Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Percentage Accurate: 61.3% → 90.7%

Time: 18.8s

Alternatives: 14

Speedup: 18.6×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = ((x * y) * z) / sqrt(((z * z) - (t * a)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = ((x * y) * z) / sqrt(((z * z) - (t * a)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-70}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.7e+86)
   (* y (- x))
   (if (<= z 1.1e-70)
     (* y (/ (* z x) (sqrt (- (* z z) (* t a)))))
     (/ (* y x) (sqrt (- 1.0 (/ a (/ (* z z) t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.7e+86) {
		tmp = y * -x;
	} else if (z <= 1.1e-70) {
		tmp = y * ((z * x) / sqrt(((z * z) - (t * a))));
	} else {
		tmp = (y * x) / sqrt((1.0 - (a / ((z * z) / t))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-6.7d+86)) then
        tmp = y * -x
    else if (z <= 1.1d-70) then
        tmp = y * ((z * x) / sqrt(((z * z) - (t * a))))
    else
        tmp = (y * x) / sqrt((1.0d0 - (a / ((z * z) / t))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.7e+86) {
		tmp = y * -x;
	} else if (z <= 1.1e-70) {
		tmp = y * ((z * x) / Math.sqrt(((z * z) - (t * a))));
	} else {
		tmp = (y * x) / Math.sqrt((1.0 - (a / ((z * z) / t))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.7e+86:
		tmp = y * -x
	elif z <= 1.1e-70:
		tmp = y * ((z * x) / math.sqrt(((z * z) - (t * a))))
	else:
		tmp = (y * x) / math.sqrt((1.0 - (a / ((z * z) / t))))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.7e+86)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.1e-70)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(t * a)))));
	else
		tmp = Float64(Float64(y * x) / sqrt(Float64(1.0 - Float64(a / Float64(Float64(z * z) / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.7e+86)
		tmp = y * -x;
	elseif (z <= 1.1e-70)
		tmp = y * ((z * x) / sqrt(((z * z) - (t * a))));
	else
		tmp = (y * x) / sqrt((1.0 - (a / ((z * z) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.7e+86], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.1e-70], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.69999999999999968e86

   1. Initial program 29.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 29.8%

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

      2. associate-*l* 25.9%

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

      3. associate-*r/ 26.0%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 26.0%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 96.5%

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

      1. neg-mul-1 96.5%

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

   6. Simplified 96.5%

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

   if -6.69999999999999968e86 < z < 1.0999999999999999e-70

   1. Initial program 73.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 73.1%

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

      2. associate-*l* 77.3%

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

      3. associate-*r/ 80.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   if 1.0999999999999999e-70 < z

   1. Initial program 56.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 60.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 60.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 60.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot \sqrt{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}}} \]

      2. sqrt-unprod 60.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{\frac{\sqrt{z \cdot z - t \cdot a}}{z} \cdot \frac{\sqrt{z \cdot z - t \cdot a}}{z}}}} \]

      3. frac-times 55.3%

         \[\leadsto \frac{x \cdot y}{\sqrt{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}}{z \cdot z}}}} \]

      4. add-sqr-sqrt 55.3%

         \[\leadsto \frac{x \cdot y}{\sqrt{\frac{\color{blue}{z \cdot z - t \cdot a}}{z \cdot z}}} \]

   5. Applied egg-rr 55.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{\frac{z \cdot z - t \cdot a}{z \cdot z}}}} \]
   6. Step-by-step derivation

      1. div-sub 55.3%

         \[\leadsto \frac{x \cdot y}{\sqrt{\color{blue}{\frac{z \cdot z}{z \cdot z} - \frac{t \cdot a}{z \cdot z}}}} \]

      2. *-inverses 97.0%

         \[\leadsto \frac{x \cdot y}{\sqrt{\color{blue}{1} - \frac{t \cdot a}{z \cdot z}}} \]

      3. *-commutative 97.0%

         \[\leadsto \frac{x \cdot y}{\sqrt{1 - \frac{\color{blue}{a \cdot t}}{z \cdot z}}} \]

