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

Percentage Accurate: 61.2% → 90.9%

Time: 15.1s

Alternatives: 13

Speedup: 18.6×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 61.2% 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: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \frac{a}{z} \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-59}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - t_1}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ a z) (/ t z))))
   (if (<= z -7.2e+22)
     (/ (* x y) (fma 0.5 t_1 -1.0))
     (if (<= z 2.7e-59)
       (* y (/ (* z x) (sqrt (- (* z z) (* a t)))))
       (/ (* x y) (sqrt (- 1.0 t_1)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (a / z) * (t / z);
	double tmp;
	if (z <= -7.2e+22) {
		tmp = (x * y) / fma(0.5, t_1, -1.0);
	} else if (z <= 2.7e-59) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = (x * y) / sqrt((1.0 - t_1));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(a / z) * Float64(t / z))
	tmp = 0.0
	if (z <= -7.2e+22)
		tmp = Float64(Float64(x * y) / fma(0.5, t_1, -1.0));
	elseif (z <= 2.7e-59)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(Float64(x * y) / sqrt(Float64(1.0 - t_1)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.2e+22], N[(N[(x * y), $MachinePrecision] / N[(0.5 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.7e-59], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.2e22

   1. Initial program 35.7%

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

      1. associate-/l* 37.3%

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

   3. Simplified 37.3%

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

   4. Taylor expanded in z around -inf 88.3%

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

      1. fma-neg 88.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot t}{{z}^{2}}, -1\right)}} \]

      2. unpow2 88.3%

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

      3. times-frac 95.0%

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

      4. metadata-eval 95.0%

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

   6. Simplified 95.0%

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

   if -7.2e22 < z < 2.6999999999999999e-59

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-*l* 81.3%

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

      3. associate-*r/ 85.8%

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

   3. Simplified 85.8%

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

   if 2.6999999999999999e-59 < z

   1. Initial program 51.8%

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

      1. associate-/l* 53.2%

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

   3. Simplified 53.2%

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

      1. add-sqr-sqrt 53.2%

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

      2. sqrt-unprod 53.2%

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

      3. frac-times 45.2%

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

      4. add-sqr-sqrt 45.2%

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

   5. Applied egg-rr 45.2%

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

      1. div-sub 45.2%

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

      2. *-inverses 93.5%

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

      3. *-commutative 93.5%

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

      4. times-frac 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{z} \cdot \frac{t}{z}, -1\right)}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-59}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{z} \cdot \frac{t}{z}}}\\ \end{array} \]

Alternative 2: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+22}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \left(\frac{a}{z} \cdot \frac{t}{z}\right) \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9e+22)
   (- (* x y))
   (if (<= z 5e+83)
     (* y (/ (* z x) (sqrt (- (* z z) (* a t)))))
     (/ (* x y) (+ 1.0 (* (* (/ a z) (/ t z)) -0.5))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9e+22) {
		tmp = -(x * y);
	} else if (z <= 5e+83) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-9d+22)) then
        tmp = -(x * y)
    else if (z <= 5d+83) then
        tmp = y * ((z * x) / sqrt(((z * z) - (a * t))))
    else
        tmp = (x * y) / (1.0d0 + (((a / z) * (t / z)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9e+22) {
		tmp = -(x * y);
	} else if (z <= 5e+83) {
		tmp = y * ((z * x) / Math.sqrt(((z * z) - (a * t))));
	} else {
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -9e+22:
		tmp = -(x * y)
	elif z <= 5e+83:
		tmp = y * ((z * x) / math.sqrt(((z * z) - (a * t))))
	else:
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9e+22)
		tmp = Float64(-Float64(x * y));
	elseif (z <= 5e+83)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(Float64(Float64(a / z) * Float64(t / z)) * -0.5)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -9e+22)
		tmp = -(x * y);
	elseif (z <= 5e+83)
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	else
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9e+22], (-N[(x * y), $MachinePrecision]), If[LessEqual[z, 5e+83], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.9999999999999996e22

