2. Taylor expanded in b around inf 44.0%

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

      1. mul-1-neg 44.0%

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

      2. distribute-rgt-neg-out 44.0%

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

   4. Simplified 44.0%

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

   5. Taylor expanded in a around 0 15.5%

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

      1. mul-1-neg 15.5%

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

      2. unsub-neg 15.5%

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

   7. Simplified 15.5%

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

   8. Taylor expanded in a around inf 20.4%

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

      1. mul-1-neg 20.4%

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

      2. associate-*r* 21.3%

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

      3. *-commutative 21.3%

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

   10. Simplified 21.3%

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

   if -4.20000000000000026e-38 < y < 3.39999999999999992e-116

   1. Initial program 95.9%

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

   2. Taylor expanded in t around inf 56.4%

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

      1. mul-1-neg 56.4%

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

      2. *-commutative 56.4%

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

   4. Simplified 56.4%

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

   5. Taylor expanded in y around 0 44.2%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-38} \lor \neg \left(y \leq 3.4 \cdot 10^{-116}\right):\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14: 26.8% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7200000000000 \lor \neg \left(y \leq 6.8 \cdot 10^{-78}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7200000000000.0) (not (<= y 6.8e-78))) (* y (* t (- x))) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7200000000000.0) || !(y <= 6.8e-78)) {
		tmp = y * (t * -x);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-7200000000000.0d0)) .or. (.not. (y <= 6.8d-78))) then
        tmp = y * (t * -x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7200000000000.0) || !(y <= 6.8e-78)) {
		tmp = y * (t * -x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -7200000000000.0) or not (y <= 6.8e-78):
		tmp = y * (t * -x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -7200000000000.0) || !(y <= 6.8e-78))
		tmp = Float64(y * Float64(t * Float64(-x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -7200000000000.0) || ~((y <= 6.8e-78)))
		tmp = y * (t * -x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -7200000000000.0], N[Not[LessEqual[y, 6.8e-78]], $MachinePrecision]], N[(y * N[(t * (-x)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -7.2e12 or 6.80000000000000023e-78 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in t around inf 69.9%

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

      1. mul-1-neg 69.9%

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

      2. *-commutative 69.9%

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

   4. Simplified 69.9%

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

   5. Taylor expanded in y around 0 24.1%

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

      1. neg-mul-1 24.1%

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

      2. distribute-rgt-neg-in 24.1%

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

   7. Simplified 24.1%

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

   8. Taylor expanded in t around inf 22.3%

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

      1. mul-1-neg 22.3%

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

      2. associate-*r* 24.9%

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

      3. *-commutative 24.9%

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

      4. *-commutative 24.9%

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

   10. Simplified 24.9%

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

   if -7.2e12 < y < 6.80000000000000023e-78

   1. Initial program 95.6%

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

   2. Taylor expanded in t around inf 56.7%

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

      1. mul-1-neg 56.7%

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

      2. *-commutative 56.7%

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

   4. Simplified 56.7%

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

   5. Taylor expanded in y around 0 39.7%

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

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7200000000000 \lor \neg \left(y \leq 6.8 \cdot 10^{-78}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15: 19.4% 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 97.6%

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

2. Taylor expanded in t around inf 64.0%

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

   1. mul-1-neg 64.0%

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

   2. *-commutative 64.0%

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

4. Simplified 64.0%

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

5. Taylor expanded in y around 0 20.2%

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

6. Final simplification 20.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023319 
(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))))))