      4. associate-/l* 98.9%

         \[\leadsto \frac{x \cdot y}{\sqrt{1 - \color{blue}{\frac{a}{\frac{z \cdot z}{t}}}}} \]

   7. Simplified 98.9%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-70}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\sqrt{1 - \frac{a}{\frac{z \cdot z}{t}}}}\\ \end{array} \]

Alternative 2: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.2e+86)
   (* y (- x))
   (if (<= z 4.3e+24)
     (* y (/ (* z x) (sqrt (- (* z z) (* t a)))))
     (/ (* y x) (+ 1.0 (* (/ a (/ (* z z) t)) -0.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e+86) {
		tmp = y * -x;
	} else if (z <= 4.3e+24) {
		tmp = y * ((z * x) / sqrt(((z * z) - (t * a))));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-5.2d+86)) then
        tmp = y * -x
    else if (z <= 4.3d+24) then
        tmp = y * ((z * x) / sqrt(((z * z) - (t * a))))
    else
        tmp = (y * x) / (1.0d0 + ((a / ((z * z) / t)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e+86) {
		tmp = y * -x;
	} else if (z <= 4.3e+24) {
		tmp = y * ((z * x) / Math.sqrt(((z * z) - (t * a))));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.2e+86:
		tmp = y * -x
	elif z <= 4.3e+24:
		tmp = y * ((z * x) / math.sqrt(((z * z) - (t * a))))
	else:
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.2e+86)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4.3e+24)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(t * a)))));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(Float64(a / Float64(Float64(z * z) / t)) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.2e+86)
		tmp = y * -x;
	elseif (z <= 4.3e+24)
		tmp = y * ((z * x) / sqrt(((z * z) - (t * a))));
	else
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.2e+86], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4.3e+24], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.1999999999999995e86

   1. Initial program 29.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 29.8%

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

      2. associate-*l* 25.9%

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

      3. associate-*r/ 26.0%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 26.0%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 96.5%

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

      1. neg-mul-1 96.5%

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

   6. Simplified 96.5%

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

   if -5.1999999999999995e86 < z < 4.29999999999999987e24

   1. Initial program 76.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 76.8%

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

      2. associate-*l* 80.2%

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

      3. associate-*r/ 82.6%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 82.6%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   if 4.29999999999999987e24 < z

   1. Initial program 45.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 51.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 51.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   4. Taylor expanded in z around inf 95.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 95.0%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{\color{blue}{z \cdot z}}} \]

      2. associate-/l* 95.1%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \color{blue}{\frac{a}{\frac{z \cdot z}{t}}}} \]

   6. Simplified 95.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 3: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-69}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-67}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7e-69)
   (* y (- x))
   (if (<= z 2.9e-67)
     (* y (/ (* z x) (sqrt (* t (- a)))))
     (/ (* y x) (+ 1.0 (* (/ a (/ (* z z) t)) -0.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e-69) {
		tmp = y * -x;
	} else if (z <= 2.9e-67) {
		tmp = y * ((z * x) / sqrt((t * -a)));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-7d-69)) then
        tmp = y * -x
    else if (z <= 2.9d-67) then
        tmp = y * ((z * x) / sqrt((t * -a)))
    else
        tmp = (y * x) / (1.0d0 + ((a / ((z * z) / t)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e-69) {
		tmp = y * -x;
	} else if (z <= 2.9e-67) {
		tmp = y * ((z * x) / Math.sqrt((t * -a)));
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7e-69:
		tmp = y * -x
	elif z <= 2.9e-67:
		tmp = y * ((z * x) / math.sqrt((t * -a)))
	else:
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7e-69)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.9e-67)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(Float64(a / Float64(Float64(z * z) / t)) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7e-69)
		tmp = y * -x;
	elseif (z <= 2.9e-67)
		tmp = y * ((z * x) / sqrt((t * -a)));
	else
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7e-69], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.9e-67], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.0000000000000003e-69

   1. Initial program 45.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 45.3%

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

      2. associate-*l* 42.8%

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

      3. associate-*r/ 44.9%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 44.9%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 82.0%