   1. Initial program 35.7%

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

      1. *-commutative 35.7%

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

      2. associate-*l* 34.2%

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

      3. associate-*r/ 35.7%

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

   3. Simplified 35.7%

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

   4. Taylor expanded in z around -inf 94.5%

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

      1. neg-mul-1 94.5%

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

   6. Simplified 94.5%

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

   if -8.9999999999999996e22 < z < 5.00000000000000029e83

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l* 83.6%

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

      3. associate-*r/ 86.9%

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

   3. Simplified 86.9%

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

   if 5.00000000000000029e83 < z

   1. Initial program 32.2%

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

      1. associate-/l* 32.6%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in z around inf 93.4%

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

      1. unpow2 93.4%

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

   6. Simplified 93.4%

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

      1. *-commutative 93.4%

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

      2. times-frac 97.0%

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

   8. Applied egg-rr 97.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+22}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \left(\frac{a}{z} \cdot \frac{t}{z}\right) \cdot -0.5}\\ \end{array} \]

Alternative 3: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+22}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-61}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{z} \cdot \frac{t}{z}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.8e+22)
   (- (* x y))
   (if (<= z 1.8e-61)
     (* y (/ (* z x) (sqrt (- (* z z) (* a t)))))
     (/ (* x y) (sqrt (- 1.0 (* (/ a z) (/ t z))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.8e+22) {
		tmp = -(x * y);
	} else if (z <= 1.8e-61) {
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	} else {
		tmp = (x * y) / sqrt((1.0 - ((a / z) * (t / z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-7.8d+22)) then
        tmp = -(x * y)
    else if (z <= 1.8d-61) then
        tmp = y * ((z * x) / sqrt(((z * z) - (a * t))))
    else
        tmp = (x * y) / sqrt((1.0d0 - ((a / z) * (t / z))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.8e+22) {
		tmp = -(x * y);
	} else if (z <= 1.8e-61) {
		tmp = y * ((z * x) / Math.sqrt(((z * z) - (a * t))));
	} else {
		tmp = (x * y) / Math.sqrt((1.0 - ((a / z) * (t / z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.8e+22:
		tmp = -(x * y)
	elif z <= 1.8e-61:
		tmp = y * ((z * x) / math.sqrt(((z * z) - (a * t))))
	else:
		tmp = (x * y) / math.sqrt((1.0 - ((a / z) * (t / z))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.8e+22)
		tmp = Float64(-Float64(x * y));
	elseif (z <= 1.8e-61)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(Float64(x * y) / sqrt(Float64(1.0 - Float64(Float64(a / z) * Float64(t / z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.8e+22)
		tmp = -(x * y);
	elseif (z <= 1.8e-61)
		tmp = y * ((z * x) / sqrt(((z * z) - (a * t))));
	else
		tmp = (x * y) / sqrt((1.0 - ((a / z) * (t / z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.8e+22], (-N[(x * y), $MachinePrecision]), If[LessEqual[z, 1.8e-61], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.80000000000000042e22

   1. Initial program 35.7%

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

      1. *-commutative 35.7%

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

      2. associate-*l* 34.2%

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

      3. associate-*r/ 35.7%

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

   3. Simplified 35.7%

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

   4. Taylor expanded in z around -inf 94.5%

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

      1. neg-mul-1 94.5%

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

   6. Simplified 94.5%

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

   if -7.80000000000000042e22 < z < 1.80000000000000007e-61

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-*l* 81.3%

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

      3. associate-*r/ 85.8%

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

   3. Simplified 85.8%

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

   if 1.80000000000000007e-61 < z

   1. Initial program 51.8%

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

      1. associate-/l* 53.2%

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

   3. Simplified 53.2%

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

      1. add-sqr-sqrt 53.2%

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

      2. sqrt-unprod 53.2%

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

      3. frac-times 45.2%

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

      4. add-sqr-sqrt 45.2%

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

   5. Applied egg-rr 45.2%

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

      1. div-sub 45.2%

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

      2. *-inverses 93.5%

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

      3. *-commutative 93.5%

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

      4. times-frac 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+22}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-61}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{a}{z} \cdot \frac{t}{z}}}\\ \end{array} \]