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

      1. neg-mul-1 82.0%

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

   6. Simplified 82.0%

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

   if -7.0000000000000003e-69 < z < 2.90000000000000005e-67

   1. Initial program 73.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 73.4%

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

      2. associate-*l* 78.9%

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

      3. associate-*r/ 80.7%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 80.7%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around 0 68.4%

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

      1. associate-*r* 68.4%

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

      2. neg-mul-1 68.4%

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

   6. Simplified 68.4%

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

   if 2.90000000000000005e-67 < z

   1. Initial program 55.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 59.8%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 59.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   4. Taylor expanded in z around inf 93.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 93.5%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{\color{blue}{z \cdot z}}} \]

      2. associate-/l* 93.6%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \color{blue}{\frac{a}{\frac{z \cdot z}{t}}}} \]

   6. Simplified 93.6%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-69}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-67}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 4: 76.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-211}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-92}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.7e-211)
   (* y (- x))
   (if (<= z 1.22e-92)
     (* y (/ (* z x) (+ z (* -0.5 (/ (* t a) z)))))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.7e-211) {
		tmp = y * -x;
	} else if (z <= 1.22e-92) {
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-1.7d-211)) then
        tmp = y * -x
    else if (z <= 1.22d-92) then
        tmp = y * ((z * x) / (z + ((-0.5d0) * ((t * a) / z))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.7e-211) {
		tmp = y * -x;
	} else if (z <= 1.22e-92) {
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.7e-211:
		tmp = y * -x
	elif z <= 1.22e-92:
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.7e-211)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.22e-92)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z)))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.7e-211)
		tmp = y * -x;
	elseif (z <= 1.22e-92)
		tmp = y * ((z * x) / (z + (-0.5 * ((t * a) / z))));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.7e-211], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.22e-92], N[(y * N[(N[(z * x), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7e-211

   1. Initial program 48.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 48.1%

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

      2. associate-*l* 47.9%

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

      3. associate-*r/ 49.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 49.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 72.3%

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

      1. neg-mul-1 72.3%

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

   6. Simplified 72.3%

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

   if -1.7e-211 < z < 1.21999999999999994e-92

   1. Initial program 75.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 75.8%

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

      2. associate-*l* 81.6%

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

      3. associate-*r/ 85.3%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 85.3%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 43.7%

      \[\leadsto y \cdot \frac{x \cdot z}{\color{blue}{z + -0.5 \cdot \frac{a \cdot t}{z}}} \]

   if 1.21999999999999994e-92 < z

   1. Initial program 59.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 59.0%

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

      2. associate-*l* 57.4%

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

      3. associate-*r/ 60.4%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 60.4%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 90.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-211}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-92}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 76.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-211}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-93}:\\ \;\;\;\;\frac{y}{\frac{z + -0.5 \cdot \frac{t \cdot a}{z}}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.6e-211)
   (* y (- x))
   (if (<= z 9.6e-93) (/ y (/ (+ z (* -0.5 (/ (* t a) z))) (* z x))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e-211) {
		tmp = y * -x;
	} else if (z <= 9.6e-93) {
		tmp = y / ((z + (-0.5 * ((t * a) / z))) / (z * x));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-1.6d-211)) then
        tmp = y * -x
    else if (z <= 9.6d-93) then
        tmp = y / ((z + ((-0.5d0) * ((t * a) / z))) / (z * x))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e-211) {
		tmp = y * -x;
	} else if (z <= 9.6e-93) {
		tmp = y / ((z + (-0.5 * ((t * a) / z))) / (z * x));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.6e-211:
		tmp = y * -x
	elif z <= 9.6e-93:
		tmp = y / ((z + (-0.5 * ((t * a) / z))) / (z * x))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.6e-211)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 9.6e-93)
		tmp = Float64(y / Float64(Float64(z + Float64(-0.5 * Float64(Float64(t * a) / z))) / Float64(z * x)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.6e-211)
		tmp = y * -x;
	elseif (z <= 9.6e-93)
		tmp = y / ((z + (-0.5 * ((t * a) / z))) / (z * x));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.6e-211], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 9.6e-93], N[(y / N[(N[(z + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.59999999999999993e-211

   1. Initial program 48.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 48.1%

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

      2. associate-*l* 47.9%

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

      3. associate-*r/ 49.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 49.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 72.3%

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

      1. neg-mul-1 72.3%

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

   6. Simplified 72.3%

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

   if -1.59999999999999993e-211 < z < 9.6000000000000004e-93

   1. Initial program 75.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 75.8%

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

      2. associate-*l* 81.6%

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

      3. associate-*r/ 85.3%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 85.3%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]
   4. Step-by-step derivation

      1. clear-num 85.3%

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

      2. un-div-inv 85.3%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

      3. *-commutative 85.3%

         \[\leadsto \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{\color{blue}{z \cdot x}}} \]

   5. Applied egg-rr 85.3%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot x}}} \]

   6. Taylor expanded in z around inf 43.7%

      \[\leadsto \frac{y}{\frac{\color{blue}{z + -0.5 \cdot \frac{a \cdot t}{z}}}{z \cdot x}} \]

   if 9.6000000000000004e-93 < z

   1. Initial program 59.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 59.0%

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

      2. associate-*l* 57.4%

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

      3. associate-*r/ 60.4%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 60.4%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 90.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-211}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-93}:\\ \;\;\;\;\frac{y}{\frac{z + -0.5 \cdot \frac{t \cdot a}{z}}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6: 75.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-209}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot \frac{x \cdot \left(z \cdot y\right)}{t \cdot \frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.95e-209)
   (* y (- x))
   (if (<= z 2.5e-162) (* 2.0 (/ (* x (* z y)) (* t (/ a z)))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.95e-209) {
		tmp = y * -x;
	} else if (z <= 2.5e-162) {
		tmp = 2.0 * ((x * (z * y)) / (t * (a / z)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-1.95d-209)) then
        tmp = y * -x
    else if (z <= 2.5d-162) then
        tmp = 2.0d0 * ((x * (z * y)) / (t * (a / z)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.95e-209) {
		tmp = y * -x;
	} else if (z <= 2.5e-162) {
		tmp = 2.0 * ((x * (z * y)) / (t * (a / z)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.95e-209:
		tmp = y * -x
	elif z <= 2.5e-162:
		tmp = 2.0 * ((x * (z * y)) / (t * (a / z)))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.95e-209)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.5e-162)
		tmp = Float64(2.0 * Float64(Float64(x * Float64(z * y)) / Float64(t * Float64(a / z))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.95e-209)
		tmp = y * -x;
	elseif (z <= 2.5e-162)
		tmp = 2.0 * ((x * (z * y)) / (t * (a / z)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.95e-209], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.5e-162], N[(2.0 * N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.95e-209

   1. Initial program 48.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 48.6%

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

      2. associate-*l* 48.4%

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

      3. associate-*r/ 49.6%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 49.6%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 72.9%

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

      1. neg-mul-1 72.9%

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

   6. Simplified 72.9%

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

   if -1.95e-209 < z < 2.50000000000000007e-162

   1. Initial program 74.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 71.1%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   3. Simplified 71.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   4. Taylor expanded in z around -inf 40.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}} \cdot z \]

   5. Taylor expanded in a around inf 37.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{0.5 \cdot \frac{a \cdot t}{z}}} \cdot z \]
   6. Step-by-step derivation

      1. associate-*l/ 37.3%

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

      2. frac-2neg 37.3%

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

      3. associate-/l* 40.4%

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

   7. Applied egg-rr 40.4%

      \[\leadsto \color{blue}{\frac{-\left(x \cdot y\right) \cdot z}{-0.5 \cdot \frac{a}{\frac{z}{t}}}} \]
   8. Step-by-step derivation

      1. neg-mul-1 40.4%

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

      2. distribute-lft-neg-in 40.4%

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

      3. metadata-eval 40.4%

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

      4. times-frac 40.4%

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

      5. metadata-eval 40.4%

         \[\leadsto \color{blue}{2} \cdot \frac{\left(x \cdot y\right) \cdot z}{\frac{a}{\frac{z}{t}}} \]