Alternative 4: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-156}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \left(\frac{a}{z} \cdot \frac{t}{z}\right) \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.25e-156)
   (- (* x y))
   (if (<= z 1.12e-60)
     (* y (/ (* z x) (sqrt (* a (- t)))))
     (/ (* x y) (+ 1.0 (* (* (/ a z) (/ t z)) -0.5))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.25e-156) {
		tmp = -(x * y);
	} else if (z <= 1.12e-60) {
		tmp = y * ((z * x) / sqrt((a * -t)));
	} else {
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-1.25d-156)) then
        tmp = -(x * y)
    else if (z <= 1.12d-60) then
        tmp = y * ((z * x) / sqrt((a * -t)))
    else
        tmp = (x * y) / (1.0d0 + (((a / z) * (t / z)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.25e-156) {
		tmp = -(x * y);
	} else if (z <= 1.12e-60) {
		tmp = y * ((z * x) / Math.sqrt((a * -t)));
	} else {
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.25e-156:
		tmp = -(x * y)
	elif z <= 1.12e-60:
		tmp = y * ((z * x) / math.sqrt((a * -t)))
	else:
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.25e-156)
		tmp = Float64(-Float64(x * y));
	elseif (z <= 1.12e-60)
		tmp = Float64(y * Float64(Float64(z * x) / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(Float64(Float64(a / z) * Float64(t / z)) * -0.5)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.25e-156)
		tmp = -(x * y);
	elseif (z <= 1.12e-60)
		tmp = y * ((z * x) / sqrt((a * -t)));
	else
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.25e-156], (-N[(x * y), $MachinePrecision]), If[LessEqual[z, 1.12e-60], N[(y * N[(N[(z * x), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25000000000000002e-156

   1. Initial program 53.8%

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

      1. *-commutative 53.8%

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

      2. associate-*l* 51.9%

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

      3. associate-*r/ 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in z around -inf 87.3%

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

      1. neg-mul-1 87.3%

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

   6. Simplified 87.3%

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

   if -1.25000000000000002e-156 < z < 1.12e-60

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-*l* 76.9%

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

      3. associate-*r/ 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in z around 0 72.3%

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

      1. associate-*r* 72.3%

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

      2. neg-mul-1 72.3%

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

   6. Simplified 72.3%

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

   if 1.12e-60 < z

   1. Initial program 51.8%

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

      1. associate-/l* 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in z around inf 86.3%

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

      1. unpow2 86.3%

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

   6. Simplified 86.3%

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

      1. *-commutative 86.3%

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

      2. times-frac 88.6%

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

   8. Applied egg-rr 88.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-156}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \left(\frac{a}{z} \cdot \frac{t}{z}\right) \cdot -0.5}\\ \end{array} \]

Alternative 5: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-156}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{a \cdot \left(-t\right)}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \left(\frac{a}{z} \cdot \frac{t}{z}\right) \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2e-156)
   (- (* x y))
   (if (<= z 1.7e-29)
     (/ (* x y) (/ (sqrt (* a (- t))) z))
     (/ (* x y) (+ 1.0 (* (* (/ a z) (/ t z)) -0.5))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e-156) {
		tmp = -(x * y);
	} else if (z <= 1.7e-29) {
		tmp = (x * y) / (sqrt((a * -t)) / z);
	} else {
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-2d-156)) then
        tmp = -(x * y)
    else if (z <= 1.7d-29) then
        tmp = (x * y) / (sqrt((a * -t)) / z)
    else
        tmp = (x * y) / (1.0d0 + (((a / z) * (t / z)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e-156) {
		tmp = -(x * y);
	} else if (z <= 1.7e-29) {
		tmp = (x * y) / (Math.sqrt((a * -t)) / z);
	} else {
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2e-156:
		tmp = -(x * y)
	elif z <= 1.7e-29:
		tmp = (x * y) / (math.sqrt((a * -t)) / z)
	else:
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2e-156)
		tmp = Float64(-Float64(x * y));
	elseif (z <= 1.7e-29)
		tmp = Float64(Float64(x * y) / Float64(sqrt(Float64(a * Float64(-t))) / z));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(Float64(Float64(a / z) * Float64(t / z)) * -0.5)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2e-156)
		tmp = -(x * y);
	elseif (z <= 1.7e-29)
		tmp = (x * y) / (sqrt((a * -t)) / z);
	else
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2e-156], (-N[(x * y), $MachinePrecision]), If[LessEqual[z, 1.7e-29], N[(N[(x * y), $MachinePrecision] / N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.00000000000000008e-156