      6. associate-*l* 40.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{x \cdot \left(y \cdot z\right)}}{\frac{a}{\frac{z}{t}}} \]

      7. *-commutative 40.6%

         \[\leadsto 2 \cdot \frac{x \cdot \color{blue}{\left(z \cdot y\right)}}{\frac{a}{\frac{z}{t}}} \]

      8. associate-/r/ 40.5%

         \[\leadsto 2 \cdot \frac{x \cdot \left(z \cdot y\right)}{\color{blue}{\frac{a}{z} \cdot t}} \]

   9. Simplified 40.5%

      \[\leadsto \color{blue}{2 \cdot \frac{x \cdot \left(z \cdot y\right)}{\frac{a}{z} \cdot t}} \]

   if 2.50000000000000007e-162 < z

   1. Initial program 60.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 60.6%

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

      2. associate-*l* 60.8%

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

      3. associate-*r/ 63.8%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 85.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-209}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot \frac{x \cdot \left(z \cdot y\right)}{t \cdot \frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7: 75.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-209}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;z \cdot \left(2 \cdot \frac{\frac{y \cdot \left(z \cdot x\right)}{a}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.6e-209)
   (* y (- x))
   (if (<= z 3.5e-160) (* z (* 2.0 (/ (/ (* y (* z x)) a) t))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e-209) {
		tmp = y * -x;
	} else if (z <= 3.5e-160) {
		tmp = z * (2.0 * (((y * (z * x)) / a) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-2.6d-209)) then
        tmp = y * -x
    else if (z <= 3.5d-160) then
        tmp = z * (2.0d0 * (((y * (z * x)) / a) / t))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e-209) {
		tmp = y * -x;
	} else if (z <= 3.5e-160) {
		tmp = z * (2.0 * (((y * (z * x)) / a) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.6e-209:
		tmp = y * -x
	elif z <= 3.5e-160:
		tmp = z * (2.0 * (((y * (z * x)) / a) / t))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.6e-209)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.5e-160)
		tmp = Float64(z * Float64(2.0 * Float64(Float64(Float64(y * Float64(z * x)) / a) / t)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.6e-209)
		tmp = y * -x;
	elseif (z <= 3.5e-160)
		tmp = z * (2.0 * (((y * (z * x)) / a) / t));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.6e-209], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.5e-160], N[(z * N[(2.0 * N[(N[(N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.59999999999999984e-209

   1. Initial program 48.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 48.6%

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

      2. associate-*l* 48.4%

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

      3. associate-*r/ 49.6%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 49.6%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 72.9%

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

      1. neg-mul-1 72.9%

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

   6. Simplified 72.9%

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

   if -2.59999999999999984e-209 < z < 3.5000000000000003e-160

   1. Initial program 74.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 71.1%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   3. Simplified 71.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   4. Taylor expanded in z around -inf 40.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}} \cdot z \]

   5. Taylor expanded in a around inf 37.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{0.5 \cdot \frac{a \cdot t}{z}}} \cdot z \]

   6. Taylor expanded in x around 0 37.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{y \cdot \left(z \cdot x\right)}{a \cdot t}\right)} \cdot z \]
   7. Step-by-step derivation

      1. associate-/r* 40.6%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{y \cdot \left(z \cdot x\right)}{a}}{t}}\right) \cdot z \]

      2. *-commutative 40.6%

         \[\leadsto \left(2 \cdot \frac{\frac{\color{blue}{\left(z \cdot x\right) \cdot y}}{a}}{t}\right) \cdot z \]