   1. Initial program 53.8%

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

      1. *-commutative 53.8%

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

      2. associate-*l* 51.9%

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

      3. associate-*r/ 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in z around -inf 87.3%

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

      1. neg-mul-1 87.3%

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

   6. Simplified 87.3%

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

   if -2.00000000000000008e-156 < z < 1.69999999999999986e-29

   1. Initial program 74.5%

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

      1. associate-/l* 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in z around 0 72.3%

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

      1. associate-*r* 70.2%

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

      2. neg-mul-1 70.2%

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

   6. Simplified 72.3%

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

   if 1.69999999999999986e-29 < z

   1. Initial program 49.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 88.7%

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

      1. unpow2 88.7%

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

   6. Simplified 88.7%

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

      1. *-commutative 88.7%

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

      2. times-frac 91.2%

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

   8. Applied egg-rr 91.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-156}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{a \cdot \left(-t\right)}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \left(\frac{a}{z} \cdot \frac{t}{z}\right) \cdot -0.5}\\ \end{array} \]

Alternative 6: 76.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-134}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-128}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.5e-134)
   (- (* x y))
   (if (<= z 5e-128) (* y (/ (* z x) (+ z (* -0.5 (/ (* a t) z))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e-134) {
		tmp = -(x * y);
	} else if (z <= 5e-128) {
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-5.5d-134)) then
        tmp = -(x * y)
    else if (z <= 5d-128) then
        tmp = y * ((z * x) / (z + ((-0.5d0) * ((a * t) / z))))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e-134) {
		tmp = -(x * y);
	} else if (z <= 5e-128) {
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.5e-134:
		tmp = -(x * y)
	elif z <= 5e-128:
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e-134)
		tmp = Float64(-Float64(x * y));
	elseif (z <= 5e-128)
		tmp = Float64(y * Float64(Float64(z * x) / Float64(z + Float64(-0.5 * Float64(Float64(a * t) / z)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.5e-134)
		tmp = -(x * y);
	elseif (z <= 5e-128)
		tmp = y * ((z * x) / (z + (-0.5 * ((a * t) / z))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.5e-134], (-N[(x * y), $MachinePrecision]), If[LessEqual[z, 5e-128], N[(y * N[(N[(z * x), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5000000000000002e-134

   1. Initial program 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-*l* 50.2%

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

      3. associate-*r/ 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in z around -inf 87.7%

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

      1. neg-mul-1 87.7%

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

   6. Simplified 87.7%

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

   if -5.5000000000000002e-134 < z < 5.0000000000000001e-128

   1. Initial program 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*l* 76.3%

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

      3. associate-*r/ 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in z around inf 52.4%

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

   if 5.0000000000000001e-128 < z

   1. Initial program 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-*l* 56.1%

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

      3. associate-*r/ 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in z around inf 84.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-134}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-128}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 75.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-148}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-145}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x}{\frac{\frac{t}{z}}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9e-148)
   (- (* x y))
   (if (<= z 5.8e-145) (* -2.0 (* (/ y a) (/ x (/ (/ t z) z)))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9e-148) {
		tmp = -(x * y);
	} else if (z <= 5.8e-145) {
		tmp = -2.0 * ((y / a) * (x / ((t / z) / z)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-9d-148)) then
        tmp = -(x * y)
    else if (z <= 5.8d-145) then
        tmp = (-2.0d0) * ((y / a) * (x / ((t / z) / z)))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9e-148) {
		tmp = -(x * y);
	} else if (z <= 5.8e-145) {
		tmp = -2.0 * ((y / a) * (x / ((t / z) / z)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -9e-148:
		tmp = -(x * y)
	elif z <= 5.8e-145:
		tmp = -2.0 * ((y / a) * (x / ((t / z) / z)))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9e-148)
		tmp = Float64(-Float64(x * y));
	elseif (z <= 5.8e-145)
		tmp = Float64(-2.0 * Float64(Float64(y / a) * Float64(x / Float64(Float64(t / z) / z))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -9e-148)
		tmp = -(x * y);
	elseif (z <= 5.8e-145)
		tmp = -2.0 * ((y / a) * (x / ((t / z) / z)));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9e-148], (-N[(x * y), $MachinePrecision]), If[LessEqual[z, 5.8e-145], N[(-2.0 * N[(N[(y / a), $MachinePrecision] * N[(x / N[(N[(t / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.00000000000000029e-148

   1. Initial program 52.6%

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

      1. *-commutative 52.6%

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

      2. associate-*l* 50.6%

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

      3. associate-*r/ 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in z around -inf 87.8%

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

      1. neg-mul-1 87.8%

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

   6. Simplified 87.8%

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

   if -9.00000000000000029e-148 < z < 5.79999999999999968e-145

   1. Initial program 72.9%

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

      1. associate-/l* 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in z around inf 49.9%

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

      1. unpow2 49.9%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in a around inf 47.7%

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

      1. associate-/l* 47.7%

         \[\leadsto -2 \cdot \color{blue}{\frac{y}{\frac{a \cdot t}{{z}^{2} \cdot x}}} \]

      2. *-commutative 47.7%

         \[\leadsto -2 \cdot \frac{y}{\frac{\color{blue}{t \cdot a}}{{z}^{2} \cdot x}} \]

      3. times-frac 49.9%

         \[\leadsto -2 \cdot \frac{y}{\color{blue}{\frac{t}{{z}^{2}} \cdot \frac{a}{x}}} \]

      4. unpow2 49.9%

         \[\leadsto -2 \cdot \frac{y}{\frac{t}{\color{blue}{z \cdot z}} \cdot \frac{a}{x}} \]

      5. associate-/r* 49.9%

         \[\leadsto -2 \cdot \frac{y}{\color{blue}{\frac{\frac{t}{z}}{z}} \cdot \frac{a}{x}} \]

   9. Simplified 49.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{y}{\frac{\frac{t}{z}}{z} \cdot \frac{a}{x}}} \]

   10. Taylor expanded in y around 0 47.7%

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

       1. times-frac 49.8%

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

       2. *-commutative 49.8%

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

       3. associate-/l* 49.8%

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

       4. unpow2 49.8%

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

       5. associate-/r* 49.9%

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

   12. Simplified 49.9%

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

   if 5.79999999999999968e-145 < z

   1. Initial program 55.8%

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

      1. *-commutative 55.8%

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

      2. associate-*l* 56.5%

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

      3. associate-*r/ 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in z around inf 84.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-148}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-145}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x}{\frac{\frac{t}{z}}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 75.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{-147}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-150}:\\ \;\;\;\;-2 \cdot \left(\left(z \cdot x\right) \cdot \frac{y}{a \cdot \frac{t}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.08e-147)
   (- (* x y))
   (if (<= z 3.5e-150) (* -2.0 (* (* z x) (/ y (* a (/ t z))))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.08e-147) {
		tmp = -(x * y);
	} else if (z <= 3.5e-150) {
		tmp = -2.0 * ((z * x) * (y / (a * (t / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-1.08d-147)) then
        tmp = -(x * y)
    else if (z <= 3.5d-150) then
        tmp = (-2.0d0) * ((z * x) * (y / (a * (t / z))))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.08e-147) {
		tmp = -(x * y);
	} else if (z <= 3.5e-150) {
		tmp = -2.0 * ((z * x) * (y / (a * (t / z))));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.08e-147:
		tmp = -(x * y)
	elif z <= 3.5e-150:
		tmp = -2.0 * ((z * x) * (y / (a * (t / z))))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.08e-147)
		tmp = Float64(-Float64(x * y));
	elseif (z <= 3.5e-150)
		tmp = Float64(-2.0 * Float64(Float64(z * x) * Float64(y / Float64(a * Float64(t / z)))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.08e-147)
		tmp = -(x * y);
	elseif (z <= 3.5e-150)
		tmp = -2.0 * ((z * x) * (y / (a * (t / z))));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.08e-147], (-N[(x * y), $MachinePrecision]), If[LessEqual[z, 3.5e-150], N[(-2.0 * N[(N[(z * x), $MachinePrecision] * N[(y / N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.07999999999999995e-147