   8. Simplified 40.6%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\left(z \cdot x\right) \cdot y}{a}}{t}\right)} \cdot z \]

   if 3.5000000000000003e-160 < z

   1. Initial program 60.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 60.6%

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

      2. associate-*l* 60.8%

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

      3. associate-*r/ 63.8%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 85.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-209}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;z \cdot \left(2 \cdot \frac{\frac{y \cdot \left(z \cdot x\right)}{a}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8: 76.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-209}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.3e-209)
   (* y (- x))
   (/ (* y x) (+ 1.0 (* (/ a (/ (* z z) t)) -0.5)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.3e-209) {
		tmp = y * -x;
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-1.3d-209)) then
        tmp = y * -x
    else
        tmp = (y * x) / (1.0d0 + ((a / ((z * z) / t)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.3e-209) {
		tmp = y * -x;
	} else {
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.3e-209:
		tmp = y * -x
	else:
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.3e-209)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 + Float64(Float64(a / Float64(Float64(z * z) / t)) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.3e-209)
		tmp = y * -x;
	else
		tmp = (y * x) / (1.0 + ((a / ((z * z) / t)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.3e-209], N[(y * (-x)), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 + N[(N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.29999999999999992e-209

   1. Initial program 48.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 48.6%

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

      2. associate-*l* 48.4%

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

      3. associate-*r/ 49.6%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 49.6%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 72.9%

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

      1. neg-mul-1 72.9%

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

   6. Simplified 72.9%

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

   if -1.29999999999999992e-209 < z

   1. Initial program 64.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-/l* 67.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   4. Taylor expanded in z around inf 73.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 73.8%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{\color{blue}{z \cdot z}}} \]

      2. associate-/l* 74.6%

         \[\leadsto \frac{x \cdot y}{1 + -0.5 \cdot \color{blue}{\frac{a}{\frac{z \cdot z}{t}}}} \]

   6. Simplified 74.6%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-209}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 9: 75.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-215}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot \left(z \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.2e-215)
   (* y (- x))
   (if (<= z 3.2e-163) (/ 1.0 (/ z (* y (* z x)))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e-215) {
		tmp = y * -x;
	} else if (z <= 3.2e-163) {
		tmp = 1.0 / (z / (y * (z * x)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-5.2d-215)) then
        tmp = y * -x
    else if (z <= 3.2d-163) then
        tmp = 1.0d0 / (z / (y * (z * x)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e-215) {
		tmp = y * -x;
	} else if (z <= 3.2e-163) {
		tmp = 1.0 / (z / (y * (z * x)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.2e-215:
		tmp = y * -x
	elif z <= 3.2e-163:
		tmp = 1.0 / (z / (y * (z * x)))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.2e-215)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.2e-163)
		tmp = Float64(1.0 / Float64(z / Float64(y * Float64(z * x))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.2e-215)
		tmp = y * -x;
	elseif (z <= 3.2e-163)
		tmp = 1.0 / (z / (y * (z * x)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.2e-215], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.2e-163], N[(1.0 / N[(z / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2e-215

   1. Initial program 49.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 49.1%

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

      2. associate-*l* 48.8%

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

      3. associate-*r/ 50.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 71.0%

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

      1. neg-mul-1 71.0%

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

   6. Simplified 71.0%

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

   if -5.2e-215 < z < 3.19999999999999988e-163

   1. Initial program 74.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 70.5%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   3. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   4. Taylor expanded in z around -inf 19.6%

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

      1. mul-1-neg 19.6%

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

      2. associate-/l* 19.3%

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

      3. distribute-neg-frac 19.3%

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

   6. Simplified 19.3%

      \[\leadsto \color{blue}{\frac{-y}{\frac{z}{x}}} \cdot z \]
   7. Step-by-step derivation

      1. associate-*l/ 28.8%

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

   8. Applied egg-rr 28.8%

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

      1. clear-num 28.8%

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

      2. inv-pow 28.8%

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

      3. *-commutative 28.8%

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

      4. add-sqr-sqrt 18.7%

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

      5. sqrt-unprod 28.7%

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

      6. sqr-neg 28.7%

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

      7. sqrt-unprod 10.2%

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

      8. add-sqr-sqrt 28.9%

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

   10. Applied egg-rr 28.9%

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

       1. unpow-1 28.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{z}{x}}{z \cdot y}}} \]