   1. Initial program 52.6%

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

      1. *-commutative 52.6%

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

      2. associate-*l* 50.6%

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

      3. associate-*r/ 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in z around -inf 87.8%

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

      1. neg-mul-1 87.8%

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

   6. Simplified 87.8%

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

   if -1.07999999999999995e-147 < z < 3.4999999999999998e-150

   1. Initial program 72.9%

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

      1. associate-/l* 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in z around inf 49.9%

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

      1. unpow2 49.9%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in a around inf 47.7%

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

      1. associate-/l* 47.7%

         \[\leadsto -2 \cdot \color{blue}{\frac{y}{\frac{a \cdot t}{{z}^{2} \cdot x}}} \]

      2. *-commutative 47.7%

         \[\leadsto -2 \cdot \frac{y}{\frac{\color{blue}{t \cdot a}}{{z}^{2} \cdot x}} \]

      3. times-frac 49.9%

         \[\leadsto -2 \cdot \frac{y}{\color{blue}{\frac{t}{{z}^{2}} \cdot \frac{a}{x}}} \]

      4. unpow2 49.9%

         \[\leadsto -2 \cdot \frac{y}{\frac{t}{\color{blue}{z \cdot z}} \cdot \frac{a}{x}} \]

      5. associate-/r* 49.9%

         \[\leadsto -2 \cdot \frac{y}{\color{blue}{\frac{\frac{t}{z}}{z}} \cdot \frac{a}{x}} \]

   9. Simplified 49.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{y}{\frac{\frac{t}{z}}{z} \cdot \frac{a}{x}}} \]
   10. Step-by-step derivation

       1. frac-times 50.2%

          \[\leadsto -2 \cdot \frac{y}{\color{blue}{\frac{\frac{t}{z} \cdot a}{z \cdot x}}} \]

   11. Applied egg-rr 50.2%

       \[\leadsto -2 \cdot \frac{y}{\color{blue}{\frac{\frac{t}{z} \cdot a}{z \cdot x}}} \]
   12. Step-by-step derivation

       1. associate-/r/ 50.2%

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

       2. *-commutative 50.2%

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

       3. *-commutative 50.2%

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

   13. Applied egg-rr 50.2%

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

   if 3.4999999999999998e-150 < z

   1. Initial program 55.8%

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

      1. *-commutative 55.8%

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

      2. associate-*l* 56.5%

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

      3. associate-*r/ 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in z around inf 84.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{-147}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-150}:\\ \;\;\;\;-2 \cdot \left(\left(z \cdot x\right) \cdot \frac{y}{a \cdot \frac{t}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 75.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-145}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-156}:\\ \;\;\;\;-2 \cdot \frac{y}{\frac{a \cdot \frac{t}{z}}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.05e-145)
   (- (* x y))
   (if (<= z 3.7e-156) (* -2.0 (/ y (/ (* a (/ t z)) (* z x)))) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e-145) {
		tmp = -(x * y);
	} else if (z <= 3.7e-156) {
		tmp = -2.0 * (y / ((a * (t / z)) / (z * x)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-1.05d-145)) then
        tmp = -(x * y)
    else if (z <= 3.7d-156) then
        tmp = (-2.0d0) * (y / ((a * (t / z)) / (z * x)))
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e-145) {
		tmp = -(x * y);
	} else if (z <= 3.7e-156) {
		tmp = -2.0 * (y / ((a * (t / z)) / (z * x)));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.05e-145:
		tmp = -(x * y)
	elif z <= 3.7e-156:
		tmp = -2.0 * (y / ((a * (t / z)) / (z * x)))
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.05e-145)
		tmp = Float64(-Float64(x * y));
	elseif (z <= 3.7e-156)
		tmp = Float64(-2.0 * Float64(y / Float64(Float64(a * Float64(t / z)) / Float64(z * x))));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.05e-145)
		tmp = -(x * y);
	elseif (z <= 3.7e-156)
		tmp = -2.0 * (y / ((a * (t / z)) / (z * x)));
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.05e-145], (-N[(x * y), $MachinePrecision]), If[LessEqual[z, 3.7e-156], N[(-2.0 * N[(y / N[(N[(a * N[(t / z), $MachinePrecision]), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.04999999999999996e-145

   1. Initial program 52.6%

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

      1. *-commutative 52.6%

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

      2. associate-*l* 50.6%

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

      3. associate-*r/ 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in z around -inf 87.8%

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

      1. neg-mul-1 87.8%

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

   6. Simplified 87.8%

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

   if -1.04999999999999996e-145 < z < 3.7e-156

   1. Initial program 72.9%

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

      1. associate-/l* 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in z around inf 49.9%

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

      1. unpow2 49.9%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in a around inf 47.7%

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

      1. associate-/l* 47.7%

         \[\leadsto -2 \cdot \color{blue}{\frac{y}{\frac{a \cdot t}{{z}^{2} \cdot x}}} \]

      2. *-commutative 47.7%

         \[\leadsto -2 \cdot \frac{y}{\frac{\color{blue}{t \cdot a}}{{z}^{2} \cdot x}} \]

      3. times-frac 49.9%

         \[\leadsto -2 \cdot \frac{y}{\color{blue}{\frac{t}{{z}^{2}} \cdot \frac{a}{x}}} \]

      4. unpow2 49.9%

         \[\leadsto -2 \cdot \frac{y}{\frac{t}{\color{blue}{z \cdot z}} \cdot \frac{a}{x}} \]

      5. associate-/r* 49.9%

         \[\leadsto -2 \cdot \frac{y}{\color{blue}{\frac{\frac{t}{z}}{z}} \cdot \frac{a}{x}} \]

   9. Simplified 49.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{y}{\frac{\frac{t}{z}}{z} \cdot \frac{a}{x}}} \]
   10. Step-by-step derivation

       1. frac-times 50.2%

          \[\leadsto -2 \cdot \frac{y}{\color{blue}{\frac{\frac{t}{z} \cdot a}{z \cdot x}}} \]

   11. Applied egg-rr 50.2%

       \[\leadsto -2 \cdot \frac{y}{\color{blue}{\frac{\frac{t}{z} \cdot a}{z \cdot x}}} \]

   if 3.7e-156 < z

   1. Initial program 55.8%

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

      1. *-commutative 55.8%

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

      2. associate-*l* 56.5%

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

      3. associate-*r/ 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in z around inf 84.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-145}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-156}:\\ \;\;\;\;-2 \cdot \frac{y}{\frac{a \cdot \frac{t}{z}}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 10: 77.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-129}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \left(\frac{a}{z} \cdot \frac{t}{z}\right) \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.6e-129)
   (- (* x y))
   (/ (* x y) (+ 1.0 (* (* (/ a z) (/ t z)) -0.5)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e-129) {
		tmp = -(x * y);
	} else {
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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-129)) then
        tmp = -(x * y)
    else
        tmp = (x * y) / (1.0d0 + (((a / z) * (t / z)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e-129) {
		tmp = -(x * y);
	} else {
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.6e-129:
		tmp = -(x * y)
	else:
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.6e-129)
		tmp = Float64(-Float64(x * y));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(Float64(Float64(a / z) * Float64(t / z)) * -0.5)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.6e-129)
		tmp = -(x * y);
	else
		tmp = (x * y) / (1.0 + (((a / z) * (t / z)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.6e-129], (-N[(x * y), $MachinePrecision]), N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6000000000000001e-129