       2. associate-/l/ 35.5%

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

       3. *-commutative 35.5%

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

       4. associate-*r* 38.5%

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

       5. *-commutative 38.5%

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

   12. Simplified 38.5%

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

   if 3.19999999999999988e-163 < z

   1. Initial program 60.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 60.9%

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

      2. associate-*l* 61.1%

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

      3. associate-*r/ 64.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 84.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-215}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot \left(z \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10: 74.8% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.7 \cdot 10^{-198}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-185}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.7e-198)
   (* y (- x))
   (if (<= z 3.2e-185) (/ (* z (* y x)) (- z)) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.7e-198) {
		tmp = y * -x;
	} else if (z <= 3.2e-185) {
		tmp = (z * (y * x)) / -z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-5.7d-198)) then
        tmp = y * -x
    else if (z <= 3.2d-185) then
        tmp = (z * (y * x)) / -z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.7e-198) {
		tmp = y * -x;
	} else if (z <= 3.2e-185) {
		tmp = (z * (y * x)) / -z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.7e-198:
		tmp = y * -x
	elif z <= 3.2e-185:
		tmp = (z * (y * x)) / -z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.7e-198)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.2e-185)
		tmp = Float64(Float64(z * Float64(y * x)) / Float64(-z));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.7e-198)
		tmp = y * -x;
	elseif (z <= 3.2e-185)
		tmp = (z * (y * x)) / -z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.7e-198], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.2e-185], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.70000000000000028e-198

   1. Initial program 49.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 49.0%

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

      2. associate-*l* 48.8%

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

      3. associate-*r/ 50.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 73.6%

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

      1. neg-mul-1 73.6%

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

   6. Simplified 73.6%

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

   if -5.70000000000000028e-198 < z < 3.1999999999999997e-185

   1. Initial program 72.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around -inf 34.9%

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

      1. neg-mul-1 34.9%

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

   4. Simplified 34.9%

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

   if 3.1999999999999997e-185 < z

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 61.1%

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

      2. associate-*l* 60.5%

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

      3. associate-*r/ 63.4%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 63.4%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 82.8%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.7 \cdot 10^{-198}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-185}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11: 74.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-221}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-184}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.5e-221)
   (* y (- x))
   (if (<= z 3.3e-184) (/ (* z (* y x)) z) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.5e-221) {
		tmp = y * -x;
	} else if (z <= 3.3e-184) {
		tmp = (z * (y * x)) / z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-4.5d-221)) then
        tmp = y * -x
    else if (z <= 3.3d-184) then
        tmp = (z * (y * x)) / z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.5e-221) {
		tmp = y * -x;
	} else if (z <= 3.3e-184) {
		tmp = (z * (y * x)) / z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.5e-221:
		tmp = y * -x
	elif z <= 3.3e-184:
		tmp = (z * (y * x)) / z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.5e-221)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.3e-184)
		tmp = Float64(Float64(z * Float64(y * x)) / z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.5e-221)
		tmp = y * -x;
	elseif (z <= 3.3e-184)
		tmp = (z * (y * x)) / z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.5e-221], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.3e-184], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.50000000000000026e-221

   1. Initial program 49.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 49.1%

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

      2. associate-*l* 48.8%

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

      3. associate-*r/ 50.1%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 71.0%

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

      1. neg-mul-1 71.0%

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

   6. Simplified 71.0%

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

   if -4.50000000000000026e-221 < z < 3.2999999999999997e-184

   1. Initial program 75.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf 38.7%

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

   if 3.2999999999999997e-184 < z

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 61.1%

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

      2. associate-*l* 60.5%

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

      3. associate-*r/ 63.4%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 63.4%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 82.8%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-221}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-184}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 12: 72.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-304}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{y \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.2e-304) (* y (- x)) (/ 1.0 (/ 1.0 (* y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e-304) {
		tmp = y * -x;
	} else {
		tmp = 1.0 / (1.0 / (y * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-5.2d-304)) then
        tmp = y * -x
    else
        tmp = 1.0d0 / (1.0d0 / (y * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e-304) {
		tmp = y * -x;
	} else {
		tmp = 1.0 / (1.0 / (y * x));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.2e-304:
		tmp = y * -x
	else:
		tmp = 1.0 / (1.0 / (y * x))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.2e-304)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(y * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.2e-304)
		tmp = y * -x;
	else
		tmp = 1.0 / (1.0 / (y * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.2e-304], N[(y * (-x)), $MachinePrecision], N[(1.0 / N[(1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.19999999999999993e-304