   1. Initial program 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-*l* 50.2%

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

      3. associate-*r/ 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in z around -inf 87.7%

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

      1. neg-mul-1 87.7%

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

   6. Simplified 87.7%

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

   if -1.6000000000000001e-129 < z

   1. Initial program 61.0%

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

      1. associate-/l* 64.8%

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

   3. Simplified 64.8%

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

   4. Taylor expanded in z around inf 73.4%

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

      1. unpow2 73.4%

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

   6. Simplified 73.4%

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

      1. *-commutative 73.4%

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

      2. times-frac 74.9%

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

   8. Applied egg-rr 74.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-129}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \left(\frac{a}{z} \cdot \frac{t}{z}\right) \cdot -0.5}\\ \end{array} \]

Alternative 11: 74.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-196}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-162}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.6e-196)
   (- (* x y))
   (if (<= z 3.7e-162) (* y (/ (* z x) z)) (* x y))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e-196) {
		tmp = -(x * y);
	} else if (z <= 3.7e-162) {
		tmp = y * ((z * x) / z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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-196)) then
        tmp = -(x * y)
    else if (z <= 3.7d-162) then
        tmp = y * ((z * x) / z)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e-196) {
		tmp = -(x * y);
	} else if (z <= 3.7e-162) {
		tmp = y * ((z * x) / z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.6e-196:
		tmp = -(x * y)
	elif z <= 3.7e-162:
		tmp = y * ((z * x) / z)
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.6e-196)
		tmp = Float64(-Float64(x * y));
	elseif (z <= 3.7e-162)
		tmp = Float64(y * Float64(Float64(z * x) / z));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.6e-196)
		tmp = -(x * y);
	elseif (z <= 3.7e-162)
		tmp = y * ((z * x) / z);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.6e-196], (-N[(x * y), $MachinePrecision]), If[LessEqual[z, 3.7e-162], N[(y * N[(N[(z * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6e-196

   1. Initial program 53.7%

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

      1. *-commutative 53.7%

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

      2. associate-*l* 52.7%

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

      3. associate-*r/ 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in z around -inf 83.2%

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

      1. neg-mul-1 83.2%

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

   6. Simplified 83.2%

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

   if -1.6e-196 < z < 3.7000000000000002e-162

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-*l* 76.4%

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

      3. associate-*r/ 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in z around inf 28.1%

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

   if 3.7000000000000002e-162 < z

   1. Initial program 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-*l* 56.1%

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

      3. associate-*r/ 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in z around inf 83.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-196}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-162}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12: 72.7% accurate, 18.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e-310) (- (* x y)) (* x y)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-310) {
		tmp = -(x * y);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-5d-310)) then
        tmp = -(x * y)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-310) {
		tmp = -(x * y);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e-310:
		tmp = -(x * y)
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e-310)
		tmp = Float64(-Float64(x * y));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5e-310)
		tmp = -(x * y);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e-310], (-N[(x * y), $MachinePrecision]), N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.999999999999985e-310

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-*l* 54.8%

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

      3. associate-*r/ 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in z around -inf 76.5%

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

      1. neg-mul-1 76.5%

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

   6. Simplified 76.5%

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

   if -4.999999999999985e-310 < z

   1. Initial program 59.5%

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

      1. *-commutative 59.5%

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

      2. associate-*l* 59.4%

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

      3. associate-*r/ 60.9%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in z around inf 71.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 13: 43.6% accurate, 37.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \cdot y \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* x y))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	return x * y

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	return Float64(x * y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z, t, a)
	tmp = x * y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(x * y), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

   1. *-commutative 57.2%

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

   2. associate-*l* 57.0%

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

   3. associate-*r/ 59.1%

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

3. Simplified 59.1%

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

4. Taylor expanded in z around inf 40.2%

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

5. Final simplification 40.2%

   \[\leadsto x \cdot y \]

Developer target: 89.3% 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 2023176 
(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)))))