   1. Initial program 53.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 53.3%

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

      2. associate-*l* 54.7%

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

      3. associate-*r/ 55.8%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 64.9%

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

      1. neg-mul-1 64.9%

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

   6. Simplified 64.9%

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

   if -5.19999999999999993e-304 < z

   1. Initial program 61.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 61.4%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   3. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   4. Taylor expanded in z around -inf 12.6%

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

      1. mul-1-neg 12.6%

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

      2. associate-/l* 12.8%

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

      3. distribute-neg-frac 12.8%

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

   6. Simplified 12.8%

      \[\leadsto \color{blue}{\frac{-y}{\frac{z}{x}}} \cdot z \]
   7. Step-by-step derivation

      1. associate-*l/ 14.8%

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

   8. Applied egg-rr 14.8%

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

      1. clear-num 14.8%

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

      2. inv-pow 14.8%

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

      3. *-commutative 14.8%

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

      4. add-sqr-sqrt 6.6%

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

      5. sqrt-unprod 28.3%

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

      6. sqr-neg 28.3%

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

      7. sqrt-unprod 31.2%

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

      8. add-sqr-sqrt 58.0%

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

   10. Applied egg-rr 58.0%

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

       1. unpow-1 58.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{z}{x}}{z \cdot y}}} \]

       2. associate-/l/ 60.6%

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

       3. *-commutative 60.6%

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

       4. associate-*r* 65.3%

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

       5. *-commutative 65.3%

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

   12. Simplified 65.3%

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

   13. Taylor expanded in z around 0 76.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{y \cdot x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-304}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{y \cdot x}}\\ \end{array} \]

Alternative 13: 72.6% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-304}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.4e-304) (* y (- x)) (* y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.4e-304) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-3.4d-304)) then
        tmp = y * -x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.4e-304) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.4e-304:
		tmp = y * -x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.4e-304)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.4e-304)
		tmp = y * -x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.4e-304], N[(y * (-x)), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3999999999999999e-304

   1. Initial program 53.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 53.3%

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

      2. associate-*l* 54.7%

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

      3. associate-*r/ 55.8%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around -inf 64.9%

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

      1. neg-mul-1 64.9%

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

   6. Simplified 64.9%

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

   if -3.3999999999999999e-304 < z

   1. Initial program 61.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. *-commutative 61.7%

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

      2. associate-*l* 61.2%

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

      3. associate-*r/ 64.8%

         \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Simplified 64.8%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   4. Taylor expanded in z around inf 76.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-304}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 14: 42.8% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = y * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.7%

   \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
2. Step-by-step derivation

   1. *-commutative 57.7%

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

   2. associate-*l* 58.1%

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

   3. associate-*r/ 60.5%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

3. Simplified 60.5%

   \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

4. Taylor expanded in z around inf 45.3%

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

5. Final simplification 45.3%

   \[\leadsto y \cdot x \]

Developer target: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< z -3.1921305903852764e+46)
   (- (* y x))
   (if (< z 5.976268120920894e+90)
     (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z < (-3.1921305903852764d+46)) then
        tmp = -(y * x)
    else if (z < 5.976268120920894d+90) then
        tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (Math.sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z < -3.1921305903852764e+46:
		tmp = -(y * x)
	elif z < 5.976268120920894e+90:
		tmp = (x * z) / (math.sqrt(((z * z) - (a * t))) / y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z < -3.1921305903852764e+46)
		tmp = Float64(-Float64(y * x));
	elseif (z < 5.976268120920894e+90)
		tmp = Float64(Float64(x * z) / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z < -3.1921305903852764e+46)
		tmp = -(y * x);
	elseif (z < 5.976268120920894e+90)
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[z, -3.1921305903852764e+46], (-N[(y * x), $MachinePrecision]), If[Less[z, 5.976268120920894e+90], N[(N[(x * z), $MachinePrecision] / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023199 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))